# Exact cube-root fluctuations in an area-constrained random walk model

Lucas D’Alimonte111Université de Fribourg<EMAIL_ADDRESS>Romain
Panis222Université de Genève<EMAIL_ADDRESS>
###### Abstract

This article is devoted to the study of the behaviour of a (1+1)-dimensional
model of random walk conditioned to enclose an area of order $N^{2}$. Such a
conditioning enforces a globally concave trajectory. We study the local
deviations of the walk from its convex hull. To this end, we introduce two
quantities — the mean facet length $\mathsf{MeanFL}$ and the mean local
roughness $\mathsf{MeanLR}$ — measuring the typical longitudinal and
transversal fluctuations around the boundary of the convex hull of the random
walk. Our main result is that $\mathsf{MeanFL}$ is of order $N^{2/3}$ and
$\mathsf{MeanLR}$ is of order $N^{1/3}$. Moreover, following the strategy of
Hammond (Ann. Prob., 2012), we identify the polylogarithmic corrections in the
scaling of the _maximal_ facet length and of the _maximal_ local roughness,
showing that the former one scales as $N^{2/3}(\log N)^{1/3}$, while the
latter scales as $N^{1/3}(\log N)^{2/3}$. The object of study is intended to
be a toy model for the interface of a two-dimensional statistical mechanics
model (such as the Ising model) in the phase separation regime — we discuss
this issue at the end of this work.

##  1 Introduction

The phase separation problem is concerned with the study of the boundary
appearing between two different phases of a statistical mechanics model, in a
regime where those two phases can coexist. In his seminal work, Wulff [Wul01]
proposed that such a boundary should macroscopically adopt a deterministic
limit shape given by the solution of a variational problem involving
thermodynamic quantities such as the surface tension. This prediction has been
an object of intense study and has by now been made rigorous in a very wide
variety of settings, see for instance the monographs [DKS92, BIV00, CP00,
Cer06].

While this macroscopic shape is dependent on the model, the _fluctuations_ of
the random phase boundary are widely believed to behave in an universal way. A
first natural candidate to measure these fluctuations is the deviations from
the limit shape. These have been shown to be Gaussian in [DH96] in the context
of area-constrained random walks, and later on in [DH97] for the 2D Ising
phase boundary at low temperatures. Two other quantities of interest are given
by the maximal facet length and the maximal local roughness of the interface,
that are respectively the length of the largest segment of the convex hull of
the interface and the maximal deviation of the interface from this convex
hull. In a seminal paper [Ale01], Alexander conjectured that the exponent
governing the scaling of the maximal local roughness should be $1/3$. In the
context of percolation models in the phase separation setting, he derived
upper bounds for an averaged version of the local roughness. Alexander and
Usun [UA03] then complemented this work by providing lower bounds for the
local roughness in the setup of Bernoulli percolation. Later on, in a
remarkable series of papers, Hammond [Ham11, Ham12a, Ham12b] was able to
identify the exact scale of the maximal facet length and the maximal local
roughness of a droplet of volume $N^{2}$ in the planar subcritical random-
cluster model. Indeed, he proved that the former is of order $N^{2/3}(\log
N)^{1/3}$, while the latter is of order $N^{1/3}(\log N)^{2/3}$, validating
the exponent derived in [Ale01]. These results are built on the identification
of $N^{2/3}$ as the scale at which the curvature effect enforced by the
conditioning has the same order of magnitude as the Gaussian fluctuations of
the interface. Finally, let us mention that in an earlier paper, Hammond and
Peres [HP08] introduced a continuous and Brownian version of the phase
separation problem, studying a two-dimensional Brownian loop conditioned to
enclose a large area. They proved results in favour of the appearance of the
cube-root fluctuations in this setting.

In this paper, we study a model of random walks, with geometrically randomised
length, conditioned to enclose an area at least equal to $N^{2}$. This model
was suggested by Hammond in [Ham12a, Section 1.0.4] and later in [Ham19]. We
prove that, as predicted by the author, his techniques can successfully be
applied to this setting, enabling us to establish the above-mentioned
polylogarithmic corrections. However, the main innovation of this paper is the
identification of the scaling of the typical facet length and local roughness
(rather than their maximal values). Call $\mathsf{MeanFL}$ (resp.
$\mathsf{MeanLR}$) the length (resp. the local roughness) of the facet
intersecting a given line. We prove that $\mathsf{MeanFL}\asymp N^{2/3}$ and
$\mathsf{MeanLR}\asymp N^{1/3}$, see Theorem 1.3.

The scaling exponents $1/3$ and $2/3$ have been shown to arise in various
related contexts in statistical mechanics. An important example is the
critical prewetting in the Ising model, which has been extensively studied in
a beautiful series of papers [HV04, Vel04, ISV15, GG21, IOSV22] to mention a
few of them. A remarkable aspect of these works is that the results are
derived without the help of any integrable feature. Let us also note that
[CLM+16, CKL23] provided strong evidence for a similar behaviour in the
context of the SOS model above a wall in $(2+1)$ dimensions.

To the best of our knowledge, it is the first time that those exponents are
identified in a context such as ours. Indeed, in the above-mentioned works,
the interface lies above a facet of length much longer than $N^{2/3}$ — this
facet being deterministic and artificially created by looking at the interface
along a side of a large box for instance. However, in our work, the facets are
not deterministic, and themselves reflect the competition between the
randomness and the curvature induced by the conditioning.

Finally, we strongly believe (supported by [Ham12a]) that our approach is
robust and should allow to derive the same result for a large variety of
models, including the fluctuations of the outermost circuit in a subcritical
random-cluster model conditioned to enclose a large area, which itself is a
good toy model for the boundary of a droplet in a supercritical Potts model.
Note that this suggests that at scale $N^{2/3}$, our model and more general
phase boundaries models should lie in the same universality class. This is
discussed in Section 5.

###  1.1 Definition of the model and statement of the main results

Let $\Lambda$ be the set of finite paths in the first quadrant of
$\mathbb{Z}^{2}$ which start on the $y$-axis and end on the $x$-axis, and
which are oriented in the sense that they only take rightward and downward
steps, see Figure 1. For $\gamma\in\Lambda$, we define $|\gamma|$ to be its
length, i.e. its number of steps (which is also the sum of the $y$-coordinate
of its starting point and the $x$-coordinate of its ending point). It will be
convenient to identify an element $\gamma\in\Lambda$ with the set of points of
$\mathbb{N}^{2}$ it passes by, i.e $\gamma=(\gamma(k))_{0\leq k\leq|\gamma|}$.
We set $\Lambda_{n}$ to be the subset of $\Lambda$ of such oriented paths of
length $n$. It is clear that $|\Lambda_{n}|=2^{n}$. If $a,b\in\mathbb{N}^{2}$,
we denote by $\Lambda^{a\rightarrow b}$ the set of downright paths from $a$ to
$b$.

Let $0<\lambda<\frac{1}{2}$. We define a probability measure
$\mathbb{P}_{\lambda}$ on $\Lambda$ by requiring that, for $\gamma\in\Lambda$,

$\mathbb{P}_{\lambda}[\gamma]=\frac{1}{Z_{\lambda}}\lambda^{|\gamma|},$ (1.1)

where $Z_{\lambda}$ is the normalisation constant given by
$Z_{\lambda}=(1-2\lambda)^{-1}$.

Figure 1: An example of path $\gamma$ (in bold black) with $|\gamma|=k+\ell$,
and its least concave majorant $\mathcal{C}(\gamma)$ (in blue). The mean facet
of $\gamma$ is coloured in orange, while the mean local roughness is the
length of the red segment. The area $\mathcal{A}(\gamma)$ is the area of the
light green shaded region.

The probability measure $\mathbb{P}_{\lambda}$ is to be seen as a _background_
measure which imposes an exponential decay of the probability of sampling a
long path. We now enforce a competing constraint of enclosing a large area.
For $\gamma\in\Lambda$, we define the area of $\gamma$, denoted by
$\mathcal{A}(\gamma)$, to be the area enclosed by the graph of $\gamma$ and
the two coordinate axes (see Figure 1). Let $N\in\mathbb{N}$ and let $\Gamma$
be a sample of $\mathbb{P}_{\lambda}$. We define a new probability measure
$\mathbb{P}_{\lambda}^{N^{2}}$ on $\Lambda$ by

$\mathbb{P}_{\lambda}^{N^{2}}\left[\>\cdot\>\right]:=\mathbb{P}_{\lambda}\big{[}\>\cdot\>|\>\mathcal{A}(\Gamma)\geq
N^{2}\big{]}.$ (1.2)

We also introduce $\Lambda^{N^{2}}\subset\Lambda$, the set of all paths of
$\Lambda$ that capture an area greater than $N^{2}$. Samples of
$\mathbb{P}_{\lambda}^{N^{2}}$ typically experience a competition between
capturing a large area and minimizing the length of the paths. As explained
before, it suggests that for large values of $N$, a typical sample of
$\mathbb{P}_{\lambda}^{N^{2}}$ will have a global curvature imposed by the
area condition and Gaussian fluctuations imposed by the background measure
(once the length is fixed, the background measure is the uniform law). The
object of interest in our work, which tracks the competition between these two
phenomena, is defined below.

######  Definition 1.1 (Least concave majorant, facets).

Let $\gamma\in\Lambda$. We set $\mathsf{Ext}(\gamma)$ to be the set of non-
zero extremal points of the convex hull of $\gamma\cup\\{(0,0)\\}$. Let
$\mathcal{C}(\gamma)$ be the least concave majorant of $\gamma$, that is, the
graph of the piecewise affine function passing through all the points of
$\mathsf{Ext}(\gamma)$. Observe that with this definition,
$\mathcal{C}(\gamma)$ becomes a finite union of line segments of
$\mathbb{R}^{2}$ which are called facets (see Figure 1), the endpoints of
these facets precisely being the elements of $\mathsf{Ext}(\gamma)$. For
$x\in\gamma$, we define its local roughness to be its Euclidean distance to
$\mathcal{C}(\gamma)$,

$\mathsf{LR}(x):=\mathrm{d}(x,\mathcal{C}(\gamma)).$ (1.3)

Our main result concerns the tightness of the length of a typical facet
intercepting a given ray emanating from the origin (resp. the fluctuations of
$\Gamma$ along the facet) at scale $N^{2/3}$ (resp. $N^{1/3}$). Let us first
define the quantities of interest.

######  Definition 1.2 ($\mathsf{MeanFL}$, $\mathsf{MeanLR}$).

Let $\gamma\in\Lambda$. We define $\mathsf{MeanFac}(\gamma)$ to be the facet
intersecting the line $\\{y=x\\}$ (if we are in the case where the ray
$\\{y=x\\}$ intercepts the endpoint of two consecutive facets, we arbitrarily
choose the leftmost one for $\mathsf{MeanFac}(\gamma)$). We call
$x_{\mathsf{mid}}(\gamma)$ the point of $\mathbb{N}^{2}\cap\gamma$ that is the
closest to the line $\\{y=x\\}$ (as previously, in case of conflict, we
arbitrarily choose the leftmost one). Assume that
$\mathsf{MeanFac}=[a,b](=\\{at+(1-t)b,\>t\in[0,1]\\})$, with
$a,b\in\mathbb{N}^{2}$. Define the following random variables,

$\mathsf{MeanFL}(\gamma):=\mathrm{d}(a,b),\qquad\mathsf{MeanLR}(\gamma):=\mathsf{LR}(x_{\mathsf{mid}}).$
(1.4)

In words, $\mathsf{MeanFL}$ is the length of $\mathsf{MeanFac}$ and
$\mathsf{MeanLR}$ is the local roughness of the “mean vertex” of $\gamma$.

The following result is the main contribution of the paper.

######  Theorem 1.3 (Tightness of $\mathsf{MeanFL}$ and $\mathsf{MeanLR}$ at
scales $N^{2/3}$ and $N^{1/3}$).

Let $0<\lambda<\frac{1}{2}$. For any $\varepsilon>0$, there exist $c,C>0$ and
$N_{0}\in\mathbb{N}$ such that for any $N\geq N_{0}$,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}cN^{\frac{2}{3}}<\mathsf{MeanFL}(\Gamma)<CN^{\frac{2}{3}}\big{]}>1-\varepsilon,$
(1.5)

and

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}cN^{\frac{1}{3}}<\mathsf{MeanLR}(\Gamma)<CN^{\frac{1}{3}}\big{]}>1-\varepsilon.$
(1.6)

We chose for convenience to formulate Theorem 1.3 in terms of the statistics
of $\mathsf{MeanFac}$. However, the proof never uses the symmetry of the model
around the line $\\{y=x\\}$ so the result is not specific to the choice of the
line $\\{y=x\\}$: for any $\alpha>0$, the result holds333Note however that the
constants $c,C$ might now depend on $\alpha$. for $\mathsf{MeanFac}_{\alpha}$
which is defined to be the unique facet intercepting the line $\\{y=\alpha
x\\}$.

######  Remark 1.4.

In the proof of Theorem 1.3, we actually derive a slightly stronger statement.
Indeed, we obtain stretch-exponential upper tails for $\mathsf{MeanFL}$ and
$\mathsf{MeanLR}$ (see Propositions 3.1 and 3.2)

Following [Ham12a] and [Ham12b], our second result identifies the logarithmic
corrections to the maximal facet length and to the maximal local roughness
along $\mathcal{C}(\gamma)$. We call these quantities $\mathsf{MaxFL}(\gamma)$
(resp. $\mathsf{MaxLR}(\gamma)$).

######  Theorem 1.5.

Let $0<\lambda<\frac{1}{2}$. There exist $c,C>0$ such that,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}cN^{\frac{2}{3}}(\log
N)^{\frac{1}{3}}<\mathsf{MaxFL}(\Gamma)<CN^{\frac{2}{3}}(\log
N)^{\frac{1}{3}}\big{]}\xrightarrow[N\rightarrow\infty]{}{1},$ (1.7)

and

$\mathbb{P}_{\lambda}^{N^{2}}[cN^{\frac{1}{3}}(\log
N)^{\frac{2}{3}}<\mathsf{MaxLR}(\Gamma)<CN^{\frac{1}{3}}(\log
N)^{\frac{2}{3}}]\xrightarrow[N\rightarrow\infty]{}{1}.$ (1.8)

###  1.2 Related works and known results

As pointed out in the introduction, similar models have been studied quite
extensively in the literature, especially in [DH96], where they are introduced
as toy models for the study of a low-temperature interface of a
(1+1)-dimensional SOS model. In this work, the authors investigate the
behaviour of the model at the macroscopic scale $N$ and at the mesoscopic
scale $N^{1/2}$. It is possible to extend their result to our setup. To
properly state it, we introduce the following parametrisation of $\Gamma$: let
$\Gamma(t)$ be the linear interpolation between the points $\Gamma(k)$ for
$0\leq k\leq|\Gamma|$. Using the discussion of [DH96, Section 1], together
with the basic estimates given by Lemma 2.1 and Proposition 2.4, one can
obtain the following result.

######  Theorem 1.6.

Let $0<\lambda<\frac{1}{2}$. There exists a deterministic, concave and
continuous function $f_{\lambda}:[0,1]\rightarrow\mathbb{R}^{+}$ such that for
any $\varepsilon>0$,

$\mathbb{P}_{\lambda}^{N^{2}}\Big{[}\sup_{t\in[0,1]}\big{|}N^{-1}\Gamma_{N}(|\Gamma_{N}|t)-f_{\lambda}(t)\big{|}>\varepsilon\Big{]}\xrightarrow[N\rightarrow\infty]{}{0}.$
(1.9)

######  Remark 1.7.

Using Lemma 2.1, the time scaling in (1.9) is linear in $N$.

Such a phenomenon is by now very well known under the name of a limit shape
phenomenon, and is known to arise in a large variety of situations (see for
instance [Oko16, Ken18]). In (1+1) dimensions such as in our setting, two
different points of view can be adopted to prove a statement such as Theorem
1.6: the first one is the classical theory of sample path large deviations
culminating with the celebrated Mogulski’i Theorem (see [DZ10, Section 5]).
The second possible point of view has a more statistical mechanics flavour and
is known as Wulff theory (see the reference monograph [DKS92]). In the work
[DH96], it is shown that both approaches can be implemented and yield the same
result. In both cases, the function $f_{\lambda}$ is identified as the
minimiser of a deterministic variational problem. Let us conclude this
discussion by noticing that actually much stronger statements than (1.9) can
be obtained, and in particular large deviations principles for the sample path
$\Gamma$, though we will not focus on results of this type.

The main result of [DH96, Theorem 2.1] focuses on the fluctuations of
$\Gamma_{N}$ around $f_{\lambda}$. Again, minor modifications of their proof
lead to the following result.

######  Theorem 1.8.

Let $0<\lambda<\frac{1}{2}$. There exists a (centered) Gaussian process
$\xi_{\lambda}$ on the space $\mathcal{C}([0,1])$ (equipped with the topology
of the uniform convergence) such that under the measure
$\mathbb{P}_{\lambda}^{N^{2}}$,

$\Big{(}\frac{1}{\sqrt{N}}(\Gamma_{N}(|\Gamma_{N}|t)-Nf_{\lambda}(t))\Big{)}_{t\in[0,1]}\xrightarrow[N\rightarrow\infty]{(d)}{\xi_{\lambda}},$
(1.10)

where the convergence holds in distribution.

These two results can be stated heuristically as follows: at the macroscopic
scale $N$, the conditioning on the event $\\{\mathcal{A}(\Gamma)\geq N^{2}\\}$
enforces a deterministic and global curvature, whereas at the mesoscopic scale
$N^{1/2}$ the conditioning has no effect on $\mathbb{P}_{\lambda}$ and the
Gaussian nature of the measure $\mathbb{P}_{\lambda}$ is unchanged. As
explained above, Theorem 1.3 identifies $N^{2/3}$ as being the scale at which
those two competing effects are of the same order.

###  1.3 A resampling strategy

The proofs below heavily relie on a particularly simple (yet crucial) feature
of the model: the so-called Brownian Gibbs property (see [CH14]). It can be
stated as follows: start from a sample $\Gamma$ of
$\mathbb{P}_{\lambda}^{N^{2}}$, choose two points $a,b\in\Gamma$ with any
random procedure “explorable from the exterior444This terminology is directly
inspired by the notion of explorable set in percolation theory. of the path”
(by this we mean that the event $\\{a=x,b=y\\}$ is measurable with respect to
$\Gamma\setminus\Gamma_{x,y}$). Then, conditionally on $(a,b)$ and
$\Gamma\setminus\Gamma_{a,b}$, the distribution of the random variable
$\Gamma_{a,b}$ is the uniform distribution on $\Lambda^{a\rightarrow b}$
conditionally on the fact that the resulting path of $\Lambda$ encloses an
area greater than $N^{2}$. This apparently naive observation allows one to
implement a strategy of resampling. Indeed, one can construct several
Markovian dynamics on $\Lambda$ leaving the distribution
$\mathbb{P}_{\lambda}^{N^{2}}$ invariant in a quite general fashion: start
from a sample $\Gamma$ of $\mathbb{P}_{\lambda}^{N^{2}}$, choose two points
$a,b\in\Gamma$ according to the above procedure and replace $\Gamma_{a,b}$ by
a sample of the uniform distribution of $\Gamma_{a,b}$ subject to the above-
mentioned conditioning. Then, it is clear, thanks to the preceding
observation, that the distribution of the output is
$\mathbb{P}_{\lambda}^{N^{2}}$. In what follows, we shall call such a dynamic
on $\Lambda$ a resampling procedure.

This point of view will be used several times in the proofs to analyse
marginals of $\mathbb{P}_{\lambda}^{N^{2}}$ in well-chosen regions of the
first quadrant. A simple illustration is given in the proof of Proposition
2.10.

###  1.4 Open problems

In light of the preceding discussions, two natural questions arise. We now
describe them.

#####

As explained above, we expect the methods developed in this article to help
the analysis of the droplet of a subcritical planar random-cluster model. We
strongly believe that our strategy could complement the results obtained by
Hammond in [Ham11, Ham12a, Ham12b]. We provide strong heuristics towards this
result in Section 5.

######  Open Problem A (Droplet boundary in the subcritical planar random-
cluster model).

Extend Theorem 1.3 to the study of a typical facet in the setup of the droplet
in the subcritical planar random-cluster model.

Once tightness is obtained in Theorem 1.3, it is quite natural to try to
identify the candidate for the scaling limit of the excursion below a typical
facet. A similar question has been answered in [IOSV22] where the authors
obtained convergence (in the bulk phase) of the object of study to the so-
called _Ferrari–Spohn diffusion_. In our setup, the excursion below the mean
facet should correspond to the excursion of a Ferrari-Spohn process. We intend
to study this question in the future.

######  Open Problem B (Scaling limit of the excursion below a facet).

Construct the Ferrari–Spohn excursion $\mathsf{(FSE)}$. Prove that the piece
of path lying below the mean facet, after a proper rescaling, converges to the
$\mathsf{FSE}$ under $\mathbb{P}_{\lambda}^{N^{2}}$ in the limit
$N\rightarrow\infty$.

Compared to [IOSV22], the study of Open Problem B requires two additional
ingredients. Indeed, one needs a fine understanding of the way that the
$O(1)N^{1/3}$ excursions of size $N^{2/3}$ interact together along the convex
hull of the droplet. Moreover, the construction of $\mathsf{FSE}$ and the
proof of the convergence of the mean excursion towards it are non-trivial as
the pinning condition takes place _at the scale of the correlation length_ of
the system.

###  1.5 Organisation of the paper

The paper is organised as follows: in Section 2, we gather some preliminary
results that are going to be our toolbox for the proofs of the main results.
Section 3 is then devoted to the proof of Theorem 1.3, while Section 4
consists in an adaptation of the arguments of [Ham12a, Ham12b] to prove
Theorem 1.5. Finally, Section 5 is devoted to a discussion regarding the
extension of the results to other statistical mechanics models in the Wulff
setting.

##### Notations and conventions.

We shall adopt Landau formalism for real valued-sequences. Namely, whenever
$(a_{n})$ and $(b_{n})$ are two real-valued sequences, we will write
$a_{n}=o(b_{n})$ when $|a_{n}|/|b_{n}|\xrightarrow[n\rightarrow\infty]{}{0}$.
We will also use the notation $a_{n}=O(b_{n})$ when there exists some constant
$C>0$ such that $|a_{n}|\leq C|b_{n}|$ for all $n$ large enough. If
$a_{n}=O(b_{n})$ and $b_{n}=O(a_{n})$, we shall write that $a_{n}\asymp
b_{n}$. Finally we shall write $a_{n}\sim b_{n}$ when
$a_{n}/b_{n}\xrightarrow[n\rightarrow\infty]{}{1}$.

If $A$ is a set, we denote by $\mathcal{P}(A)$ the power set of $A$. For
$x=(x_{1},x_{2})\in\mathbb{R}^{2}$, $\|x\|:=\sqrt{x_{1}^{2}+x_{2}^{2}}$
denotes the Euclidean norm of $x$. If $S\subset\mathbb{R}^{2}$ is a Borel set,
we denote by $|S|$ its Lebesgue measure. Moreover, for $x\in\mathbb{Z}^{2}$ we
will write $\arg(x)\in[0,2\pi)$ to denote the complex argument of $x$ seen as
an element of $\mathbb{C}$. For $t\in\mathbb{R}$, $\lfloor t\rfloor$ will
denote the integer part of $t$ and $\lceil
t\rceil:=\inf\\{k\in\mathbb{Z},\>k\geq t\\}$.

######  Acknowledgements.

We warmly thank Alan Hammond, who suggested the problem, for numerous
stimulating discussions and precious writing advices. We also warmly thank
Ivan Corwin for suggesting this collaboration and for stimulating discussions.
We thank Raphaël Cerf, Trishen S. Gunaratnam, Ioan Manolescu and Yvan Velenik
for inspiring discussions at various stages of the project. LD was supported
by the Swiss National Science Foundation grant n°182237. RP was supported by
the NSF through DMS-1811143, the Swiss National Science Foundation and the
NCCR SwissMAP.

##  2 Preliminary results

In the rest of this work, we fix $0<\lambda<\frac{1}{2}$.

###  2.1 Basic statistics of typical samples of $\mathbb{P}_{\lambda}^{N^{2}}$

In this subsection, we study some basic properties of a typical sample of
$\mathbb{P}_{\lambda}^{N^{2}}$.

######  Lemma 2.1 (Tail estimates for the length of a sample of
$\mathbb{P}_{\lambda}^{N^{2}}$).

There exist $c,C>0$ such that for any $N\geq 1$, any $t\geq 0$,

$\mathbb{P}_{\lambda}^{N^{2}}\left[|\Gamma|\geq tN\right]\leq
C\mathrm{e}^{-cN(t-2\sqrt{2})}.$ (2.1)

###### Proof.

It is clear that

$\mathbb{P}_{\lambda}^{N^{2}}\left[|\Gamma|\geq
tN\right]\leq\frac{\mathbb{P}_{\lambda}\left[|\Gamma|\geq
tN\right]}{\mathbb{P}_{\lambda}\left[\mathcal{A}(\Gamma)\geq N^{2}\right]}.$
(2.2)

An easy computation yields

$\mathbb{P}_{\lambda}\left[|\Gamma|\geq
tN\right]=\frac{1}{Z_{\lambda}}\sum_{k\geq\lfloor
tN\rfloor}(2\lambda)^{k}=(2\lambda)^{\lfloor tN\rfloor}.$ (2.3)

It remains to lower bound $\mathbb{P}_{\lambda}\left[\mathcal{A}(\Gamma)\geq
N^{2}\right]$. Let $\mathsf{y}(\Gamma)$ (resp. $\mathsf{x}(\Gamma)$) be the
$y$-coordinate (resp. $x$-coordinate) of the first (resp. last) vertex of
$\Gamma$. The measure $\mathbb{P}_{\lambda}$ conditioned on
$(\mathsf{y}(\Gamma),\mathsf{x}(\Gamma))$ is exactly the uniform measure over
path starting at $(0,\mathsf{y}(\Gamma))$ and ending at
$(\mathsf{x}(\Gamma),0)$. We claim that

$\mathbb{P}_{\lambda}\big{[}\mathcal{A}(\Gamma)\geq
N^{2}\>|\>(\mathsf{y}(\Gamma),\mathsf{x}(\Gamma))=(\lceil\sqrt{2}N\rceil,\lceil\sqrt{2}N\rceil)\big{]}\geq\frac{1}{2}.$
(2.4)

Indeed, the square formed by the vertices
$(0,\lceil\sqrt{2}N\rceil),(\lceil\sqrt{2}N\rceil,\lceil\sqrt{2}N\rceil),(\lceil\sqrt{2}N\rceil,0)$
and $(0,0)$ has an area at least equal to $2N^{2}$. Hence, a symmetry argument
shows that the proportion of oriented paths starting at
$(0,\lceil\sqrt{2}N\rceil)$ and ending at $(\lceil\sqrt{2}N\rceil,0)$ that
fulfill the requirement $\left\\{\mathcal{A}(\Gamma)\geq N^{2}\right\\}$ is at
least $1/2$. By a standard computation, we find $c_{1}=c_{1}(\lambda)>0$ such
that, for all $N\geq 1$,

$\mathbb{P}_{\lambda}\left[(\mathsf{y}(\Gamma),\mathsf{x}(\Gamma))=(\lceil\sqrt{2}N\rceil,\lceil\sqrt{2}N\rceil)\right]=\frac{1}{Z_{\lambda}}\lambda^{2\lceil\sqrt{2}N\rceil}\binom{2\lceil\sqrt{2}N\rceil}{\lceil\sqrt{2}N\rceil}\geq
c_{1}(2\lambda)^{2\sqrt{2}N}N^{-1/2}.$ (2.5)

Putting all the pieces together

$\mathbb{P}_{\lambda}^{N^{2}}[|\Gamma|\geq tN]\leq
c_{1}^{-1}\sqrt{N}(2\lambda)^{tN}(2\lambda)^{-2\sqrt{2}N}.$ (2.6)

The proof follows readily. ∎

This tail estimate allows us to argue that a typical sample of
$\mathbb{P}_{\lambda}^{N^{2}}$ stays confined between two balls of linear
radii with very high probability. For $K\geq 0$, let $B_{K}$ be the Euclidean
ball of radius $K$ centered at $0$.

######  Lemma 2.2 (Confinement lemma).

There exist $K_{1},K_{2},c,C>0$ such that for all $N\geq 1$,

$\mathbb{P}_{\lambda}^{N^{2}}[\Gamma\subset B_{K_{1}N}\setminus
B_{K_{2}N}]\geq 1-C\mathrm{e}^{-cN}.$ (2.7)

###### Proof.

Using Lemma 2.1 and a simple geometric observation, we get that for
$K_{1}\in\mathbb{N}_{>0}$,

$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}\left[\Gamma\cap(B_{K_{1}N})^{c}\neq\emptyset\right]$
$\displaystyle\leq$ $\displaystyle\mathbb{P}_{\lambda}^{N^{2}}[|\Gamma|\geq
K_{1}N]$ $\displaystyle\leq$ $\displaystyle
C\mathrm{e}^{-c(K_{1}-2\sqrt{2})N},$

which gives that $\Gamma\subset B_{K_{1}N}$ with high probability for
$K_{1}>2\sqrt{2}$.

For the second part of the statement, notice that the area of a path that is
contained in $B_{K_{1}N}$ and intersects $B_{KN}$ (for $K<K_{1}$) is
necessarily smaller than $2K_{1}KN^{2}$. Choosing $K_{2}<1/(2K_{1})$ so that
the preceding quantity is smaller than $N^{2}$ yields that

$\mathbb{P}_{\lambda}^{N^{2}}\left[\Gamma\cap
B_{K_{2}N}\neq\emptyset,\Gamma\cap(B_{K_{1}N})^{c}=\emptyset\right]=0.$ (2.8)

Hence,

$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}[\Gamma\cap
B_{K_{2}N}\neq\emptyset]$ $\displaystyle=$
$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}\left[\Gamma\cap
B_{K_{2}N}\neq\emptyset,\Gamma\cap(B_{K_{1}N})^{c}\neq\emptyset\right]+0$
$\displaystyle\leq$
$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}\left[\Gamma\cap(B_{K_{1}N})^{c}\neq\emptyset\right],$

and the result follows. ∎

In what follows, $K_{1},K_{2}$ will always denote the constants given by Lemma
2.2.

The following quantity will be of particular interest for the rest of this
work.

######  Definition 2.3 (Excess area).

The excess area of a path $\gamma\in\Lambda^{N^{2}}$ is the quantity
$\mathsf{ExcessArea}(\gamma)$ defined by

$\mathsf{ExcessArea}(\gamma):=\mathcal{A}(\gamma)-N^{2}.$ (2.9)

Since $\mathbb{P}_{\lambda}^{N^{2}}$exponentially penalizes long paths, we may
expect the typical area of a path to be close to $N^{2}$. The following result
quantifies this observation and will be very useful later.

######  Proposition 2.4 (Tail behaviour of $\mathsf{ExcessArea}$).

There exists $c>0$ such that for all $0\leq t\leq N$,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{ExcessArea}(\Gamma)\geq tN]\leq
2\mathrm{e}^{-ct}.$ (2.10)

The idea of the proof is simple. By Lemma 2.1, a sample $\Gamma$ of
$\mathbb{P}_{\lambda}^{N^{2}}$ has a length of order $N$. Assume that this
path satisfies $\mathcal{A}(\Gamma)>N^{2}+tN$. If we remove its first $t$
steps we obtain a path of area (roughly) at least $N^{2}$ and which is
exponentially (in $t$) favoured by $\mathbb{P}_{\lambda}^{N^{2}}$.

###### Proof.

If $A$ is any subset of $\Lambda^{N^{2}}$, we shall write
$Z_{\lambda}^{N^{2}}\left[A\right]=\sum_{\gamma\in A}\lambda^{|\gamma|}$ for
the partition function of $A$. We will also write
$Z_{\lambda}^{N^{2}}:=Z_{\lambda}^{N^{2}}[\Lambda^{N^{2}}]$. By definition,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{ExcessArea}(\Gamma)\geq tN,\text{
}\Gamma\subset B_{K_{1}N}\setminus
B_{K_{2}N}]=\frac{1}{Z_{\lambda}^{N^{2}}}\sum_{K_{2}N\leq a,b\leq
K_{1}N}\lambda^{a+b}\sum_{\begin{subarray}{c}\gamma:(0,a)\rightarrow(b,0)\\\
\mathcal{A}(\gamma)\geq N^{2}+tN\end{subarray}}1.$ (2.11)

If $\gamma:(0,a)\rightarrow(b,0)$ with $K_{2}N\leq a,b\leq K_{1}N$, call
$\mathsf{c}=\mathsf{c}(\gamma)$ the $y$-coordinate of the point of $\gamma$ of
$x$-coordinate $\lfloor t/K_{1}\rfloor$. Splitting the path $\gamma$ at the
point of coordinates $(\lfloor t/K_{1}\rfloor,\mathsf{c})$ splits $\gamma$
into a pair of elements of $\Lambda$ (after translation) that we denote by
$(\gamma_{1},\gamma_{2})$, where $\gamma_{1}$ is a path from
$(0,a-\mathsf{c})$ to $(\lfloor t/K_{1}\rfloor,0)$ and $\gamma_{2}$ is an
element of $\Lambda^{N^{2}}$ from $(0,\mathsf{c})$ to $(b-\lfloor
t/K_{1}\rfloor,0)$. As a result,

$\sum_{\begin{subarray}{c}\gamma:(0,a)\rightarrow(b,0)\\\
\mathcal{A}(\gamma)\geq
N^{2}+tN\end{subarray}}1\leq\sum_{c=0}^{a}\big{|}\Lambda^{(0,a-c)\rightarrow(\lfloor
t/K_{1}\rfloor},0)\big{|}\cdot\big{|}\Lambda^{(0,c)\rightarrow(b-\lfloor
t/K_{1}\rfloor),0)}_{N^{2}}\big{|},$ (2.12)

where $\Lambda^{x\rightarrow y}$ is the set of path of starting at $x$ and
ending at $y$, and the subscript $N^{2}$ accounts for the condition of
enclosing an area of at least $N^{2}$. Now, notice that

$\sum_{b=K_{2}N}^{K_{1}N}\lambda^{c+b-\lfloor
t/K_{1}\rfloor}\big{|}\Lambda^{(0,c)\rightarrow(b-\lfloor
t/K_{1}\rfloor,0)}_{N^{2}}\big{|}\leq
Z_{\lambda}^{N^{2}}[\mathsf{y}(\Gamma)=c],$ (2.13)

where we recall that $\mathsf{y}(\gamma)$ is the $y$-coordinate of the
starting point of $\gamma$. Recall also that $\mathsf{x}(\gamma)$ is the
$x$-coordinate of the last point of $\gamma$. Hence,

$\displaystyle\sum_{K_{2}N\leq a,b\leq K_{1}N}\lambda^{a+b}$
$\displaystyle\sum_{\begin{subarray}{c}\gamma:(0,a)\rightarrow(b,0)\\\
\mathcal{A}(\gamma)\geq N^{2}+tN\end{subarray}}1$
$\displaystyle\leq\sum_{a=K_{2}N}^{K_{1}N}\sum_{c=0}^{a}\big{|}\Lambda^{(0,a-c)\rightarrow(\lfloor
t/K_{1}\rfloor,0)}\big{|}\lambda^{a-c+\lfloor
t/K_{1}\rfloor}Z_{\lambda}^{N^{2}}[\mathsf{y}(\Gamma)=c]$
$\displaystyle\leq\sum_{c=0}^{K_{1}N}Z_{\lambda}^{N^{2}}[\mathsf{y}(\Gamma)=c]\sum_{a\geq
c}|\Lambda^{(0,a-c)\rightarrow(\lfloor
t/K_{1}\rfloor,0)}\big{|}\lambda^{a-c+\lfloor t/K_{1}\rfloor}$
$\displaystyle\leq\sum_{c=0}^{K_{1}N}Z_{\lambda}^{N^{2}}[\mathsf{y}(\Gamma)=c]Z_{\lambda}[\mathsf{x}(\Gamma)=\lfloor
t/K_{1}\rfloor]$ $\displaystyle\leq
Z_{\lambda}^{N^{2}}Z_{\lambda}[|\Gamma|\geq\lfloor t/K_{1}\rfloor]$
$\displaystyle\leq Z_{\lambda}^{N^{2}}(2\lambda)^{\lfloor t/K_{1}\rfloor}.$

The proof follows readily. ∎

###  2.2 Non-existence of large flat sections of a typical sample of
$\mathbb{P}_{\lambda}^{N^{2}}$

In the analysis of samples of $\mathbb{P}_{\lambda}^{N^{2}}$ in given angular
sectors, we will sometimes need to ensure that the marginal is not
“degenerate” in the sense that it is not supported on “flat” paths (i.e almost
horizontal / vertical paths). Because of the oriented feature of the model,
this is an additional difficulty in comparison to the setup of subcritical
statistical mechanics models, where it is often known thanks to the
Ornstein–Zernike theory that the surface tension is analytic and bounded away
from 0 and infinity (see for instance [CIV08, Theorem A]).

Let $\mathbf{A}$ be a cone of apex the origin in the first quadrant of angle
$\Theta_{\mathbf{A}}\in(0,\pi/2]$. For $\gamma\in\Lambda^{N^{2}}$, let
$x_{\mathbf{A}}=x_{\mathbf{A}}(\gamma)$ (resp
$y_{\mathbf{A}}=y_{\mathbf{A}}(\gamma)$) be the left-most (resp. right-most)
point of $\gamma\cap\mathbf{A}$. Also, recall that the set
$\Lambda^{x\rightarrow y}\subset\Lambda$ consists of all the oriented path
going from $x$ to $y$. Observe that if $\gamma$ is a path of $\Lambda$
containing $x$ and $y$, then there is a natural notion of restriction of
$\gamma$ between $x$ and $y$: it is the only element of $\Lambda^{x\rightarrow
y}$ which coincides with $\gamma$ between $x$ and $y$. We also denote by
$\theta(x_{\mathbf{A}},y_{\mathbf{A}})\in[0,\pi/2]$ the angle formed by the
segment $[x_{\mathbf{A}},y_{\mathbf{A}}]$ and the horizontal line going trough
$x_{\mathbf{A}}$. For $\varepsilon>0$, we define the following events

$\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}:=\big{\\{}\gamma\in\Lambda^{N^{2}}\>,\>\theta(x_{\mathbf{A}},y_{\mathbf{A}})\in[0,\varepsilon]\big{\\}},$
(2.14)

and,

$\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{-}:=\big{\\{}\gamma\in\Lambda^{N^{2}}\>,\>\theta(x_{\mathbf{A}},y_{\mathbf{A}})\in[\tfrac{\pi}{2}-\varepsilon,\tfrac{\pi}{2}]\big{\\}}.$
(2.15)

######  Proposition 2.5.

There exist $\varepsilon>0$ and $c=c(\varepsilon),C=C(\varepsilon)>0$ such
that, for any cone $\mathbf{A}$ as above,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\Gamma\in\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}\cup\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{-}\big{]}\leq
C\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathrm{e}^{-c\|x_{\mathbf{A}}(\Gamma)-y_{\mathbf{A}}(\Gamma)\|}\big{]}.$
(2.16)

For the proof of Proposition 2.5, we will use a probabilistic version of the
multi-valued map principle, stated and proved in the appendix (Lemma B.1).

###### Proof of Proposition 2.5.

We are going to create an appropriate multi-valued map $T$, transforming a
path belonging to $\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}$ to another path
of $\Lambda^{N^{2}}\setminus\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}$, in
such a way that this map has a lot of possible images but very few pre-images,
and such that the probability of one element of the image of a path
$\gamma\in\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}$ by $T$ is not too small
compared to $\mathbb{P}_{\lambda}^{N^{2}}[\gamma]$.

Figure 2: An illustration of the effect of the surgery map
$\mathsf{Surg}(\gamma,\mathcal{H})$. The cone $\mathbf{A}$ is the light green
region. The $\eta k$ edges of $\mathcal{H}$ that have been selected are the
blue ones, and the vertical edges that have been added are depicted in red. We
observe that the angle $\theta(x_{\mathbf{A}},y_{\mathbf{A}})$ increases in
$\mathsf{Surg}(\gamma,\mathcal{H})$.

Let $\eta>0$ be a small parameter to be fixed later. For
$\gamma\in\Lambda^{N^{2}}$, we denote by
$\ell_{\mathbf{A}}=\ell_{\mathbf{A}}(\gamma)$ the length of the section of the
path enclosed by $\mathbf{A}$, that is,
$\ell_{\mathbf{A}}(\gamma)=|\gamma_{x_{\mathbf{A}},y_{\mathbf{A}}}|$. We first
define a surgical procedure modifying $\gamma$: if $\mathcal{H}$ is some
subset of the set of the horizontal steps of $\gamma$ inside $\mathbf{A}$, we
set $\mathsf{Surg}\left(\gamma,\mathcal{H}\right)$ to be the new path obtained
by adding a vertical step just before each step belonging to $\mathcal{H}$
(see Figure 2). We are going to apply this procedure for $\mathcal{H}$ being a
proportion $\eta$ of the horizontal steps that $\gamma$ makes in $\mathbf{A}$.
Of course to be able to do so we need to ensure that the paths $\gamma$ we
apply the map to already contain a proportion at least $\eta$ of horizontal
steps in $\mathbf{A}$. Since we will apply the map to elements of
$\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}$, we expect the proportion of
horizontal steps to be quite large. Indeed, if
$\gamma\in\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}$, and if $\alpha(\gamma)$
is the proportion of vertical steps in the piece of path of $\gamma$ in
$\mathbf{A}$, then one has that

$\varepsilon\geq\arctan\left(\frac{\alpha(\gamma)}{1-\alpha(\gamma)}\right),$
(2.17)

so that $\alpha(\gamma)\leq\tan(\varepsilon)\leq 2\varepsilon$ (if
$\varepsilon$ is small enough). For technical reasons, it will also be
convenient to work on a set of paths for which $\ell_{\mathbf{A}}$ is fixed.
Introduce, for $k\geq 1$,

$A_{k}:=\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}\cap\\{\ell_{\mathbf{A}}=k\\}\cap\Lambda^{N^{2}},$
(2.18)

and

$B_{k}:=\\{k\leq\ell_{\mathbf{A}}\leq(1+\eta)k\\}\cap\Lambda^{N^{2}}.$ (2.19)

Define the multi-valued map $T_{\eta,k}:A_{k}\rightarrow\mathcal{P}(B_{k})$ by
setting for any $\gamma\in A_{k}$,

$T_{\eta,k}(\gamma)=\\{\mathsf{Surg}(\gamma,\mathcal{H}),\left|\mathcal{H}\right|=\eta
k\\}.$ (2.20)

We notice that for $\gamma\in A_{k}$,
$T_{\eta,k}(\gamma)\subset\Lambda^{N^{2}}$. Indeed, the area can only increase
under the action of $\mathsf{Surg}(\gamma,\mathcal{H})$, for any $\mathcal{H}$
being a subset of the horizontal steps of $\gamma$. Moreover, after the
surgery, the length of a path inside $\mathbf{A}$ has increased by at most
$\eta k$. Thus, the map $T_{\eta,k}$ is well-defined. We will consider the
probability measures $\mathbb{P}_{\lambda}^{N^{2}}$ on $A_{k}$ and $B_{k}$.
This way, we are exactly in the setting of Lemma B.1. Introducing

$\varphi(T_{\eta,k}):=\max_{a\in A_{k}}\max_{b\in
T_{\eta,k}(a)}\frac{\mathbb{P}_{\lambda}^{N^{2}}[a]}{\mathbb{P}_{\lambda}^{N^{2}}[b]},\qquad\psi(T_{\eta,k}):=\frac{\max_{b\in\tilde{B}_{k}}\big{|}T_{\eta,k}^{-1}(b)\big{|}}{\min_{a\in
A_{k}}\big{|}T_{\eta,k}(a)\big{|}},$ (2.21)

where $\tilde{B}_{k}:=\textup{Im}(T_{\eta,k})$. A direct application of Lemma
B.1 yields that

$\mathbb{P}_{\lambda}^{N^{2}}[A_{k}]\leq\varphi(T_{\eta,k})\psi(T_{\eta,k})\mathbb{P}_{\lambda}^{N^{2}}[B_{k}].$
(2.22)

We are now left with estimating $\varphi(T_{\eta,k})$ and $\psi(T_{\eta,k})$.

We start by observing that $\mathsf{Surg}(\gamma,\mathcal{H})$ increases the
length of a path $\gamma$ by $\left|\mathcal{H}\right|$ units, so that for any
$\gamma\in A_{k}$ and $\tilde{\gamma}\in T_{\eta,k}(\gamma)$, the ratio
defining $\varphi$ is constant, and given by

$\varphi(T_{\eta,k})=\lambda^{-\eta k}.$ (2.23)

Now, our task is to bound $\psi(T_{\eta,k})$. Let $\gamma\in A_{k}$. Then, it
is clear that

$\big{|}T_{\eta,k}(\gamma)\big{|}\geq\binom{(1-\alpha(\gamma))k}{\eta
k}\geq\binom{(1-2\varepsilon)k}{\eta k},$ (2.24)

where $\alpha(\gamma)$ is the proportion of vertical steps in the piece of
path of $\gamma$ in $\mathbf{A}$ and where we used (2.17) to get that
$\alpha(\gamma)\leq 2\varepsilon$ (if $\varepsilon$ is small enough). Next, we
have to bound $\left|T_{\eta,k}^{-1}(\widetilde{\gamma})\right|$, for
$\widetilde{\gamma}\in\tilde{B}_{k}$. Some vertical steps may have been pushed
outside the sector $\mathbf{A}$ by the effect of $T_{\eta,k}$, and we must
explore them to reconstruct $\gamma$ knowing $T_{\eta,k}(\gamma)$. Since at
most $\eta k$ of these steps disappeared in $\mathbf{A}$, we obtain

$\big{|}T^{-1}_{\eta,k}(\widetilde{\gamma})\big{|}\leq\binom{(2\varepsilon+2\eta)k}{\eta
k}.$ (2.25)

Putting all the pieces together and using (2.22), we obtain

$\mathbb{P}_{\lambda}^{N^{2}}\left[A_{k}\right]\leq\lambda^{-\eta
k}\frac{\binom{(2\varepsilon+2\eta)k}{\eta k}}{\binom{(1-2\varepsilon)k}{\eta
k}}\mathbb{P}_{\lambda}^{N^{2}}[B_{k}].$ (2.26)

Standard estimates555For any $c_{1}<c_{2}$, as $k\rightarrow\infty$,
$\binom{c_{2}k}{c_{1}k}\sim\mathrm{e}^{c_{2}kI\left(\frac{c_{1}}{c_{2}}\right)}.$
(2.27) yield that, as $k\rightarrow\infty$,

$\lambda^{-\eta k}\frac{\binom{(2\varepsilon+2\eta)k}{\eta
k}}{\binom{(1-2\varepsilon)k}{\eta
k}}\sim\exp(-k[(1-2\varepsilon)I\left(\frac{\eta}{1-2\varepsilon}\right)-(2\varepsilon+2\eta)I\left(\frac{\eta}{2\varepsilon+2\eta}\right)+\eta\log\lambda]),$
(2.28)

where for $x\in(0,1)$, $I(x):=-x\log x-(1-x)\log(1-x)$.

Set

$c_{\varepsilon,\eta}=(1-2\varepsilon)I\left(\frac{\eta}{1-2\varepsilon}\right)-(2\varepsilon+2\eta)I\left(\frac{\eta}{2\varepsilon+2\eta}\right)+\eta\log\lambda.$
(2.29)

It is easy to check that one can find a pair $\varepsilon,\eta(\varepsilon)>0$
satisfying $c_{\varepsilon,\eta(\varepsilon)}>0$. Let us fix such an
$\varepsilon>0$ and drop $\eta$ from the notations. We obtained that there
exists $C=C(\varepsilon)>0$ such that for $k\geq 1$,

$\lambda^{-\eta k}\frac{\binom{(2\varepsilon+2\eta)k}{\eta
k}}{\binom{(1-2\varepsilon)k}{\eta k}}\leq C\mathrm{e}^{-c_{\varepsilon}k}.$
(2.30)

Hence,

$\mathbb{P}_{\lambda}^{N^{2}}[\Gamma\in\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+},\>\ell_{A}=k]\leq
C\mathrm{e}^{-c_{\varepsilon}k}\mathbb{P}_{\lambda}^{N^{2}}[k\leq\ell_{\mathbf{A}}\leq(1+\eta)k].$
(2.31)

Summing over $k$ we finally obtain the existence of
$c_{1}=c_{1}(\varepsilon),C_{1}=C_{1}(\varepsilon)$ such that

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\Gamma\in\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{+}\big{]}\leq
C_{1}\mathbb{E}^{N^{2}}_{\lambda}\big{[}e^{-c_{1}\ell_{\mathbf{A}}}\big{]}\leq
C_{1}\mathbb{E}^{N^{2}}_{\lambda}\big{[}\mathrm{e}^{-c_{1}\|x_{\mathbf{A}}(\Gamma)-y_{\mathbf{A}}(\Gamma)\|}\big{]}.$
(2.32)

A similar argument holds for $\mathsf{Bad}_{\varepsilon,\mathbf{A}}^{-}$. This
concludes the proof. ∎

###  2.3 Couplings between $\mathbb{P}_{\lambda}^{N^{2}}$ and a conditioned
random walk

This subsection is devoted to the description of two couplings between the
measure $\mathbb{P}_{\lambda}^{N^{2}}$ conditioned on a pinning event and the
law of a random walk bridge. We first describe and prove the existence of
these couplings and then explain what they will be useful for.

Let $a,b\in\mathbb{N}^{2}$ such that $\arg(a)>\arg(b)$. We introduce the event
$\mathsf{Fac}(a,b)$ defined by

$\mathsf{Fac}(a,b):=\left\\{\gamma\in\Lambda,\>\text{The segment }[a,b]\text{
is a facet of }\gamma\right\\}.$ (2.33)

Recall that the set $\Lambda^{a\rightarrow b}\subset\Lambda$ consists of all
the oriented path going from $a$ to $b$. We introduce the following partial
order in the set $\Lambda^{a\rightarrow b}$: for
$\gamma^{1},\gamma^{2}\in\Lambda^{a\rightarrow b}$, we shall say that
$\gamma^{1}\geq\gamma^{2}$ when

$\forall(x_{1},y_{1})\in\gamma^{1},(x_{2},y_{2})\in\gamma^{2},x_{1}=x_{2}\Rightarrow
y_{1}\geq y_{2}.$ (2.34)

Finally, we denote by $\mathbb{P}_{a,b}^{-}$ the uniform measure over all the
paths of $\Lambda^{a\rightarrow b}$ that remain under the segment $[a,b]$.

With these definitions in hand, we are now able to describe the first
coupling.

######  Proposition 2.6.

Let $a$ and $b$ be two vertices of $\mathbb{N}^{2}$ such that
$\arg(a)>\arg(b)$. There exists a probability measure $\Psi$ on the space
$\Lambda^{a\rightarrow b}\times\Lambda^{a\rightarrow b}$ such that:

  1. (i)

the marginal of $\Psi$ on its first coordinate has the law of $\Gamma_{a,b}$
where $\Gamma$ is sampled according to
$\mathbb{P}_{\lambda}^{N^{2}}[\>\cdot\>|\>\mathsf{Fac}(a,b)]$,

  2. (ii)

the marginal of $\Psi$ on its second coordinate has law
$\mathbb{P}^{-}_{a,b}$,

  3. (iii)

$\Psi\big{[}\\{(\gamma^{1},\gamma^{2})\in\Lambda^{a\rightarrow
b}\times\Lambda^{a\rightarrow b},\gamma^{1}\geq\gamma^{2}\\}\big{]}=1.$

###### Proof.

The proof relies on a dynamical Markov chain argument and is, for percolation
aficionados, very similar to that of Holley’s inequality (see for instance
[Gri10]).

We shall start from the restriction of a sample of
$\mathbb{P}_{\lambda}^{N^{2}}[\>\cdot\>|\>\mathsf{Fac}(a,b)]$ to
$\Lambda^{a\rightarrow b}$ that we denote $\Gamma_{a,b}$, and then describe
two Markovian dynamics $(\Gamma^{1}_{k})_{k\geq 0}$ and
$(\Gamma^{2}_{k})_{k\geq 0}$ on the state space $\Lambda^{a\rightarrow b}$
with the following properties:

  1. $(\mathbf{P1})$

$\Gamma_{0}^{1}=\Gamma_{0}^{2}=\Gamma_{a,b}$,

  2. $(\mathbf{P2})$

$\forall k\geq 0$, $\Gamma_{k}^{1}\geq\Gamma_{k}^{2}$,

  3. $(\mathbf{P3})$

the law of $\Gamma_{a,b}$ is the stationary measure of the Markov chain
$(\Gamma_{k}^{1})_{k\geq 0}$,

  4. $(\mathbf{P4})$

$\mathbb{P}^{-}_{a,b}$ is the stationary measure of the Markov chain
$(\Gamma_{k}^{2})_{k\geq 0}$.

Now, let us describe the Markovian dynamics. Let $p=\|a-b\|_{1}$ be the length
of any path of $\Lambda^{a\rightarrow b}$. Let $(\ell_{k})_{k\geq 0}$ be a
family of independent random variables all having a uniform distribution on
$\\{1,\dots,p-1\\}$, and $(X_{k})_{k\geq 0}$ a family of independent random
variables all having a uniform distribution on $\\{0,1\\}$. At step $k$, the
random variable $\ell_{k}$ is the position of the vertex on which we will
proceed a random modification.

Looking at the two edges incident to the vertex $\Gamma^{i}_{k}(\ell_{k})$ for
$i\in\\{1,2\\}$, three different configurations can arise:

  1. $1)$

If these two edges are co-linear, we do not modify anything:
$\Gamma^{i}_{k+1}=\Gamma^{i}_{k}$ for $i\in\\{1,2\\}$.

  2. $2)$

If these edges form a corner of the type

    1. $(i)$

For $\Gamma^{1}$ we flip the corner into the opposite corner  if and only if
we have $X_{k}=1$ and the modified path does not go above the facet $[a,b]$.
If any of these two conditions is not verified, we do not modify
$\Gamma_{k}^{1}$.

    2. $(ii)$

We apply the same procedure for $\Gamma^{2}$.

  3. $3)$

If these edges form a corner of the type

    1. $(i)$

For $\Gamma^{1}$, we flip this corner into the opposite corner  if and only if
$X_{k}=1$ and the area enclosed by this new path is still greater than
$N^{2}$. If any of these two conditions is not verified we do not modify
$\Gamma^{1}_{k}$.

    2. $(ii)$

For $\Gamma^{2}$, we flip the corner into the opposite one if and only if
$X_{k}=1$.

It is straightforward that these Markov dynamics are irreducible, aperiodic,
and reversible. Then, $\Gamma^{1}$ (resp. $\Gamma^{2}$) admits a stationnary
measure and converges towards it: it is straightforward to see that this
stationary measure is the law of $\Gamma_{a,b}$ (resp $\mathbb{P}^{-}_{a,b}$).
Moreover, by construction, one always has $\Gamma^{1}\geq\Gamma^{2}$. This
yields the desired stochastic domination. ∎

This coupling will be very efficient when computing local statistics of a
sample of $\mathbb{P}_{\lambda}^{N^{2}}$ along a facet (e.g. the local
roughness along a facet) as it allows us to compare them to the ones of a
random walk excursion on which we know much more information. More precisely,
Proposition 2.6 has the following consequence: an upper bound on the length of
a facet automatically converts into an upper bound on the local roughness
under this facet.

######  Lemma 2.7.

There exist $c,C,N_{0}>0$ such that the following holds. For any
$a,b\in\mathbb{N}^{2}$ with $\arg(a)>\arg(b)$ and $\|b-a\|\geq N_{0}$ and any
$t\geq 0$,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\max_{x\in\Gamma_{a,b}}\mathsf{LR}(x)>t\|b-a\|^{1/2}\>\big{|}\>\mathsf{Fac}(a,b)\big{]}\leq
C\exp(-ct^{2}).$ (2.35)

###### Proof.

Observe that thanks to Proposition 2.6, the random variable
$\max_{x\in\Gamma_{a,b}}\mathsf{LR}(x)$ where $\Gamma$ is sampled according to
$\mathbb{P}_{\lambda}^{N^{2}}\left[\>\cdot\>|\>\mathsf{Fac}(a,b)\right]$ is
stochastically dominated by the random variable
$\max_{x\in\omega_{a,b}}\mathsf{LR}(x)$ where $\omega_{a,b}$ is sampled
according to $\mathbb{P}^{-}_{a,b}$.

The statement then follows by classical random walks arguments. Indeed, the
results of [CC13] imply that a lattice random walk bridge conditioned to stay
below a line segment with increments having exponential tails, after
subtraction of the linear term corresponding to the equation of that segment,
converges towards the standard Brownian excursion. As a consequence, knowing
the fluctuations of the Brownian excursion (see [Chu75]) on an interval of
size $\|b-a\|$ leads to the bound

$\mathbb{P}^{-}_{a,b}\big{[}\max_{x\in\omega_{a,b}}\mathsf{LR}(x)\geq
t\|b-a\|^{1/2}\big{]}\leq C\exp({-ct^{2}}),$ (2.36)

when $\|b-a\|$ is large enough. ∎

######  Remark 2.8.

In Section 4, we will use the preceding result in the regime
$t=\left(\log\|b-a\|\right)^{\alpha}$ for some power $0<\alpha<1$. We claim
that the result still holds true and is a consequence of classical moderate
deviations estimates for random walks (see [dA92]).

The second coupling we shall need is closely related to the first one with the
only exception that we do not assume that $[a,b]$ is a facet anymore. This
time, we will build an “increasing coupling” (in the sense of the partial
order of $\Lambda^{a\rightarrow b}$) between the law of $\Gamma_{a,b}$ where
$\Gamma$ is sampled according to
$\mathbb{P}_{\lambda}^{N^{2}}[\>\cdot\>|\>a,b\in\Gamma]$ and the distribution
of the random walk bridge between $a$ and $b$. We denote by $\mathbb{P}_{a,b}$
the uniform distribution on $\Lambda^{a\rightarrow b}$.

######  Proposition 2.9.

Let $a$ and $b$ be two vertices of $\mathbb{N}^{2}$ such that
$\arg(a)>\arg(b)$. There exists a probability measure $\overline{\Psi}$ on the
space $\Lambda^{a\rightarrow b}\times\Lambda^{a\rightarrow b}$ such that:

  1. (i)

the marginal of $\overline{\Psi}$ on its first coordinate has the law of
$\Gamma_{a,b}$ where $\Gamma$ is sampled according to
$\mathbb{P}_{\lambda}^{N^{2}}[\>\cdot\>|\>a,b\in\Gamma]$,

  2. (ii)

the marginal of $\overline{\Psi}$ on its second coordinate has law
$\mathbb{P}_{a,b}$,

  3. (iii)

$\overline{\Psi}\big{[}\\{(\gamma^{1},\gamma^{2})\in\Lambda^{a\rightarrow
b}\times\Lambda^{a\rightarrow b},\gamma^{1}\geq\gamma^{2}\\}\big{]}=1.$

###### Proof.

The proof follows the strategy used to obtain Proposition 2.6 except that we
do not have the condition of not going above the segment $[a,b]$ anymore. ∎

###  2.4 A rough upper bound on the length of a typical facet

The goal of this section is to give an illustration of the resampling strategy
which will be used throughout the paper. The result proved in this section
will also be useful for the proof of the upper bounds of Theorem 1.3.

######  Proposition 2.10.

Let $\epsilon>0$. There exist $c=c(\epsilon),C=C(\epsilon)>0$ such that, for
all $1\leq t\leq N^{2/3-\epsilon}$.

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanFL}(\Gamma)\geq
tN^{\frac{2}{3}+\epsilon}\big{]}\leq C\mathrm{e}^{-ct^{3/2}N^{3\epsilon/2}}.$
(2.37)

Moreover, there exists $c^{\prime},C^{\prime}>0$ such that, for all $t\geq 1$

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanFL}(\Gamma)\geq
tN^{\frac{4}{3}}\big{]}\leq C^{\prime}\mathrm{e}^{-c^{\prime}tN^{4/3}}.$
(2.38)

Before proving this result, we need to define an event which we will ask the
resamplings to satisfy in order to ensure they capture a large enough area.

Let us first recall and introduce some notations. For $x,y\in\mathbb{N}^{2}$,
let $\mathbf{A}_{x,y}$ be the cone of apex $0$ bounded by $x$ and $y$ and let
$\mathbf{T}_{0,x,y}$ be the triangle of apexes $0,x$ and $y$. For a cone
$\mathbf{A}_{x,y}$ and $\gamma\in\Lambda$, we define
$\textup{Enclose}(\gamma\cap\mathbf{A}_{x,y})$ to be the region delimited by
$\gamma$ and the two boundary axes of $\mathbf{A}_{x,y}$. The following event,
illustrated in Figure 3, will be the one we will require our resampling to
satisfy.

######  Definition 2.11 (Good area capture).

Let $\gamma\in\Lambda$ be a path, $\eta>0$ and $x,y\in\mathbb{N}^{2}$. We say
that $\gamma$ realises the event $\mathsf{GAC}\left(x,y,\eta\right)$ (meaning
“good area capture”) if

  1. $(i)$

$\gamma\cap\mathbf{A}_{x,y}\in\Lambda^{x\rightarrow y}$, or in words, $\gamma$
connects $x$ and $y$ by a oriented path,

  2. $(ii)$

$|\textup{Enclose}(\gamma\cap\mathbf{A}_{x,y})|-|\mathbf{T}_{0,x,y}|\geq\eta||x-y||^{\frac{3}{2}}$.

Figure 3: An illustration of the event $\mathsf{GAC}\left(x,y,\eta\right)$.
The light green shaded region corresponds to
$\textup{Enclose}(\gamma\cap\mathbf{A}_{x,y})$. The event that the path
$\gamma$ satisfies $\mathsf{GAC}\left(x,y,\eta\right)$ means that the
(algebraic) area of the orange hatched region exceeds $\eta\|x-y\|^{3/2}$.

###### Proof of Proposition 2.10.

Let $1\leq t\leq N^{2/3-\epsilon}$. Introduce the event

$\mathsf{BigMeanFL}(t):=\big{\\{}\mathsf{MeanFL}(\Gamma)\geq
tN^{\frac{2}{3}+\epsilon}\big{\\}}.$ (2.39)

We pick uniformly at random, and independently of $\Gamma$, two points
$\mathbf{x},\mathbf{y}$ (with $\arg(\textbf{x})>\arg(\textbf{y})$) in
$B_{K_{1}N}$ and call $\mathsf{GoodHit}$ the event that these two points
coincide with the extremities of $\mathsf{MeanFac}$. The measure associated
with this procedure is denoted $\mathbb{P}$.

Let us call $\mathsf{Bad}_{\varepsilon}^{+,-}$ the event that
$\theta(\textbf{x},\textbf{y})\in[0,\varepsilon)\cup(\pi/2-\varepsilon,\pi/2]$
for the $\varepsilon>0$ given by Proposition 2.5. Let
$\mathcal{E}:=\mathsf{GoodHit}\cap(\mathsf{Bad}_{\varepsilon}^{+,-})^{c}\cap\\{\Gamma\subset
B_{K_{1}N}\setminus B_{K_{2}N}\\}$. Resample $\Gamma$ between $\mathbf{x}$ and
$\mathbf{y}$ according to the uniform law among paths
$\gamma\in\Lambda^{\mathbf{x}\rightarrow\mathbf{y}}$ such that
$(\Gamma\setminus\Gamma_{\mathbf{x},\mathbf{y}})\cup\gamma$ satisfies the area
condition and call $\tilde{\Gamma}$ the resampling. Note that $\tilde{\Gamma}$
has law $\mathbb{P}_{\lambda}^{N^{2}}$.

Since we are resampling along a facet, if $(\Gamma,(\mathbf{x},\mathbf{y}))$
realises $\mathcal{E}$, then $\Gamma_{\mathbf{x},\mathbf{y}}$ lies underneath
the segment joining x and y. Thus, if
$(\Gamma,(\mathbf{x},\mathbf{y}))\in\mathcal{E}$, replacing
$\Gamma_{\mathbf{x},\mathbf{y}}$ by any path satisfying
$\mathsf{GAC}(\mathbf{x},\mathbf{y},\eta)$ still satisfies the area condition
and666We use the fact that
$\mathsf{GAC}\left(\mathbf{x},\mathbf{y},\eta\right)$ can be seen as an event
on $\Lambda^{\mathbf{x}\rightarrow\mathbf{y}}$ and for convenience we keep the
same notation when looking at this event under the two measures of interest.

$\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\tilde{\Gamma}\in\mathsf{GAC}(\mathbf{x},\mathbf{y},\eta)\text{
}|\text{
}(\Gamma,(\mathbf{x},\mathbf{y}))\in\mathsf{BigMeanFL}(t)\cap\mathcal{E}]\geq\mathbb{P}_{\mathbf{x},\mathbf{y}}[\mathsf{GAC}(\mathbf{x},\mathbf{y},\eta)].$
(2.40)

where we recall that $\mathbb{P}_{a,b}$ is the uniform law on
$\Lambda^{a\rightarrow b}$. Indeed, conditioning on
$(\Gamma,(\mathbf{x},\mathbf{y}))\in\mathsf{BigMeanFL}(t)\cap\mathcal{E}$, the
above inequality is equivalent to

$\frac{|\\{\gamma\in\Lambda^{\mathbf{x}\rightarrow\mathbf{y}},\text{
}\gamma\in\mathsf{GAC}(\mathbf{x},\mathbf{y},\eta)\\}|}{|\\{\gamma\in\Lambda^{\mathbf{x}\rightarrow\mathbf{y}},\text{
}(\Gamma\setminus\Gamma_{\mathbf{x},\mathbf{y}})\cup\gamma\in\Lambda^{N^{2}}\\}|}\geq\frac{|\\{\gamma\in\Lambda^{\mathbf{x}\rightarrow\mathbf{y}},\text{
}\gamma\in\mathsf{GAC}(\mathbf{x},\mathbf{y},\eta)\\}|}{|\Lambda^{\mathbf{x}\rightarrow\mathbf{y}}|},$
(2.41)

which is trivially true. Now, if
$\theta(\mathbf{x},\mathbf{y})\in[\varepsilon,\pi/2-\varepsilon]$, standard
computations on the simple random walk bridge (see Lemma 3.4 for more details)
ensure that for $\eta>0$ sufficiently small there exists
$c_{1}=c_{1}(\eta,\varepsilon)>0$ such that,

$\mathbb{P}_{\mathbf{x},\mathbf{y}}[\mathsf{GAC}(\mathbf{x},\mathbf{y},\eta)]\geq
c_{1}.$ (2.42)

Notice also that

$\\{\tilde{\Gamma}\in\mathsf{GAC}(\mathbf{x},\mathbf{y},\eta),(\Gamma,(\mathbf{x},\mathbf{y}))\in\mathsf{BigMeanFL}(t)\cap\mathcal{E}\\}\subset\big{\\{}\mathsf{ExcessArea}(\tilde{\Gamma})\geq\eta
t^{\frac{3}{2}}N^{1+\frac{3}{2}\epsilon}\big{\\}}.$ (2.43)

Putting all the pieces together, and using the fact that $\tilde{\Gamma}$ has
law $\mathbb{P}_{\lambda}^{N^{2}}$,

$\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}(t)\cap\mathcal{E}]\leq
c_{1}^{-1}\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{ExcessArea}(\Gamma)\geq\eta
t^{\frac{3}{2}}N^{1+\frac{3}{2}\epsilon}\big{]}.$ (2.44)

Finally, notice that for some $c_{2}>0$,

$\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}\left[\mathsf{GoodHit}\text{
}|\text{ }\mathsf{BigMeanFL}(t)\cap\\{\Gamma\subset B_{K_{1}N}\setminus
B_{K_{2}N}\\}\right]\geq\frac{c_{2}}{N^{4}}.$ (2.45)

Putting all the pieces together,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}(t)]\leq\frac{N^{4}}{c_{2}}\cdot\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}(t)\cap\mathcal{E}]\\\
+\frac{N^{4}}{c_{2}}\cdot\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}(t)\cap\mathsf{GoodHit}\cap(\mathsf{Bad}_{\varepsilon}^{+,-})\cap\\{\Gamma\subset
B_{K_{1}N}\setminus B_{K_{2}N}\\}]\\\
+\mathbb{P}_{\lambda}^{N^{2}}\left[\mathsf{BigMeanFL}(t)\cap\\{\Gamma\subset
B_{K_{1}N}\setminus B_{K_{2}N}\\}^{c}\right].$ (2.46)

To bound the second term we use Proposition 2.5 above. Notice that,

$\mathsf{BigMeanFL}(t)\cap\mathsf{GoodHit}\cap(\mathsf{Bad}_{\varepsilon}^{+,-})\cap\\{\Gamma\subset
B_{K_{1}N}\setminus B_{K_{2}N}\\}\\\
\subset\bigcup_{\begin{subarray}{c}{u,v}\in B_{K_{1}N}\setminus B_{K_{2}N}\\\
\|u-v\|\geq
tN^{\frac{2}{3}+\epsilon}\end{subarray}}\big{\\{}\Gamma\in\mathsf{Bad}^{+}_{\varepsilon,\mathbf{A}_{u,v}}\cup\mathsf{Bad}^{+}_{\varepsilon,\mathbf{A}_{u,v}}\big{\\}},$

where $\mathbf{A}_{u,v}$ is the cone of apex $0$ bounded by $u$ and $v$.
Finally, apply Lemma 2.2, Proposition 2.4, and Proposition 2.5 to get that for
some $c=c(\varepsilon)>0$,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}(t)]\leq\frac{N^{4}}{c_{2}}\cdot
c_{1}^{-1}\cdot\exp(-ct^{\frac{3}{2}}N^{\frac{3}{2}\epsilon})+\frac{N^{4}}{c_{2}}\cdot(K_{1}N)^{4}\cdot
2\exp(-ctN^{\frac{2}{3}+\epsilon})+\exp(-cN),$ (2.47)

and thus the first inequality (recall that $t\leq N^{2/3-\epsilon}$).

For $t\geq 1$777In fact, one can take $t\geq 3\sqrt{2}N^{-1/3}$ here. notice
that

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanFL}(\Gamma)\geq
tN^{\frac{4}{3}}\big{]}\leq\mathbb{P}_{\lambda}^{N^{2}}\big{[}|\Gamma|\geq(tN^{1/3})\cdot
N\big{]}.$ (2.48)

Using Lemma 2.1 we get the result. ∎

##  3 Analysis of the mean facet length and the mean local roughness

###  3.1 Upper bounds

In the following subsection, we actually prove a stronger statement than the
upper bounds of Theorem 1.3. Recall that $\mathsf{MeanFL}$ and
$\mathsf{MeanLR}$ where defined in Definition 1.2. The goal of this subsection
is to prove upper tail estimates on $\mathsf{MeanFL}$ and $\mathsf{MeanLR}$.
Our first result is a refinement of Proposition 2.10.

######  Proposition 3.1 (Upper tail of $\mathsf{MeanFL}$).

There exist $\tilde{c},c,C>0$ such that, for any $\tilde{c}\leq t\leq
N^{2/3}$,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanFL}(\Gamma)\geq
tN^{\frac{2}{3}}\big{]}\leq C\mathrm{e}^{-ct^{3/2}}.$ (3.1)

######  Proposition 3.2 (Upper tail of $\mathsf{MeanLR}$).

There exist $\tilde{c},c,C>0$ such that for any $\tilde{c}\leq t\leq N^{5/6}$,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanLR}(\Gamma)\geq
tN^{\frac{1}{3}}\big{]}\leq C\mathrm{e}^{-ct^{6/5}}.$ (3.2)

######  Remark 3.3.

For larger values of $t$, we can use Lemma 2.1 and Proposition 2.10 to get
explicit tails for $\mathsf{MeanFL}$ and $\mathsf{MeanLR}$. In particular, we
obtain that

$\limsup_{N\rightarrow\infty}\mathbb{E}^{N^{2}}_{\lambda}\left[\frac{\mathsf{MeanFL}(\Gamma)}{N^{2/3}}\right]<\infty,\qquad\limsup_{N\rightarrow\infty}\mathbb{E}^{N^{2}}_{\lambda}\left[\frac{\mathsf{MeanLR}(\Gamma)}{N^{1/3}}\right]<\infty.$
(3.3)

We keep the notations of Section 2.4. We will again use the event
$\mathsf{GAC}\left(x,y,\eta\right)$ introduced in Definition 2.11. While it is
difficult to estimate the probability of the event
$\mathsf{GAC}\left(x,y,\eta\right)$ under $\mathbb{P}_{\lambda}^{N^{2}}$, it
is much simpler to estimate it under $\mathbb{P}_{x,y}$ which is the uniform
law over $\Lambda^{x\rightarrow y}$ (assuming that we chose
$x,y\in\mathbb{N}^{2}$ such that $\Lambda^{x\rightarrow y}\neq\emptyset$). The
following result, whose proof is postponed to the appendix, gives us a lower
bound on $\mathbb{P}_{x,y}[\mathsf{GAC}\left(x,y,\eta\right)]$. Denote by
$\theta(x,y)\in[0,\pi/2]$ the angle formed by the horizontal axis and the
segment joining $x$ and $y$.

######  Lemma 3.4.

Let $\varepsilon,\eta>0$. There exist $c=c(\varepsilon,\eta)>0$ and
$N_{0}=N_{0}(\varepsilon,\eta)>0$ such that for any $x,y\in\mathbb{N}^{2}$
satisfying $\Lambda^{x\rightarrow y}\neq\emptyset$,
$\varepsilon\leq\theta(x,y)\leq\pi/2-\varepsilon$ and $\|x-y\|\geq N_{0}$,

$\mathbb{P}_{x,y}[\mathsf{GAC}\left(x,y,\eta\right)]\geq c.$ (3.4)

######  Remark 3.5.

  1. 1.

The constant $c(\varepsilon,\eta)$ in the result above degenerates as
$\varepsilon\rightarrow 0$. However, as we saw in Proposition 2.5, up to an
event of small probability, we will be able to assume that $\theta$ stays
bounded away from $0$ and $\pi/2$.

  2. 2.

The constant $\tilde{c}$ in the statement of Propositions 3.1 and 3.2 will be
chosen in terms of $N_{0}=N_{0}(\varepsilon,\eta)$ for $\varepsilon$ given by
Proposition 2.5 and a properly chosen $\eta>0$.

####  3.1.1 Proof of Proposition 3.1

As in the proof of Proposition 2.10, we would like to resample along the mean
facet. However, as we saw, to pick the extremities of the mean facet without
revealing any information about the path $\Gamma$ comes with a cost of
$O(N^{4})$. This cost was previously handled by the requirement that $t\geq
N^{\epsilon}$, but now we allow $t$ to be of smaller order. To tackle this
difficulty, we will not resample between the exact extremities of the mean
facet but between points which approximate them (see Figure 4).

###### Proof of Proposition 3.1.

Let $t>0$ be fixed, and let $\delta,\varepsilon>0$ be two small parameters
that will be fixed later. Thanks to Proposition 2.10, we can additionally
assume that $t\leq N^{\epsilon}$ for some $\epsilon\in(0,1/3)$. Indeed, if
$N^{\epsilon}\leq t\leq N^{2/3}$, using (2.37), we get

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanFL}(\Gamma)\geq(tN^{-\epsilon})N^{2/3+\epsilon}\big{]}\leq
C\mathrm{e}^{-ct^{3/2}}.$ (3.5)

We now assume $t\leq N^{\epsilon}$. Let us introduce the following event

$\mathsf{BigMeanFL}(t)=\Big{\\{}\mathsf{MeanFL}(\Gamma)\geq
tN^{\frac{2}{3}}\Big{\\}}.$ (3.6)

We also introduce, for any integer $k\geq 1$,

$\mathsf{BigMeanFL}_{k}(t)=\Big{\\{}ktN^{\frac{2}{3}}\leq\mathsf{MeanFL}(\Gamma)<(k+1)tN^{\frac{2}{3}}\Big{\\}},$
(3.7)

so that the family $(\mathsf{BigMeanFL}_{k}(t))_{k\geq 1}$ partitions the
event $\mathsf{BigMeanFL}$.

Notice that for large values of $k$, we can use Proposition 2.10 again to
conclude. If $k_{N}(\epsilon,t)$ is the smallest $k\geq 1$ such that $tk\geq
N^{\epsilon}$, using (2.37) again,

$\mathbb{P}_{\lambda}^{N^{2}}\Big{[}\mathsf{MeanFL}(\Gamma)\geq
tk_{N}(\epsilon,t)N^{\frac{2}{3}}\Big{]}\leq Ce^{-c(tk_{N})^{3/2}}.$ (3.8)

In particular it suffices to control the probabilities of
$\mathsf{BigMeanFL}_{k}(t)$ for $1\leq k\leq k_{N}(\epsilon,t)$. In fact, it
is sufficient to control the probabilities of $\mathsf{BigMeanFL}_{k}(t/3)$
for $3\leq k\leq k_{N}(\epsilon,t/3)=3k_{N}(\epsilon,t)$. We now let $s=t/3$
and $k_{N}=k_{N}(\epsilon,s)$.

Fix $3\leq k\leq k_{N}$. For $j\geq 0$, define the angles

$\theta^{+}_{j}=\theta^{+}_{j}(\delta,k):=\frac{\pi}{4}+\arctan\left(\frac{j\delta
ks}{N^{1/3}}\right),\qquad\theta^{-}_{j}=\theta^{-}_{j}(\delta,k):=\frac{\pi}{4}-\arctan\left(\frac{j\delta
ks}{N^{1/3}}\right),$ (3.9)

and denote $\ell_{j}^{+}=\ell_{j}^{+}(\delta,k)$ (resp.
$\ell_{j}^{-}=\ell_{j}^{-}(\delta,k)$) the half-line rooted at the origin and
of argument $\theta^{+}_{j}$ (resp. $\theta^{-}_{j}$).

When $\mathsf{BigMeanFL}_{k}(s)$ occurs, there are at most $2/\delta$ indices
$j$ such that $\ell_{j}^{+}$ (resp. $\ell_{j}^{-}$) intersects the mean facet.
For a sample $\Gamma$ and $0\leq j\leq 2/\delta$, we call
$a_{j}=a_{j}(\Gamma,\delta,k)$ (resp. $b_{j}=b_{j}(\Gamma,\delta,k)$) the
points of $\mathbb{N}^{2}$ that are the closest888There might be ambiguity in
the choice of such points and we solve this issue by asking $a_{j}$ (resp.
$b_{j}$) to lie on the right (resp. on the left) of $\ell_{j}^{+}$ (resp.
$\ell_{j}^{-}$). to the intersection between $\Gamma$ and $\ell_{j}^{+}$
(resp. $\ell_{j}^{-}$). Observe that these points are defined from $\Gamma$ by
a deterministic procedure which means that resampling a sample $\Gamma$
between two of these points yields an output distributed according to
$\mathbb{P}_{\lambda}^{N^{2}}$. In particular, we can define
$\mathbf{a}=\mathbf{a}(\Gamma,\delta,k)$ (resp.
$\mathbf{b}=\mathbf{b}(\Gamma,\delta,k)$) to be the point of $\Gamma$ defined
as follows: if $j_{0}$ (resp. $j_{1}$) is the largest $j$ such that
$\ell_{j}^{+}$ (resp. $\ell_{j}^{-}$) intersects the mean facet, then
$\mathbf{a}=a_{j_{0}}$ (resp. $\mathbf{b}=b_{j_{1}}$).

In order to be able to pick $\mathbf{a}$ and $\mathbf{b}$ and resample between
them without revealing the portion $\Gamma_{\mathbf{a},\mathbf{b}}$ of
$\Gamma$ between $\mathbf{a}$ and $\mathbf{b}$, we introduce some extra
randomness. We pick uniformly, a pair of points $(\mathbf{x},\mathbf{y})$ in
$\\{a_{j},\>0\leq j\leq 2/\delta\\}\times\\{b_{j},\>0\leq j\leq 2/\delta\\}$
and call $\mathbb{P}$ the measure associated with this random procedure. Let
$\mathsf{GoodHit}_{k}$ be the event that
$(\mathbf{x},\mathbf{y})=(\mathbf{a},\mathbf{b})$. We are going to show that
with uniformly positive probability, if we resample between $\mathbf{a}$ and
$\mathbf{b}$, the resampled path captures a linear excess of area, an event
which is exponentially unlikely by Proposition 2.4.

Figure 4: Illustration of the proof of Proposition 3.1. The original is path
depicted in black. With large probability, the new resampled blue path
captures an amount of area sufficient to compensate the area that could have
been lost (in turquoise) due to the fact that $\mathbf{a}$ and $\mathbf{b}$ do
not exactly coincide with the endpoints of the red facet. The event
$\mathsf{SD}$ ensures that $\mathbf{a}$ and $\mathbf{b}$ are not atypically
far from these two endpoints, allowing us to control the area of the turquoise
region.

As in the proof of Proposition 2.10, let us call
$\mathsf{Bad}_{\varepsilon}^{+,-}$ the event that
$\theta(\textbf{x},\textbf{y})\in[0,\varepsilon)\cup(\pi/2-\varepsilon,\pi/2]$
for the $\varepsilon>0$ given by Proposition 2.5. Write
$\mathcal{E}_{k}:=\mathsf{GoodHit}_{k}\cap(\mathsf{Bad}_{\varepsilon}^{+,-})^{c}\cap\\{\Gamma\subset
B_{K_{1}N}\setminus B_{K_{2}N}\\}$. Resample $\Gamma$ between $\mathbf{x}$ and
$\mathbf{y}$ according to the uniform law among paths
$\gamma\in\Lambda^{\mathbf{x}\rightarrow\mathbf{y}}$ such that
$(\Gamma\setminus\Gamma_{\mathbf{x},\mathbf{y}})\cup\gamma$ satisfies the area
condition and call $\tilde{\Gamma}$ the resampling. Note that $\tilde{\Gamma}$
has law $\mathbb{P}_{\lambda}^{N^{2}}$. In the proof of Proposition 2.10,
since we were resampling along the mean facet, it was clear that replacing the
path below the facet by any path having a positive area (above the facet) kept
the total area condition. However, in our case, the approximate “mean facet
resampling” comes with a potential area loss. This possibility is ruled out
for sufficiently small values of $\delta$. For $\eta>0$, introduce the (small
deviation) event

$\mathsf{SD}(\eta):=\big{\\{}\max(\mathsf{LR}(\mathbf{a}),\mathsf{LR}(\mathbf{b}))<\eta(\delta
s(k+1))^{1/2}N^{1/3}\big{\\}}.$ (3.10)

######  Claim 3.6.

There exist $\delta_{0}>0$ sufficiently small and
$c_{0}=c_{0}(\delta_{0},\eta)>0$ such that the following holds: if
$((\mathbf{x},\mathbf{y}),\Gamma)\in\mathsf{BigMeanFL}_{k}(s)\cap\mathcal{E}_{k}\cap\mathsf{SD}(\eta)$,
then, if we replace $\Gamma_{\mathbf{x},\mathbf{y}}$ by a path
$\gamma\in\mathsf{GAC}\left(\mathbf{x},\mathbf{y},\eta\right)$, the excess
area of the resulting path is at least $c_{0}(sk)^{3/2}N$. In particular, the
new path lies in $\Lambda^{N^{2}}$.

###### Proof of Claim 3.6.

Notice that by definition $(\mathbf{x},\mathbf{y})=(\mathbf{a},\mathbf{b})$
are located below the mean facet. Moreover, these points are at distance at
most $\delta(k+1)tN^{2/3}$ from the endpoints of the mean facet. Changing
$\Gamma$ between $\mathbf{a}$ and $\mathbf{b}$ by an element of
$\mathsf{GAC}\left(\mathbf{a},\mathbf{b},\eta\right)$ gives a path
$\tilde{\Gamma}$ which gained an area at least
$\eta\|\mathbf{a}-\mathbf{b}\|^{3/2}-\mathcal{A}(\mathbf{a},\mathbf{b},\Gamma)$
where $\mathcal{A}(\mathbf{a},\mathbf{b},\Gamma)$ is the area of the
quadrilateral delimited by the extremities of the mean facet of $\Gamma$ and
the pair $(\mathbf{a},\mathbf{b})$, see Figure 4. Given the hypothesis on
$\Gamma$, we find that

$\mathcal{A}(\mathbf{a},\mathbf{b},\Gamma)\leq\eta\sqrt{\delta}(s(k+1))^{3/2}N.$
(3.11)

Hence, for $\delta=\delta_{0}>0$ sufficiently small, there exists
$c_{0}=c_{0}(\delta_{0},\eta)>0$ such that (recall that $k\geq 3$)

$\mathsf{ExcessArea}(\tilde{\Gamma})\geq\eta
s^{3/2}(k-2)^{3/2}N-\eta\sqrt{\delta_{0}}(s(k+1))^{3/2}N\geq
c_{0}(sk)^{3/2}N.$ (3.12)

∎

We now fix $\delta=\delta_{0}$ and let $\eta>0$ to be fixed later. Using the
same reasoning as in (2.40) we get that for $\|\mathbf{a}-\mathbf{b}\|\geq
N_{0}$,

$\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\tilde{\Gamma}\in\mathsf{GAC}\left(\mathbf{x},\mathbf{y},\eta\right)\text{
}|\text{
}((\mathbf{x},\mathbf{y}),\Gamma)\in\mathsf{BigMeanFL}_{k}(s)\cap\mathcal{E}_{k}\cap\mathsf{SD}(\eta)]\\\
\geq\mathbb{P}_{\mathbf{a},\mathbf{b}}[\mathsf{GAC}\left(\mathbf{a},\mathbf{b},\eta\right)]\geq
c_{1},$

where $c_{1},N_{0}>0$ are given by Lemma 3.4. Since
$\|\mathbf{a}-\mathbf{b}\|\geq s(k-2)N\geq sN$, the above bound always hold as
soon as $s\geq N_{0}/N$. Assume that $s\geq N_{0}$. By the claim above, one
has

$\big{\\{}\tilde{\Gamma}\in\mathsf{GAC}\left(\mathbf{x},\mathbf{y},\eta\right),\>\Gamma\in\mathsf{BigMeanFL}_{k}(s)\cap\mathcal{E}_{k}\cap\mathsf{SD}(\eta)\big{\\}}\subset\big{\\{}\mathsf{ExcessArea}(\tilde{\Gamma})\geq
c_{0}(tk)^{3/2}N\big{\\}}.$ (3.13)

Putting all the pieces together, and using the fact that $\tilde{\Gamma}$ has
law $\mathbb{P}_{\lambda}^{N^{2}}$,

$\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}_{k}(s)\cap\mathcal{E}_{k}\cap\mathsf{SD}(\eta)]\leq
c_{1}^{-1}\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{ExcessArea}(\Gamma)\geq
c_{0}(tk)^{3/2}N].$ (3.14)

We now argue that $\mathsf{SD}(\eta)$ occurs with uniform non-zero probability
under $\mathbb{P}_{\lambda}^{N^{2}}$.

######  Claim 3.7.

There exist $\eta>0$ and $c_{2}=c_{2}(\eta)>0$ such that

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{SD}(\eta)\>|\>\mathsf{BigMeanFL}_{k}(s)]\geq
c_{2}.$ (3.15)

###### Proof of Claim 3.7..

The claim is a consequence of the coupling obtained in Proposition 2.6.
Indeed, $\mathbf{a}$ and $\mathbf{b}$ belong to the excursion of $\Gamma$
under the mean facet. Thus, conditioning on the extremities of the mean facet,
which we call $\mathbf{a}_{\mathsf{fac}}$ and $\mathbf{b}_{\mathsf{fac}}$, the
local roughness of $\mathbf{a}$ (resp. $\mathbf{b}$) is dominated by the
height of a negative random walk excursion between $\mathbf{a}_{\mathsf{fac}}$
and $\mathbf{b}_{\mathsf{fac}}$ after $\mathrm{d}_{\mathbf{a}}$ steps, where
$\mathrm{d}_{\mathbf{a}}$ is the number of steps from
$\mathbf{a}_{\mathsf{fac}}$ to $\mathbf{a}$ (resp $\mathrm{d}_{\mathbf{b}}$
steps, where $\mathrm{d}_{\mathbf{b}}$ is the number of steps from
$\mathbf{b}$ to $\mathbf{b}_{\mathsf{fac}}$) which is of order
$\sqrt{\mathrm{d}_{\mathbf{a}}}$ (resp. $\sqrt{\mathrm{d}_{\mathbf{b}}}$).
Since by choice of $\mathbf{a}$ and $\mathbf{b}$ we can deterministically
bound $\mathrm{d}_{\mathbf{a}}$ and $\mathrm{d}_{\mathbf{b}}$ by $O(\delta
skN^{2/3})$, we get the claim choosing $\eta$ large enough. ∎

We are now able to conclude. Let $\eta>0$ be given by the claim above. Notice
that

$\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{Goodhit}_{k}~{}\Big{|}~{}\mathsf{BigMeanFL}_{k}(s)\cap\mathsf{SD}(\eta)]\geq\left(\tfrac{\delta}{2}\right)^{2}.$
(3.16)

Write

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}_{k}(s)]\leq\left(\tfrac{2}{\delta}\right)^{2}\cdot
c_{2}^{-1}\cdot\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}_{k}(s)\cap\mathcal{E}_{k}\cap\mathsf{SD}(\eta)]\\\
+\left(\tfrac{2}{\delta}\right)^{2}\cdot\mathbb{P}\otimes\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}_{k}(s)\cap\mathsf{GoodHit}_{k}\cap(\mathsf{Bad}_{\varepsilon}^{+,-})\cap\\{\Gamma\subset
B_{K_{1}N}\setminus B_{K_{2}N}\\}]\\\
+\mathbb{P}_{\lambda}^{N^{2}}\left[\mathsf{BigMeanFL}_{k}(s)\cap\\{\Gamma\subset
B_{K_{1}N}\setminus B_{K_{2}N}\\}^{c}\right].$ (3.17)

To bound the first term we use (3.14) together with Proposition 2.4. To bound
the second term we use Proposition 2.5 and proceed as in the proof of
Proposition 2.10. Finally, the last term is bounded using Lemma 2.2. Putting
all the pieces together, we get the existence of $c,C>0$ such that for $3\leq
k\leq k_{N}$ (recall that $s=t/3$),

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}_{k}(s)]\leq
Ce^{-c(tk)^{3/2}}.$ (3.18)

As a consequence, setting $\tilde{c}=3N_{0}$, there exists some constants
$c^{\prime},C^{\prime}>0$ such that, for all $\tilde{c}\leq t\leq
N^{\epsilon}$,

$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{MeanFL}(\Gamma)\geq
tN^{\frac{2}{3}}]$ $\displaystyle\leq$
$\displaystyle\sum_{k=3}^{k_{N}}\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{BigMeanFL}_{k}(t/3)]+\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanFL}(\Gamma)\geq(t/3)k_{N}N^{\frac{2}{3}}\big{]}$
$\displaystyle\leq$ $\displaystyle C^{\prime}e^{-c^{\prime}t^{3/2}}.$

∎

####  3.1.2 Proof of Proposition 3.2

From Proposition 3.1, it is easy to deduce the bound of Proposition 3.2 using
the coupling introduced in Subsection 2.3.

###### Proof of Proposition 3.2.

Let $t>0$ and $\delta>0$ be a (small) constant to be fixed later . We split
the event $\\{\mathsf{MeanLR}>tN^{1/3}\\}$ according to the value of
$\mathsf{MeanFL}$,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanLR}(\Gamma)>tN^{\frac{1}{3}}\big{]}=\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanLR}(\Gamma)>tN^{\frac{1}{3}},\mathsf{MeanFL}(\Gamma)\leq
t^{2-\delta}N^{\frac{2}{3}}\big{]}\\\
+\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanLR}(\Gamma)>tN^{\frac{1}{3}},\mathsf{MeanFL}(\Gamma)\geq
t^{2-\delta}N^{\frac{2}{3}}\big{]}.$

By Proposition 3.1, we know that, for $\tilde{c}\leq t^{2-\delta}\leq
N^{2/3}$,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{MeanLR}(\Gamma)>tN^{\frac{1}{3}},\mathsf{MeanFL}(\Gamma)\geq
t^{2-\delta}N^{\frac{2}{3}}]\leq C\exp(-ct^{\frac{3}{2}(2-\delta)}).$ (3.19)

Now for $j\geq 0$, let $A_{j}$ be the following event:

$A_{j}:=\\{\mathsf{MeanLR}(\Gamma)>tN^{\frac{1}{3}},\mathsf{MeanFL}(\Gamma)=j\\}.$
(3.20)

We can now make use of Lemma 2.7 to argue that for every $j\leq
t^{2-\delta}N^{2/3}$,

$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}\left[A_{j}\right]$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}\|a-b\|=j\\\
\arg(a)>\frac{\pi}{4}\geq\arg(b)\end{subarray}}\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{MeanLR}(\Gamma)\geq
tN^{\frac{1}{3}}~{}|~{}\mathsf{Fac}_{a,b}]\mathbb{P}_{\lambda}^{N^{2}}\left[\mathsf{Fac}_{a,b}\right]$
$\displaystyle\leq$ $\displaystyle
C\exp\Big{(}-c\frac{t^{2}N^{2/3}}{j}\Big{)}\mathbb{P}_{\lambda}^{N^{2}}\left[\mathsf{MeanFL}(\Gamma)=j\right]$
$\displaystyle\leq$ $\displaystyle
C\exp(-ct^{\delta})\mathbb{P}_{\lambda}^{N^{2}}\left[\mathsf{MeanFL}(\Gamma)=j\right].$

The first equality comes from the fact that there exists exactly one pair
$a,b$ such that $\arg(a)>\frac{\pi}{4}\geq\arg(b)$ and $\mathsf{Fac}(a,b)$
occurs. Moreover, if $\mathsf{Fac}(a,b)$ occurs with such a choice of $a$ and
$b$, then $[a,b]$ will automatically be the mean facet, so Lemma 2.7 holds for
the second line. Thus, we obtained,

$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{MeanLR}(\Gamma)>tN^{\frac{1}{3}}]$
$\displaystyle\leq$ $\displaystyle
C\exp(-ct^{\frac{3}{2}(2-\delta)})+\sum_{j=0}^{t^{2-\delta}N^{2/3}}C\exp(-ct^{\delta})\mathbb{P}_{\lambda}^{N^{2}}\left[\mathsf{MeanFL}(\Gamma)=j\right]$
$\displaystyle\leq$ $\displaystyle
C\exp(-ct^{\frac{3}{2}(2-\delta)})+C\exp(-ct^{\delta}).$

Equating the exponents yields an optimal value of $\delta=\frac{6}{5}$ and the
result. ∎

###  3.2 Lower bounds

Recall the setting of Theorem 1.3. Our target estimates are the following.

######  Proposition 3.8.

There exists a function $F_{1}:\mathbb{R}^{+}\rightarrow[0,1]$ satisfying
$\lim_{t\rightarrow 0^{+}}F_{1}(t)=0$, such that for any $t>0$ small enough,

$\limsup_{N\rightarrow\infty}\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanLR}(\Gamma)<tN^{\frac{1}{3}}\big{]}\leq
F_{1}(t).$ (3.21)

######  Proposition 3.9.

There exists a function $F_{2}:\mathbb{R}^{+}\rightarrow[0,1]$ satisfying
$\lim_{t\rightarrow 0^{+}}F_{2}(t)=0$, such that for any $t>0$ small enough,

$\limsup_{N\rightarrow\infty}\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanFL}(\Gamma)<tN^{\frac{2}{3}}\big{]}\leq
F_{2}(t).$ (3.22)

######  Remark 3.10.

As before, we obtain

$\liminf_{N\rightarrow\infty}\mathbb{E}^{N^{2}}_{\lambda}\left[\frac{\mathsf{MeanFL}(\Gamma)}{N^{2/3}}\right]>0,\qquad\liminf_{N\rightarrow\infty}\mathbb{E}^{N^{2}}_{\lambda}\left[\frac{\mathsf{MeanLR}(\Gamma)}{N^{1/3}}\right]>0.$
(3.23)

Let us briefly describe the strategy of the proof. We carefully analyse the
marginal of $\mathbb{P}_{\lambda}^{N^{2}}$ in the cone $\mathbf{A}_{N}$ of
apex 0 and angular opening $N^{-1/3}$. Due to the Brownian Gibbs property,
conditionally on the outside of the cone, this marginal is the law of a random
walk bridge conditioned on capturing a random amount of area. We first prove
in Lemma 3.12 that this random area is of order $N$, with Gaussian tails.
Hence, the marginal of $\mathbb{P}_{\lambda}^{N^{2}}$ in $\mathbf{A}_{N}$ is
absolutely continuous with respect to the law of a random walk bridge without
the area conditioning, in the large $N$ limit. The strategy of proof is thus
clear: we prove the corresponding statements for the unconditioned random walk
bridge (see Lemmas 3.16 and 3.17), and transfer them to
$\mathbb{P}_{\lambda}^{N^{2}}$ thanks to the latter observation.

Let us introduce a few quantities that we will be useful in the remainder of
this section.

######  Definition 3.11.

For $N\geq 1$, we define $\theta_{N}:=N^{-1/3}$. We set $\mathbf{A}_{N}$ to be
the cone of apex 0 and angular opening of $\theta_{N}$ centered around the
line $\\{y=x\\}$. If $\gamma\in\Lambda^{N^{2}}$, let us define
$\mathbf{x}_{N}(\gamma)$ (resp. $\mathbf{y}_{N}(\gamma)$) to be point of
$\gamma$ that lies in $\mathbf{A}_{N}$ and that is the closest to the line
constituting the left (resp. right) boundary of $\mathbf{A}_{N}$. Finally, we
define

$\mathcal{A}_{N}(\gamma):=\left|\textup{Enclose}(\gamma_{\mathbf{x}_{N}(\gamma),\mathbf{y}_{N}(\gamma)}\cap\mathbf{A}_{\mathbf{x}_{N}(\gamma),\mathbf{y}_{N}(\gamma)})\right|-\left|\mathbf{T}_{0,\mathbf{x}_{N}(\gamma),\mathbf{y}_{N}(\gamma)}\right|.$
(3.24)

We shall also make a slight abuse of notation in considering
$\mathcal{A}_{N}(\widetilde{\gamma})$ for some oriented path
$\widetilde{\gamma}$ linking $\mathbf{x}_{N}(\gamma)$ to
$\mathbf{y}_{N}(\gamma)$ (for some $\gamma\in\Lambda^{N^{2}}$). In that case,
the meaning of $\mathcal{A}_{N}(\widetilde{\gamma})$ is clear: it is the
corresponding $\mathcal{A}_{N}(\gamma^{\prime})$ for any
$\gamma^{\prime}\in\Lambda^{N^{2}}$ which extends $\widetilde{\gamma}$ outside
the cone $\mathbf{A}_{\mathbf{x}_{N}(\gamma),\mathbf{y}_{N}(\gamma)}$.

The following lemma will play a central role in the proofs of Propositions 3.8
and 3.9.

######  Lemma 3.12.

There exist two constants $c,C>0$ such that for any $\beta>0$, and for $N$
large enough,

$\mathbb{P}_{\lambda}^{N^{2}}\left[\mathcal{A}_{N}(\Gamma)\geq\beta
N\right]\leq C\mathrm{e}^{-c\beta^{2}}.$ (3.25)

Figure 5: One possible output of the map $T_{x,y}$. The shift of $(x,y)$ is
denoted $(\tilde{x},\tilde{y})$. The initial path is the black path (after a
shift to the left). The two modifications involved in the procedure described
above are represented in red. The red path in the light green shaded region
has been sampled uniformly amongst the paths that capture a non-negative area
(computed with respect to the black dashed line). The area captured by the
turquoise region is sufficient to compensate the possible area loss due to the
resampling of the path between $x$ and $y$.

###### Proof.

The proof of this statement heavily relies on the probabilistic multi-valued
map principle stated in Lemma B.1. For $\alpha>0$, we introduce

$\mathsf{BadArea}^{N}_{\alpha}:=\big{\\{}\gamma\in\Lambda^{N^{2}},\>\mathcal{A}_{N}(\gamma)\in[\alpha
N,(\alpha+1)N]\big{\\}}.$ (3.26)

Define for $x,y\in\mathbb{N}^{2}$,

$A_{x,y}:=\mathsf{BadArea}^{N}_{\alpha}\cap\big{\\{}\Gamma\subset
B_{K_{1}N}\setminus
B_{K_{2}N}\big{\\}}\cap\big{\\{}\mathbf{x}_{N}(\Gamma)=x,\>\mathbf{y}_{N}(\Gamma)=y\big{\\}}\\\
\cap\big{\\{}\Gamma\notin\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N}(\Gamma),\mathbf{y}_{N}(\Gamma)}\big{\\}},$

where
$\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N}(\Gamma),\mathbf{y}_{N}(\Gamma)}:=\mathsf{Bad}^{+}_{\varepsilon,\mathbf{A}_{\mathbf{x}_{N}(\Gamma),\mathbf{y}_{N}(\Gamma)}}\cup\mathsf{Bad}^{-}_{\varepsilon,\mathbf{A}_{\mathbf{x}_{N}(\Gamma),\mathbf{y}_{N}(\Gamma)}}$,
and $\varepsilon$ is given by Proposition 2.5.

Fix $x,y\in\mathbb{N}^{2}$ such that
$\mathbb{P}_{\lambda}^{N^{2}}[A_{x,y}]\neq 0$.

Let us define a multi-valued map
$T_{x,y}:A_{x,y}\rightarrow\mathcal{P}(\Lambda^{N^{2}})$ via the following
two-step procedure (see Figure 5 for an illustration of the procedure). From
some path $\gamma\in\mathsf{BadArea}^{N}_{\alpha}$, we can create a new
oriented path of the first quadrant by erasing the portion of $\gamma$ lying
between $x$ and $y$ (previously called $\gamma_{x,y}$) and replacing it by any
oriented path $\widetilde{\gamma}_{x,y}$ linking $x$ to $y$ thus obtaining an
intermediate path $\widetilde{\gamma}$ which we require to satisfy
$\mathcal{A}_{N}(\widetilde{\gamma})\geq 0$, and finally by adding
$\lceil\frac{\alpha+1}{K_{2}}\rceil$ horizontal steps at the beginning of
$\widetilde{\gamma}$ ($K_{2}$ is the constant defined in 2.2). We define
$T_{x,y}(\gamma)$ to be the subset of $\Lambda$ of the paths that can be
obtained from $\gamma$ by this procedure. Notice that $T$ is well defined
since if $\gamma\in A_{x,y}$, the intermediate path $\widetilde{\gamma}$
defined above satisfies $\mathcal{A}(\widetilde{\gamma})\geq
N^{2}-(\alpha+1)N$, and it is easy to check that the first
$\lceil\frac{\alpha+1}{K_{2}}\rceil$ horizontal steps that have been added to
$\widetilde{\gamma}$ capture an area at least equal to $(\alpha+1)N$ (this
comes from the fact that $\gamma\in\\{\Gamma\subset B_{K_{1}N}\setminus
B_{K_{2}N}\\}$). Hence, for any $\gamma\in A_{x,y}$,
$T_{x,y}(\gamma)\subset\Lambda^{N^{2}}$.

We are going to apply Lemma B.1 to $T_{x,y}$. Before doing so, let us notice
that we may co-restrict the map defined above to the set

$B_{x,y}:=\\{\mathbf{x}_{N}(\Gamma^{\textup{cut}})=x,\mathbf{y}_{N}(\Gamma^{\textup{cut}})=y\\}\cap\\{\textup{The
first }\lceil\tfrac{\alpha+1}{K_{2}}\rceil\textup{ steps are
horizontal}\\}\subset\Lambda^{N^{2}},$ (3.27)

where $\Gamma^{\textup{cut}}$ is the path $\Gamma$ minus the first
$\lceil\frac{\alpha+1}{K_{2}}\rceil$ steps, translated of
$\lceil\frac{\alpha+1}{K_{2}}\rceil$ towards the left direction, see Figure 5.
Now, applying Lemma B.1 yields

$\mathbb{P}_{\lambda}^{N^{2}}[A_{x,y}]\leq\varphi(T_{x,y})\psi(T_{x,y})\mathbb{P}_{\lambda}^{N^{2}}[B_{x,y}],$
(3.28)

where the quantities $\varphi(T_{x,y})$ and $\psi(T_{x,y})$ are defined in
Lemma B.1. It is very easy to check that

$\varphi(T_{x,y})=\left(\frac{1}{\lambda}\right)^{\lceil\frac{\alpha+1}{K_{2}}\rceil},\qquad\psi(T_{x,y})=\frac{\left|\\{\gamma\in\Lambda^{x\rightarrow
y},\mathcal{A}_{N}(\gamma)\in[\alpha
N,(\alpha+1)N]\\}\right|}{\left|\\{\gamma\in\Lambda^{x\rightarrow
y},\mathcal{A}_{N}(\gamma)\geq 0\\}\right|}.$ (3.29)

Recall that $\mathbb{P}_{x,y}$ is the uniform measure on
$\Lambda^{x\rightarrow y}$. We can write

$\psi(T_{x,y})=\mathbb{P}_{x,y}\big{[}\mathcal{A}_{N}(\gamma)\in[\alpha
N,(\alpha+1)N]~{}\big{|}~{}\mathcal{A}_{N}(\gamma)\geq 0\big{]}.$ (3.30)

Let us call $\theta(x,y)$ the positive angle formed by the segment $[x,y]$ and
the horizontal line passing through $x$. Since we assumed that
$\mathbb{P}_{\lambda}^{N^{2}}[A_{x,y}]\neq 0$, we have
$\theta(x,y)\in[\varepsilon,\pi/2-\varepsilon]$. Let $h_{N}:=\|y-x\|$. As it
turns out, $h_{N}$ is of order $N^{2/3}$.

######  Claim 3.13.

There exist two constants $c=c(\varepsilon),C=C(\varepsilon)>0$ such that

$h_{N}\in[cN^{2/3},CN^{2/3}].$ (3.31)

###### Proof of Claim 3.13.

The constraint $\Gamma\subset B_{K_{1}N}\setminus B_{K_{2}N}$ enforces that
for $N$ large enough
$h_{N}\geq\frac{1}{2}K_{2}N\theta_{N}=\frac{1}{2}K_{2}N^{2/3}$. Then, since we
restricted ourselves to the case
$\varepsilon\leq\theta(x,y)\leq\pi/2-\varepsilon$, there exists $C_{1}>0$ such
that

$h_{N}\leq\frac{C_{1}}{\sin\varepsilon}N\theta_{N}=\frac{C_{1}}{\sin\varepsilon}N^{2/3}.$
(3.32)

∎

By translation invariance, $\mathbb{P}_{x,y}$ only depends on
$\theta=\theta(x,y)$ and $h_{N}$. Since these parameters are more relevant we
write $\mathbb{P}_{h_{N}}^{\theta}:=\mathbb{P}_{x,y}$. Moreover, with this
observation and to the cost of rotating the picture by $\pi/4$, we can always
assume that samples of $\mathbb{P}^{\theta}_{h_{N}}$ start at $0$ and end at
$e^{i\pi/4}(y-x)$, see Figure 6. We will denote by $X^{(\theta)}$ a sample of
$\mathbb{P}^{\theta}_{h_{N}}$.

Figure 6: An illustration of a sample $X^{(\vartheta)}$ of
$\mathbb{P}^{\vartheta}_{h_{N}}$ (in blue). This path corresponds to a
symmetric random walk that does $\pm 2^{-1/2}$ jumps at times $k2^{-1/2}$ for
$1\leq k\leq\sqrt{2}h_{N}\cos(\vartheta-\pi/4)$.

Now,

$\displaystyle\mathbb{P}^{\theta}_{h_{N}}\big{[}\mathcal{A}_{N}(X^{(\theta)})\in[\alpha
N,(\alpha+1)N]\big{]}$ $\displaystyle=$
$\displaystyle\mathbb{P}^{\theta}_{h_{N}}\big{[}h_{N}^{-3/2}\mathcal{A}_{N}(X^{(\theta)})\in[\alpha
Nh_{N}^{-3/2},(\alpha+1)Nh_{N}^{-3/2}]\big{]}$ $\displaystyle\leq$
$\displaystyle\mathbb{P}^{\theta}_{h_{N}}\big{[}h_{N}^{-3/2}\mathcal{A}_{N}(X^{(\theta)})\in[C^{-3/2}\alpha,c^{-3/2}(\alpha+1)]\big{]}$
$\displaystyle\leq$
$\displaystyle\sup_{\vartheta\in[\varepsilon,\frac{\pi}{2}-\varepsilon]}\mathbb{P}^{\vartheta}_{h_{N}}\big{[}h_{N}^{-3/2}\mathcal{A}_{N}(X^{(\vartheta)})\in[C^{-3/2}\alpha,c^{-3/2}(\alpha+1)]\big{]},$

where the second inequality follows by Claim 3.13.

Classical invariance theorems for random walks (started at $0$) conditioned to
reach $\alpha k$ after $k$ steps (see [Lig68] or [CC13, Remark 2.6]) yield the
following convergence in distribution999The convergence holds in the space
$\mathcal{C}([0,1],\mathbb{R})$ of continuous functions on $[0,1]$ equipped
with the topology of uniform convergence.: calling
$(X_{th_{N}}^{(\pi/4)})_{0\leq t\leq 1}$ the trajectory101010The sample is a
priori only defined for the discrete times $k/\sqrt{2}$ for $0\leq
k\leq\sqrt{2}h_{N}$ but we extend it to $[0,h_{N}]$ by linear interpolation.
of a sample of $\mathbb{P}^{\pi/4}_{h_{N}}$, one has that under
$\mathbb{P}^{\pi/4}_{h_{N}}$

$\lim_{N\rightarrow\infty}\left(\frac{X_{th_{N}}^{(\pi/4)}}{\sigma\sqrt{2^{1/2}h_{N}}}\right)_{t\in[0,1]}=(\mathsf{BB}_{t})_{t\in[0,1]},$
(3.33)

where $(\mathsf{BB}_{t})_{t\in[0,1]}$ is the standard Brownian bridge on
$[0,1]$, and $\sigma=2^{-1/2}$. This observation, together with the strategy
described in [CC13, Remark 2.6], yields a similar invariance principle for the
trajectory $(X_{th_{N}\cos(\vartheta-\pi/4)}^{(\vartheta)})_{0\leq t\leq 1}$
of a sample of $\mathbb{P}^{\vartheta}_{h_{N}}$. For $\vartheta\in(0,\pi/2)$
and $t\in[0,1]$, denote

$f^{(\vartheta)}_{N}(t):=\frac{1}{\sigma(\vartheta)\sqrt{2^{1/2}h_{N}\cos(\vartheta-\pi/4)}}(X_{th_{N}\cos(\vartheta-\pi/4)}^{(\vartheta)}-th_{N}\sin(\vartheta-\pi/4)),$
(3.34)

where $\sigma(\vartheta):=2^{-1/2}\sqrt{1-\tan^{2}(\vartheta-\pi/4)}$. Then,
for $\vartheta\in(0,\pi/2)$, under $\mathbb{P}^{\vartheta}_{h_{N}}$,

$\lim_{N\rightarrow\infty}\left(f_{N}^{(\vartheta)}(t)\right)_{t\in[0,1]}=(\mathsf{BB}_{t})_{t\in[0,1]},$
(3.35)

where the convergence holds in distribution. In particular, one has that

$\displaystyle h_{N}^{-3/2}\mathcal{A}_{N}(X^{(\vartheta)})$
$\displaystyle=h_{N}^{-3/2}\int_{0}^{h_{N}\cos(\vartheta-\pi/4)}(X^{(\vartheta)}_{t}-\tan(\vartheta-\pi/4)t)\mathrm{d}t$
$\displaystyle=2^{1/4}\sigma(\vartheta)[\cos(\vartheta-\pi/4)]^{3/2}\int_{0}^{1}f_{N}^{(\vartheta)}(t)\mathrm{d}t.$

Hence, (3.35) implies the following convergence in distribution under
$\mathbb{P}^{\vartheta}_{h_{N}}$:

$\lim_{N\rightarrow\infty}h_{N}^{-3/2}\mathcal{A}_{N}(X^{(\vartheta)})=c_{0}(\vartheta)\int_{0}^{1}\mathsf{BB}_{t}\mathrm{d}t,$
(3.36)

where
$c_{0}(\vartheta):=2^{1/4}\sigma(\vartheta)[\cos(\vartheta-\pi/4)]^{3/2}$.

######  Remark 3.14.

It follows from [CC13, Remark 2.6] that the latter convergence is uniform (for
instance at the level of the convergence of cumulative distribution function)
in $\vartheta\in[\varepsilon,\pi/2-\varepsilon]$. Indeed, when the underlying
random walk has increments with exponential moments, the authors identify the
law of the random walk bridge conditioned to stay above a line of slope
$\vartheta$ with a unconditioned random walk whose increments are given by a
suitable exponential tilt of the increments of the former walk. Hence the
Radon-Nikodym derivative of the $\vartheta$-tilted walk with respect to the
$\pi/4$-tilted walk is explicit and is a continuous function of $\vartheta$.
In particular, it is uniformly bounded on a compact interval such as
$[\varepsilon,\pi/2-\varepsilon]$. We shall use this observation several times
in what follows.

Call $\mathsf{P}$ the law of the Brownian bridge
$(\mathsf{BB}_{t})_{t\in[0,1]}$. The above observations yield that for $N$
large enough,

$\mathbb{P}^{\theta}_{h_{N}}\big{[}\mathcal{A}_{N}(X^{\theta})\in[\alpha
N,(\alpha+1)N]\big{]}\leq
2\mathsf{P}\Big{[}\int_{0}^{1}\mathsf{BB}_{t}\mathrm{d}t\geq
c_{0}(\varepsilon)^{-1}C^{-3/2}\alpha\Big{]}.$ (3.37)

A similar reasoning yields the lower bound, for $N$ large enough,

$\mathbb{P}^{\theta}_{h_{N}}[\mathcal{A}_{N}(X^{\theta})\geq
0]\geq\frac{1}{2}\mathsf{P}\left[\int_{0}^{1}\mathsf{BB}_{t}\mathrm{d}t\geq
0\right]=:\eta.$ (3.38)

Gathering (3.37) and (3.38), we get

$\psi(T_{x,y})\leq
2\eta^{-1}\mathsf{P}\Big{[}\int_{0}^{1}\mathsf{BB}_{t}\mathrm{d}t\geq
c_{0}(\varepsilon)^{-1}C^{-3/2}\alpha\Big{]}.$ (3.39)

Using the following estimate on the tail area of a Brownian bridge (see [LJ07,
Theorem 1.2]), we obtain $c_{1}=c_{1}(\varepsilon),C_{1}=C_{1}(\varepsilon)>0$
such that

$\mathsf{P}\Big{[}\int_{0}^{1}\mathsf{BB}_{t}\mathrm{d}t\geq
c_{0}(\varepsilon)^{-1}C^{-3/2}\alpha\Big{]}\leq
C_{1}\mathrm{e}^{-c_{1}\alpha^{2}}.$ (3.40)

Using (3.28), (3.38), and (3.40), we get that for $N$ large enough,

$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}\left[A_{x,y}\right]$
$\displaystyle\leq$ $\displaystyle
2C_{1}\eta^{-1}\mathrm{e}^{-c_{1}\alpha^{2}}\lambda^{-\lceil\frac{\alpha+1}{K_{2}}\rceil}\mathbb{P}_{\lambda}^{N^{2}}[B_{x,y}]$
$\displaystyle\leq$ $\displaystyle
C_{2}\mathrm{e}^{-c_{2}\alpha^{2}}\mathbb{P}_{\lambda}^{N^{2}}[B_{x,y}].$

To obtain the lemma, it remains to sum over $\alpha\geq\beta$ and over all
possible values for $x$ and $y$,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathcal{A}_{N}(\Gamma)\geq\beta
N,\>\Gamma\subset B_{K_{1}N}\setminus
B_{K_{2}N},\>\Gamma\notin\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N}(\Gamma),\mathbf{y}_{N}(\Gamma)}]\leq
C_{2}\sum_{\alpha=\lfloor\beta\rfloor}^{+\infty}\exp(-c_{2}\alpha^{2}).$
(3.41)

Then, for some $C_{3}>0$,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathcal{A}_{N}(\Gamma)\geq\beta
N,\>\Gamma\subset B_{K_{1}N}\setminus
B_{K_{2}N},\>\Gamma\notin\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N}(\Gamma),\mathbf{y}_{N}(\Gamma)}]\leq
C_{3}\mathrm{e}^{-c_{2}\beta^{2}}.$ (3.42)

Using Lemma 2.2 and Proposition 2.5, there exists $C_{4}>0$ such that for $N$
large enough (namely $N\geq\beta^{3}$),

$\mathbb{P}_{\lambda}^{N^{2}}\left[\mathcal{A}_{N}(\Gamma)\geq\beta
N\right]\leq C_{4}\mathrm{e}^{-c_{2}\beta^{2}}.$ (3.43)

∎

######  Definition 3.15.

If $\gamma_{u,v}\in\Lambda^{u\rightarrow v}$ and $z\in\gamma_{u,v}$, we define
$\mathsf{LR}(z)(\gamma_{u,v})$, the local roughness of the vertex
$z\in\gamma_{u,v}$, to be the local roughness of $z$ in the path
$\gamma_{u,v}$ extended to the left with only horizontal steps to up to the
$y$-axis and to the right with only vertical steps up to the $x$-axis. We
define the mean facet length of $\gamma_{u,v}$ similarly.

The two next lemmas study the typical behaviours of the mean facet length and
of the local roughness of a uniform path between $\mathbf{x}_{N}(\Gamma)$ and
$\mathbf{y}_{N}(\Gamma)$ when $\Gamma$ is sampled according to
$\mathbb{P}_{\lambda}^{N^{2}}$. For simplicity, in the rest of the section,
when $\Gamma$ is a sample of $\mathbb{P}_{\lambda}^{N^{2}}$, we shall
abbreviate $\mathbf{x}_{N}(\Gamma)$ (resp. $\mathbf{y}_{N}(\Gamma)$) by
$\mathbf{x}_{N}$ (resp. $\mathbf{y}_{N}$). The proofs of these lemmas rely on
Gaussian computations and are not specific to the considered model. They are
deferred to the end of the section.

We will write $\mathcal{G}_{N}:=\\{\Gamma\in B_{K_{1}N}\setminus
B_{K_{2}N}\\}\cap(\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N},\mathbf{y}_{N}})^{c}$.

######  Lemma 3.16.

There exists a function $\Psi$ independent of every parameter of the problem
with $\Psi(t)\xrightarrow[t\rightarrow 0^{+}]{}{0}$ such that, for $t>0$, if
$N$ is large enough,

$\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}[\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}]\big{]}\leq\Psi(t).$
(3.44)

The corresponding statement for the mean facet length is the following

######  Lemma 3.17.

There exists a function $\Phi$ independent of every parameter of the problem
with $\Phi(t)\xrightarrow[t\rightarrow 0^{+}]{}{0}$ such that, for $t>0$, if
$N$ is large enough,

$\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}[\mathsf{MeanFL}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{2/3}]\big{]}\leq\Phi(t).$
(3.45)

These two lemmas in hand, we can now prove the lower bound of Proposition 3.8.
Proposition 3.9 will be proved by the same method.

###### Proof of Proposition 3.8.

Fix some $t>0$, and some $\beta>0$ that is going to be chosen later as a
function of $t$. We implement the following resampling procedure: let us start
with a sample $\Gamma$ of $\mathbb{P}_{\lambda}^{N^{2}}$. As usual we
condition on the configuration outside of the cone
$\mathbf{A}_{\mathbf{x}_{N},\mathbf{y}_{N}}$ that we name
$\Gamma_{\mathsf{ext}}:=\Gamma\setminus\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}}$.
By the Brownian Gibbs property, the distribution of
$\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}}$ is uniform among the oriented paths
linking $\mathbf{x}_{N}$ and $\mathbf{y}_{N}$ that enclose at least a certain
amount of area which is measurable with respect to the exterior configuration.
Let us call $A_{\mathsf{ext}}$ the area already enclosed by
$\Gamma_{\mathsf{ext}}$ (namely,
$A_{\mathsf{ext}}=\textup{Enclose}(\Gamma\cap\mathbf{A}_{\mathbf{x}_{N},\mathbf{y}_{N}}^{c}\cap\mathbb{N}^{2})$),
so that the conditional distribution of
$\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}}$ is nothing but
$\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\>\cdot\>|\>\mathcal{A}_{N}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\geq
N^{2}-A_{\mathsf{ext}}\big{]}$.

Before starting, let us point out a crucial fact. With our definition of
$\mathsf{LR}(z)(\Gamma_{u,v})$, it is a deterministic fact that for any
$u,v\in\Gamma,z\in\Gamma_{u,v}$,

$\mathsf{LR}(z)\left(\Gamma_{u,v}\right)\leq\mathsf{LR}(z)\left(\Gamma\right)~{}~{}\text{
and thus
}~{}~{}\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\leq\mathsf{MeanLR}(\Gamma).$
(3.46)

Thus, we have

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MeanLR}(\Gamma)<tN^{1/3}\big{]}\\\
\leq\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}~{}|~{}\mathcal{A}_{N}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\geq
N^{2}-A_{\mathsf{ext}}\big{]}\big{]}.$

Introduce

$\mathcal{F}_{N}:=\\{N^{2}-A_{\mathsf{ext}}\leq\beta N\\}\cap\\{\Gamma\subset
B_{K_{1}N}\setminus
B_{K_{2}N}\\}\cap(\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N},\mathbf{y}_{N}})^{c},$
(3.47)

where $\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N},\mathbf{y}_{N}}$ was
introduced in the proof of Lemma 3.12. Note that by Lemma 2.2, Proposition 2.5
and Lemma 3.12, one has that for $N\geq\beta^{3}$,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathcal{F}_{N}^{c}]\leq Ce^{-c\beta^{2}}.$
(3.48)

Hence,

$\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}[\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}\>|\>\mathcal{A}_{N}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\geq
N^{2}-A_{\mathsf{ext}}\big{]}\big{]}\\\
\leq\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}\>|\>\mathcal{A}_{N}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\geq
N^{2}-A_{\mathsf{ext}}\big{]}\mathds{1}_{\mathcal{F}_{N}}\big{]}\\\
+Ce^{-c\beta^{2}}.$

Now, note that

$\mathds{1}_{\mathcal{F}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}\>|\>\mathcal{A}_{N}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\geq
N^{2}-A_{\mathsf{ext}}\big{]}\\\
\leq\mathds{1}_{\mathcal{F}_{N}}\frac{\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}[\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}]}{\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}[\mathcal{A}_{N}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\geq\beta
N]}.$

Using the methods developed above111111This is again an application of
Donsker’s Theorem and the use of the asymptotic results of [LJ07], we do not
write the full proof but rather refer to the proof of Lemmas 3.16 and 3.17 for
details. we could also show that for some $c^{\prime},C^{\prime}>0$, if $N$ is
large enough,

$\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\left[\mathcal{A}_{N}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\geq\beta
N\right]\geq C^{\prime}\mathrm{e}^{-c^{\prime}\beta^{2}}.$ (3.49)

By Lemma 3.16, we obtain that, integrating over $\Gamma_{\mathsf{ext}}$, and
putting everything together,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{MeanLR}(\Gamma)<tN^{1/3}]\leq
C\mathrm{e}^{-c\beta^{2}}+\frac{\Psi(t)}{C^{\prime}\mathrm{e}^{-c^{\prime}\beta^{2}}}.$
(3.50)

It remains to carefully chose $\beta$ in terms of $t$. We optimize the above
equation by setting

$\beta(t)=\sqrt{(c+c^{\prime})^{-1}\log(\tfrac{c}{c^{\prime}}\tfrac{CC^{\prime}}{\Psi(t)})}.$
(3.51)

Observe that $\beta(t)\xrightarrow[t\rightarrow 0]{}{+\infty}$. Choosing $N$
large enough, we obtain that for some $C^{\prime\prime}>0$,

$\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{MeanLR}(\Gamma)<tN^{1/3}]\leq
C^{\prime\prime}(\Psi(t))^{\frac{c}{c+c^{\prime}}}$ (3.52)

which is the announced result. ∎

Proposition 3.9 is a consequence of the exact same argument.

###### Proof of Proposition 3.9.

We proceed exactly as in the proof of Proposition 3.8. Indeed, observe that as
previously,

$\mathsf{MeanFL}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})\leq\mathsf{MeanFL}(\Gamma).$
(3.53)

Hence, the proof can be reproduced mutatis mutandi using this time Lemma 3.17.
∎

We now turn to the proofs of the two Gaussian lemmas, namely Lemma 3.16 and
Lemma 3.17. The process of the concave majorant of a Brownian bridge has been
extensively studied in [Gro83, Sui01, BP11]. Let us briefly summarise their
results.

  * •

Groeneboom showed in [Gro83] that conditionally on the convex majorant of a
Brownian motion, the difference process was a succession of independent
Brownian excursions between the extremal points of the concave majorant. The
result was later extended to the standard Brownian bridge in [BP11] via a Doob
tranformation.

  * •

Groeneboom also gave an explicit representation of the distribution of the
process of the slopes of the concave majorant. Later, Suidan [Sui01] was able
to derive the joint law of the ordered lengths of the segments of $[0,1]$ in
the partition of $[0,1]$ induced by the extremal points of the concave
majorant of a Brownian bridge.

We refer to the introduction of [BP11] for a clear and detailed presentation
of these results. In particular, we import the following statement from the
three cited articles.

######  Theorem 3.18.

Let $\left(\mathsf{BB}_{t}\right)_{0\leq t\leq 1}$ be the standard Brownian
bridge and let $\left(\mathcal{C}(t)\right)_{0\leq t\leq 1}$ be its smallest
concave majorant. Let $s\in(0,1)$. Let us define the two following random
variables:

  * •

$L(s)$ is the length of the almost surely unique facet of $\mathcal{C}$
intersecting the line $\\{x=s\\}$,

  * •

$R(s)$ is the difference process defined by $R(s):=\mathcal{C}(s)-B(s)$.

Fix $\epsilon>0$. Then, one has

$\sup_{s\in[\epsilon,1-\epsilon]}\mathbb{P}\left[L(s)<r\right]\xrightarrow[r\rightarrow
0^{+}]{}{0},$ (3.54)

and

$\sup_{s\in[\epsilon,1-\epsilon]}\mathbb{P}\left[R(s)<r\right]\xrightarrow[r\rightarrow
0^{+}]{}{0}.$ (3.55)

We briefly sketch the proof of Theorem 3.18 and refer to [Gro83, Sui01, BP11]
for more details.

###### Sketch of proof of Theorem 3.18.

Fix some $s\in[\epsilon,1-\epsilon]$. For $L(s)$, we remark that the results
of [Gro83] imply that $\mathcal{C}$ almost surely has a finite number of slope
changes (or extremal points) in $[\epsilon,1-\epsilon]$, and [BP11] gives an
explicit description of their lengths in terms of a size-biased uniform stick-
breaking process. In particular, it implies that the distribution of $L(s)$ is
absolutely continuous with respect to the Lebesgue measure on
$\mathbb{R}^{+}$, which implies that
$\mathbb{P}\left[L(s)<r\right]\xrightarrow[r\rightarrow 0]{}{0}$. The
statement follows by the observation that
$s\in(0,1)\mapsto\mathbb{P}\left[L(s)<r\right]$ is continuous.

For $R(s)$, let us denote by $a\leq s\leq b$ the horizontal coordinates of the
endpoints of the facet intersecting the line $\\{x=s\\}$. The exact same
argument as above implies that
$\mathbb{P}\left[\min(s-a,b-s)<\varepsilon\right]$ goes to 0 when
$\varepsilon$ goes to 0. On the complementary of the latter event, by [Gro83],
$R(s)$ is the height of a Brownian excursion at a positive distance of its
endpoints, and is indeed continuous with respect to the Lebesgue measure on
$\mathbb{R}^{+}$. We conclude as previously, observing that
$s\in(0,1)\mapsto\mathbb{P}\left[R(s)<r\right]$ is continuous. ∎

Now observe that in Lemmas 3.16 and 3.17, $\|\mathbf{y}_{N}-\mathbf{x}_{N}\|$
is of order $N^{2/3}$ so that the scaling is Gaussian. Hence, Donsker’s
Theorem suggests that we may use the above-mentioned results to obtain lower
tails for the random variables $N^{-1/3}\mathsf{MeanLR}$ and
$N^{-2/3}\mathsf{MeanFL}$.

###### Proof of Lemma 3.16.

As in the proof of Lemma 3.12, we write
$h_{N}=\|\mathbf{y}_{N}-\mathbf{x}_{N}\|$. Moreover, let us call
$\mathbf{t}_{N}^{0}$ the intersection between the line segment
$[\mathbf{x}_{N},\mathbf{y}_{N}]$ and the line $\\{y=x\\}$. Remember that we
work under the event $\\{\Gamma\in B_{K_{1}N}\setminus
B_{K_{2}N}\\}\cap(\mathsf{Bad}^{+,-}_{\varepsilon,\mathbf{x}_{N},\mathbf{y}_{N}})^{c}$,
which allows us to reuse Claim 3.13, namely

$cN^{2/3}\leq h_{N}\leq CN^{2/3}.$ (3.56)

Moreover, there exists a constant $c_{\varepsilon}>0$ such that

$\min(\|\mathbf{t}^{0}_{N}-\mathbf{x}_{N}\|_{2},\|\mathbf{y}_{N}-\mathbf{t}^{0}_{N}\|)\geq
c_{\varepsilon}h_{N}.$ (3.57)

Recall the parametrisation of the probability measures
$\mathbb{P}^{\theta}_{h_{N}}=\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}$
introduced in the proof of Lemma 3.12. Similarly as before,

$\displaystyle\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}\big{]}\big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}^{\theta}_{h_{N}}\big{[}h_{N}^{-1/2}\mathsf{LR}(\mathbf{t}_{N}^{0})<th_{N}^{-1/2}N^{1/3}\big{]}\big{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}^{\theta}_{h_{N}}\big{[}h_{N}^{-1/2}\mathsf{LR}(\mathbf{t}_{N}^{0})<c^{-1/2}t\big{]}\big{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\sup_{\vartheta\in[\varepsilon,\pi/2-\varepsilon]}\mathbb{P}^{\vartheta}_{h_{N}}[h_{N}^{-1/2}\mathsf{LR}(\mathbf{t}_{N}^{0})<c^{-1/2}t\big{]}\big{]},$

where $c$ is the constant appearing in (3.56). Now, observe that
$\mathbf{t}_{0}^{N}$ is independent of the random path sampled according to
$\mathbb{P}^{\vartheta}_{h_{N}}$, and we can write

$\mathbb{P}^{\vartheta}_{h_{N}}\big{[}h_{N}^{-1/2}\mathsf{LR}(\mathbf{t}^{0}_{N})<c^{-1/2}t\big{]}\leq\sup_{s\in[c_{\varepsilon},1-c_{\varepsilon}]}\mathbb{P}^{\vartheta}_{h_{N}}\big{[}h_{N}^{-1/2}\mathsf{LR}(h_{N}s)<c^{-1/2}t\big{]},$
(3.58)

Thus, we obtained that

$\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}\big{]}\big{]}\\\
\leq\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\sup_{\vartheta\in[\varepsilon,\pi/2-\varepsilon]}\sup_{s\in[c_{\varepsilon},1-c_{\varepsilon}]}\mathbb{P}^{\vartheta}_{h_{N}}\big{[}h_{N}^{-1/2}\mathsf{LR}(sh_{N})<c^{-1/2}t\big{]}\big{]},$
(3.59)

where $c_{\varepsilon}$ is the constant appearing in (3.57). Reasoning as
above, one has the following convergence in distribution (the random variable
$R$ was introduced in Theorem 3.18), under $\mathbb{P}^{\vartheta}_{h_{N}}$
and assuming $\mathcal{G}_{N}$ occurs, for
$s\in[c_{\varepsilon},1-c_{\varepsilon}]$,

$\lim_{N\rightarrow\infty}h_{N}^{-1/2}\mathsf{LR}(sh_{N})=\gamma(\vartheta)R(s),$
(3.60)

where $\gamma(\vartheta)=\sigma(\vartheta)\sqrt{2^{1/2}\cos(\vartheta-\pi/4)}$
and $\sigma(\vartheta)$ was defined below (3.34). Once again, we claim that
this convergence is uniform in $\vartheta\in[\varepsilon,\pi/2-\varepsilon]$
(see Remark 3.14). Hence, the limsup of the right-hand side of (3.59) is
bounded by

$\sup_{s\in[c_{\varepsilon},1-c_{\varepsilon}]}\mathsf{P}[R(s)<tc^{-1/2}\gamma(\varepsilon)^{-1}].$
(3.61)

We argue as in the proof of Lemma 3.12 to get for $N$ large enough

$\displaystyle\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanLR}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{1/3}\big{]}\big{]}$
$\displaystyle\leq$ $\displaystyle
C_{1}\sup_{s\in[c_{\varepsilon},1-c_{\varepsilon}]}\mathsf{P}\big{[}R(s)<tc^{-1/2}\gamma(\varepsilon)^{-1}\big{]}$
$\displaystyle=:$ $\displaystyle\Psi(t).$

The fact that $\Psi$ goes to 0 is a consequence of Theorem 3.18. ∎

We turn to the proof of Lemma 3.17.

###### Proof of Lemma 3.17.

The proof follows the exact same strategy as above. Indeed, keeping the
notations of the proof of Lemma 3.16, we may write

$\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanFL}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{2/3}\big{]}\big{]}\\\
\leq\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\sup_{\vartheta\in[\varepsilon,\pi/2-\varepsilon]}\mathbb{P}^{\vartheta}_{h_{N}}\big{[}h_{N}^{-1}\mathsf{MeanFL}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tc^{-1}\big{]}\big{]}.$
(3.62)

We are going to use the same method as in the proof of Lemma 3.16. We refer to
[Sui01, Theorems 4 and 5] to observe that the facet length scales as the time
$h_{N}$ in a random walk bridge. Some extra care is needed due to the fact
that the function
$\phi\in\mathcal{C}([0,1],\mathbb{R})\mapsto\mathsf{MeanFL}(\phi)\in\mathbb{R}^{+}$
is not continuous in $\phi$; however it is easy to check that its points of
discontinuity are included in the set of continuous functions having at least
three aligned local maxima. Using the fact that the set of such functions has
measure 0 under $\mathsf{P}$ and the above-mentioned uniformity in
$\vartheta$, we obtain from arguments similar to that of proof of Lemma 3.16
that there exists a constant $\delta(\varepsilon)>0$ such that, for $N$ large
enough,

$\displaystyle\mathbb{E}_{\lambda}^{N^{2}}\big{[}\mathds{1}_{\mathcal{G}_{N}}\mathbb{P}_{\mathbf{x}_{N},\mathbf{y}_{N}}\big{[}\mathsf{MeanFL}(\Gamma_{\mathbf{x}_{N},\mathbf{y}_{N}})<tN^{2/3}\big{]}\big{]}$
$\displaystyle\leq$ $\displaystyle
C_{1}^{\prime}\sup_{s\in[c_{\varepsilon},1-c_{\varepsilon}]}\mathsf{P}[L(s)<t\delta(\varepsilon)c^{-1}]$
$\displaystyle=:$ $\displaystyle\Phi\left(t\right).$

We conclude using Theorem 3.18. ∎

##  4 Analysis of the maximal facet length and of the maximal local roughness

This section is devoted to the proof of Theorem 1.5. The corresponding
statements in the context of the random-cluster have been obtained by Hammond
in [Ham12a, Ham12b] (see Section 5 for a detailed discussion). We will
essentially adapt the robust arguments developed by Hammond in these papers
although some new difficulties, coming from the oriented feature of the model,
will emerge.

The proof of the upper bound of (1.7) is in spirit the same as the one of
Proposition 2.10 observing that the proof can be reproduced (with care)
replacing the $N^{\epsilon}$ by $C(\log N)^{1/3}$ with a large $C$.

The proof of the lower bound is more technical and follows the lines of
[Ham12b]. We use a different strategy of resampling consisting in successively
resampling $\Gamma$ in deterministic angular sectors. We then show that each
one of these resampling has a sufficiently large probability of producing a
favourable output (meaning with a large local roughness) — and thus that the
final output has a large local roughness with very high probability. However,
we will encounter several technical issues, the main one being the fact that a
favourable ouptut could be destroyed by further resamplings. This technical
point will be solved by carefully analysing the Markovian dynamic induced by
the successive resamplings in Section 4.2. The bound on the maximal facet
length will be a simple byproduct of the bound on the maximal local roughness.

###  4.1 Upper bounds

We start by proving the upper bound on $\mathsf{MaxFL}$. The correct control
is given by the following proposition.

######  Proposition 4.1.

There exist constants $\tilde{c},c,C>0$ such that, for any $\tilde{c}\leq
t\leq N^{2/3}(\log N)^{-2/3}$,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MaxFL}(\Gamma)\geq
tN^{\frac{2}{3}}(\log N)^{\frac{1}{3}}\big{]}\leq Ce^{-ct^{3/2}\log N}.$ (4.1)

The proof of this proposition follows the same lines as the proof of
Proposition 2.10 with the exception that we modify the event
$\mathsf{GAC}\left(x,y,\eta\right)$ of Subsection 3.1 to require a poly-
logarithmic deviation of the area from its typical value.

######  Definition 4.2 (Logarithmic good area capture).

Let $\gamma\in\Lambda$ be a path, $\eta>0$ and $x,y\in\mathbb{N}^{2}$. We say
that $\gamma$ realizes the event $\mathsf{LogGAC}(x,y,\eta)$ (meaning
“logarithmic good area capture”) if

  1. $(i)$

$\gamma\cap\mathbf{A}_{x,y}\in\Lambda^{x\rightarrow y}$, or in words, $\gamma$
connects $x$ and $y$ by a oriented path,

  2. $(ii)$

$|\textup{Enclose}(\gamma\cap\mathbf{A}_{x,y})|-|\mathbf{T}_{0,x,y}|\geq\eta||x-y||^{\frac{3}{2}}\left(\log||x-y||\right)^{\frac{1}{2}}$.

As previously, we need to estimate the probability of
$\mathsf{LogGAC}(x,y,\eta)$ under $\mathbb{P}_{x,y}$, the uniform measure on
$\Lambda^{x\rightarrow y}$. The following result, whose proof is postponed to
the Appendix A, gives us a lower bound on
$\mathbb{P}_{x,y}[\mathsf{LogGAC}(x,y,\eta)]$. Recall that
$\theta(x,y)\in[0,\pi/2]$ the angle formed by the horizontal axis and the
segment joining $x$ and $y$.

######  Lemma 4.3.

Let $\varepsilon,\eta>0$. There exist $C=C(\varepsilon)>0$ and
$N_{0}=N_{0}(\varepsilon,\eta)>0$ such that for any $x,y\in\mathbb{N}^{2}$
satisfying $\Lambda^{x\rightarrow y}\neq\emptyset$,
$\varepsilon\leq\theta(x,y)\leq\pi/2-\varepsilon$ and $\|x-y\|\geq N_{0}$,

$\mathbb{P}_{x,y}\left[\mathsf{LogGAC}(x,y,\eta)\right]\geq\|x-y\|^{-C\eta^{2}}.$
(4.2)

Proposition 2.5 will ensure that the condition on $\theta(x,y)$ holds with
high probability.

###### Proof of Proposition 4.1.

We repeat the same strategy as in the proof of Proposition 2.10 considering
this time the event

$\mathsf{BigFacet}(t):=\big{\\{}\mathsf{MaxFL}(\Gamma)\geq
tN^{\frac{2}{3}}(\log N)^{\frac{1}{3}}\big{\\}}$ (4.3)

instead of $\mathsf{BigMeanFL}(t)$, $\mathsf{LogGAC}$ instead of
$\mathsf{GAC}$, and replacing $\mathsf{GoodHit}$ by the event that the random
pair $(\mathbf{x},\mathbf{y})$ hits the extremities of the largest facet. The
control on the probability of $\mathsf{LogGAC}$ is given by Lemma 4.3 above. ∎

With this result, it is now easy to estimate the correct scale for
$\mathsf{MaxLR}$. The statement is given in the following proposition.

######  Proposition 4.4.

There exist constants $\tilde{c},c,C>0$ such that, for any
$\displaystyle\tilde{c}\leq t\leq N^{5/6}(\log N)^{-5/6}$, then,

$\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{MaxLR}(\Gamma)\geq
tN^{\frac{1}{3}}(\log N)^{\frac{2}{3}}\big{]}\leq Ce^{-ct^{6/5}\log N}.$ (4.4)

###### Proof.

Denote by $\mathsf{MaxFac}(\Gamma)$ the facet where $\mathsf{MaxLR}(\Gamma)$
is attained, and by $\mathsf{MLRF}(\Gamma)$ its length. It is clear that

$\mathsf{MLRF}(\Gamma)\leq\mathsf{MaxFL}(\Gamma).$ (4.5)

With this remark and Proposition 4.1, we can conclude using the exact same
method as in the proof of Proposition 3.2, the only difference being that we
now use the coupling along the facet which has the largest local roughness.
More precisely, let $\delta>0$ to be fixed later. Let $t>0$, then

$\big{\\{}\mathsf{MaxLR}(\Gamma)\geq tN^{\frac{1}{3}}(\log
N)^{\frac{2}{3}}\big{\\}}\\\ \subset\big{\\{}\mathsf{MLRF}(\Gamma)\geq
t^{2-\delta}N^{\frac{2}{3}}(\log N)^{\frac{1}{3}}\big{\\}}\cup
A\cup\big{\\{}\Gamma\subset(B_{K_{1}N})^{c}\big{\\}}\cup\bigcup_{j=N^{\frac{1}{3}}}^{t^{2-\delta}N^{\frac{2}{3}}(\log
N)^{\frac{1}{3}}}A_{j},$ (4.6)

where

$A_{j}=\big{\\{}\mathsf{MaxLR}(\Gamma)\geq tN^{\frac{1}{3}}(\log
N)^{\frac{2}{3}},~{}\mathsf{MLRF}(\Gamma)=j,~{}\Gamma\subset
B_{K_{1}N}\big{\\}},$ (4.7) $A=\big{\\{}\mathsf{MaxLR}(\Gamma)\geq
tN^{\frac{1}{3}}(\log
N)^{\frac{2}{3}},~{}\mathsf{MLRF}(\Gamma)<N^{\frac{1}{3}}\big{\\}},$ (4.8)

and $K_{1}$ is defined as in Lemma 2.2. It is easy to check that $A=\emptyset$
for $t\geq 1$, which is now our assumption. Thanks to Lemma 2.2 and
Proposition 4.1, we already know how to control the probability of the first
three terms in the right-hand side of (4.6). The proof will follow from
obtaining an upper bound on $\mathbb{P}_{\lambda}^{N^{2}}[A_{j}]$. This bound
is (again) given by the Lemma 2.7 (see Remark 2.8).

Let us denote by $B(j,x)$ the event that $x$ belong to $\Gamma$ and the facet
above $x$ has length $j$. Then, for $N^{1/3}\leq j\leq
t^{2-\delta}N^{2/3}(\log N)^{1/3}$,

$\displaystyle\mathbb{P}_{\lambda}^{N^{2}}[A_{j}]$ $\displaystyle\leq$
$\displaystyle\sum_{x\in
B_{K_{1}N}}\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{LR}(x)\geq
tN^{\frac{1}{3}}(\log N)^{\frac{2}{3}},\>B(j,x)\big{]}$ $\displaystyle\leq$
$\displaystyle\sum_{x\in B_{K_{1}N}}\sum_{\begin{subarray}{c}a,b\in
B_{K_{1}N}\\\ \|b-a\|=j\\\
\arg(a)>\arg(x)\geq\arg(b)\end{subarray}}\mathbb{P}_{\lambda}^{N^{2}}\big{[}\mathsf{LR}(x)\geq
tN^{\frac{1}{3}}(\log
N)^{\frac{2}{3}}~{}\big{|}~{}\mathsf{Fac}(a,b)\big{]}\mathbb{P}_{\lambda}^{N^{2}}[\mathsf{Fac}(a,b)]$
$\displaystyle\leq$ $\displaystyle
O(N^{4}j)\exp\Big{(}-c\frac{t^{2}N^{2/3}(\log N)^{4/3}}{2j}\Big{)}$
$\displaystyle\leq$ $\displaystyle\mathrm{e}^{-ct^{\delta}\log N}$

where the third inequality is a consequence of Lemma 2.7 (and more
particularly Remark 2.8). The proof follows from setting $\delta=\frac{6}{5}$
and taking $t_{0}\leq t\leq N^{5/6}(\log N)^{-5/6}$, where $t_{0}$ is large
enough. ∎

###  4.2 Lower bounds

We start with the proof of the lower bound for $\mathsf{MaxLR}$ and then see
how we can deduce the lower bound for $\mathsf{MaxFL}$.

####  4.2.1 The strategy

We rely again on the strategy of resampling. We will divide the first quadrant
of $\mathbb{Z}^{2}$ in angular sectors of angle

$\theta_{N}=\theta_{N}(\chi):=\chi N^{-\frac{1}{3}}(\log N)^{\frac{1}{3}},$
(4.9)

where $\chi>0$ is a small constant that we will specify later. Then, we will
randomly process a resampling in some of these sectors. We will argue that
each one of these sectors has a probability of producing a “favourable” output
(i.e. with a large local roughness) which is decaying slowly (see Lemma 4.19)
in $N$, i.e. roughly like $N^{-\delta}$ for some (small) $\delta>0$. By
iterating these resamplings on (sufficiently many) different sectors we ensure
to recover a new path which presents at least one sector with a large local
roughness. There is a crucial point we will have to deal with: if a given
sector produces a favorable output, this could be undone by a further
resample. That is why we will resample on sufficiently distant sectors (see
Proposition 4.20).

Let us first fix some notations. Define

$m_{N}:=\left\lfloor\dfrac{\pi}{2\theta_{N}}\right\rfloor$ (4.10)

to be the number of sectors subject to resampling. Note that

$m_{N}\underset{N\rightarrow+\infty}{\sim}\dfrac{\pi}{2\chi}N^{\frac{1}{3}}(\log
N)^{-\frac{1}{3}}.$ (4.11)

Denote by $\mathbf{A}_{j}(N)$ the sector of the first quadrant defined by the
angles $(j-1)\theta_{N}$ and $j\theta_{N}$ for $j\in\\{1,\ldots,m_{N}\\}$.
Note that the first quadrant may contain a narrower sector which we will not
make use of. Finally, we introduce the following notation: given two points
$a$ and $b$ in the first quadrant, we denote by $\angle(a,b)$ the (non
negative121212We will always choose $\angle(a,b)\in[0,\pi/2]$.) angle between
the associated vectors $\overrightarrow{a}$ and
$\overrightarrow{b}$131313Sometimes we will also write
$\angle(\overrightarrow{a},\overrightarrow{b})$, depending on the context..
For a closed shape $S$, $\textup{Enclose}(S)$ is the region of the first
quadrant enclosed by $S$ (and if needed the coordinate axes).

####  4.2.2 Defining the resampling procedure

We want to resample in the sectors $\mathbf{A}_{j}(N)$. However, choosing the
beginning and ending points of the pieces subject to resampling in each sector
must be done precociously.

######  Definition 4.5.

Let $x,y\in\mathbb{N}^{2}$ be such that $\Lambda^{x\rightarrow
y}\neq\emptyset$. Denote by $\mathbf{A}_{x,y}$ the cone of apex $0$ bounded by
$x$ and $y$. For all $\gamma\in\Lambda^{N^{2}}$, $\Psi_{x,y}(\gamma)$ is the
random element of $\Lambda$ that is equal to $\gamma$ outside
$\mathbf{A}_{x,y}$, and that is random on $\mathbf{A}_{x,y}$ having the
marginal law of $\mathbb{P}_{\lambda}^{N^{2}}$ on the considered sector, given
the area condition and the path outside the sector.

######  Remark 4.6.

In words, the formation of $\Psi_{x,y}(\gamma)$ can be summed up in two steps:

  1. -

Step A: we condition on the marginal of $\gamma$ on $\mathbf{A}_{x,y}$ by the
information of $\gamma\cap\mathbf{A}_{x,y}^{c}$.

  2. -

Step B: we condition on the area:
$|\textup{Enclose}((\gamma\cap\mathbf{A}_{x,y}^{c})\cup\gamma_{x,y})|\geq
N^{2}.$

The strategy is now the following: for each sector $\mathbf{A}_{j}(N)$, we
pick some points $x_{j},y_{j}$ that belong to this sector, and to the random
path we are considering, and we apply $\Psi_{x_{j},y_{j}}$.

######  Definition 4.7 (Definition of $\mathsf{RES}_{j}$).

Let $j\in\\{1,\dots,m_{N}\\}$. We define a procedure
$\mathsf{RES}_{j}:\Lambda^{N^{2}}\rightarrow\Lambda$ that only acts on
$\mathbf{A}_{j}(N)$. Let $\mathbf{B}_{j}(N)$ be the cone of apex $0$ defined
by the angles $(j-1)\theta_{N}+\theta_{N}/4$ and $j\theta_{N}-\theta_{N}/4$.
Notice that $\mathbf{B}_{j}(N)\subset\mathbf{A}_{j}(N)$. Let
$\gamma\in\Lambda^{N^{2}}$. Let $x_{j}=x_{j}(\gamma)$ (resp.
$y_{j}=y_{j}(\gamma)$) be the left-most (resp. right-most) point of
$\gamma\cap\mathbf{B}_{j}(N)$. The procedure $\mathsf{RES}_{j}$ consists in
resampling $\gamma$ between $x_{j}$ and $y_{j}$. More precisely, we set
$\mathsf{RES}_{j}(\gamma):=\Psi_{x_{j},y_{j}}(\gamma)$.

We will denote by $(\Omega,\mathcal{F},\mathbb{P})$ the probability space in
which $\mathsf{RES}_{j}$ acts on an input $\Gamma$ having the distribution
$\mathbb{P}_{\lambda}^{N^{2}}$.

The following proposition is a straightforward consequence of the definition
of $\Psi_{x,y}$,

######  Proposition 4.8 ($\mathbb{P}_{\lambda}^{N^{2}}$ is invariant under the
resampling procedure).

Let $j\in\\{1,\dots,m_{N}\\}$. Then, if
$\Gamma\sim\mathbb{P}_{\lambda}^{N^{2}}$,
$\mathsf{RES}_{j}(\Gamma)\sim\mathbb{P}_{\lambda}^{N^{2}}$. In words, the law
$\mathbb{P}_{\lambda}^{N^{2}}$ is invariant under the map $\mathsf{RES}_{j}$.

We have now properly defined a resampling procedure on each sector
$\mathbf{A}_{j}(N)$. As we saw, this procedure can be realized in two steps.
Our next objective is to find a sufficient condition for step B to be
realized: how can we ensure that the resampled path captures enough area.

####  4.2.3 A sufficient condition to capture enough area

Let $a$ and $b$ be two points in $\mathbb{N}^{2}$ (with
$\text{arg}(a)>\text{arg}(b)$). Assume we are given $\gamma\in\Lambda^{N^{2}}$
that passes through $a$ and $b$. Modify $\gamma$ between these two points
replacing the original piece of path $\gamma_{a,b}$ by a new element
$\tilde{\gamma}_{a,b}\in\Lambda^{a\rightarrow b}$. We are looking for a
condition on $\tilde{\gamma}_{a,b}$ that is sufficient for the modified path
$\tilde{\gamma}$ to belong to $\Lambda^{N^{2}}$. It is clear that a sufficient
condition for $\tilde{\gamma}$ to enclose a sufficiently large area is

$\left|\textup{Enclose}\left(\tilde{\gamma}_{a,b}\cup[0,a]\cup[0,b]\right)\right|\geq\left|\textup{Enclose}(\gamma)\cap\mathbf{A}_{a,b}\right|.$
(4.12)

As in [Ham12b], we obtain a quantitative sufficient condition (see Corollary
4.13) for (4.12) to be satisfied. We directly import without a proof the
result from there as they trivially extend to our setting.

We will need a notation: for $x,y\in\mathbb{N}^{2}$ distinct, we denote by
$\ell_{x,y}$ the unique line that passes through $x$ and $y$. The following
result is illustrated in Figure 7.

######  Lemma 4.9.

Let $\gamma\in\Lambda^{N^{2}}$. Let $j\in\\{1,\ldots,m_{N}\\}$. Denote by
$z_{j}$ the point of $\mathcal{C}(\gamma)$ of argument $j\theta_{N}$. Let
$\ell_{j}$ be the tangent line of $\mathcal{C}(\gamma)$ at $z_{j}$. Let
$x_{j},y_{j}\in\gamma\cap\mathbf{A}_{z_{j-1},z_{j}}$ and
$\mathbf{E}_{x_{j},y_{j}}:=\mathbf{E}_{j}$ be the pentagon delimited by the
lines $\ell_{j-1},\ell_{j},\ell_{x_{j},y_{j}},\ell_{0,x_{j}}$ and
$\ell_{0,y_{j}}$. Then,

$(\textup{Enclose}(\gamma)\cap\mathbf{A}_{x_{j},y_{j}})\subset\mathbf{T}_{0,x_{j},y_{j}}\cup\mathbf{E}_{j},$
(4.13)

where $\mathbf{T}_{0,x_{j},y_{j}}$ is the triangle of apexes $0,x_{j},y_{j}$.

Figure 7: Illustration of Lemma 4.9. The light orange shaded region
corresponds to $\mathbf{E}_{j}^{0}$, and the dark orange shaded region
corresponds to $\mathbf{E}_{j}^{1}$.

An immediate consequence of the lemma is the fact that $(\ref{sufficient area
2})$ is satisfied if

$\left|\textup{Enclose}\left(\tilde{\gamma}_{a,b}\cup[0,a]\cup[0,b]\right)\right|\geq\left|\mathbf{T}_{0,a,b}\right|+\left|\mathbf{E}_{a,b}\right|.$
(4.14)

We now write $\mathbf{E}_{j}=\mathbf{E}_{j}^{0}\cup\mathbf{E}_{j}^{1}$ (where
$\mathbf{E}_{j}^{0}$ denotes the part of $\mathbf{E}_{j}$ that lies below
$\ell_{z_{j-1},z_{j}}$), see Figure 7. It seems natural to seek an upper bound
for $|\mathbf{E}_{j}|$. We start by determining an upper bound on
$|\mathbf{E}_{j}^{1}|$. This is what we perform in the next lemma.

######  Lemma 4.10 ( [Ham12b, Lemma 3.9]).

Let $\gamma\in\Lambda^{N^{2}}$. Assume $\gamma\subset B_{K_{1}N}$. We keep the
notations of the preceding lemma. Let $\overrightarrow{w_{j}}$ be the unit
vector tangent to $\mathcal{C}(\gamma)$ at $z_{j}$ oriented in the
counterclockwise sense. We define the set of sectors with moderate boundary
turning to be

$\textup{MBT}=\Big{\\{}j\in\\{1,\ldots,m_{N}\\},~{}\|z_{j}-z_{j-1}\|\leq\dfrac{10\pi
K_{1}N}{m_{N}}\textup{ and
}\angle(\overrightarrow{w_{j}},\overrightarrow{w_{j-1}})\leq\dfrac{10\pi}{m_{N}}\Big{\\}}.$
(4.15)

Then,

$|\textup{MBT}|\geq\dfrac{9m_{N}}{10},$ (4.16)

and if $j\in\textup{MBT}$,

$|\mathbf{E}_{j}^{1}|\leq\dfrac{1}{2}(20)^{3}K_{1}^{2}\chi^{3}N\log N.$ (4.17)

We now need to find an upper bound on $|\mathbf{E}_{j}^{0}|$. It is clear that
such bound can only be obtained under the assumption that the points $x_{j}$
and $y_{j}$ defined above are not to far from $\mathcal{C}(\gamma)$. This
remark motivates the following definition. Recall that if $\gamma\in\Lambda$
and $v\in\gamma$, $\mathsf{LR}(v,\gamma)=\mathrm{d}(v,\mathcal{C(\gamma)})$
denotes the local roughness of $v$ in $\gamma$.

######  Definition 4.11 (Favourable sectors).

Let $\gamma\in\Lambda^{N^{2}}$. Let $\varphi$ be a function that maps
$(0,\infty)$ to itself and that satisfies, as $t\rightarrow 0$,

$\varphi(t)=o(\sqrt{t}).$ (4.18)

For $\chi>0$, we say that the sector $\mathbf{A}_{j}(N)$ is favourable under
$\gamma$ if there exists $v\in\gamma\cap\mathbf{A}_{j}(N)$ such that

$\mathsf{LR}(v,\gamma)\geq\varphi(\chi)N^{\frac{1}{3}}(\log N)^{\frac{2}{3}}.$
(4.19)

We define $\textup{UNFAV}(\gamma,\chi)=\textup{UNFAV}$ to be the set of
$j\in\\{1,\ldots,m_{N}\\}$ such that $\mathbf{A}_{j}(N)$ is not favourable
under $\gamma$.

######  Lemma 4.12 ( [Ham12b, Lemma 3.10]).

Let $\gamma\in\Lambda^{N^{2}}$. Assume that $\gamma\cap B_{K_{2}N}=\emptyset$.
Let $j\in\textup{MBT}\cap\textup{UNFAV}$. Then,

$|\mathbf{E}_{j}^{0}|\leq 20K_{1}\chi\varphi(\chi)N\log N.$ (4.20)

From the preceding results, we can deduce the sufficient condition to capture
enough area we were looking for,

######  Corollary 4.13 (Sufficient condition to capture a large area).

Let $\gamma\in\Lambda^{N^{2}}$ be such that $\gamma\subset B_{K_{1}N}\setminus
B_{K_{2}N}$. Let $j\in\\{1,\ldots,m_{N}\\}$. Assume
$j\in\textup{MBT}\cap\textup{UNFAV}$. Let
$x_{j},y_{j}\in\gamma\cap\mathbf{A}_{j}(N)$ be chosen accordingly to the
procedure described in Definition 4.7. Then, if we resample $\gamma$ between
$x_{j}$ and $y_{j}$, replacing the path $\gamma_{x_{j},y_{j}}$ by a new path
$\tilde{\gamma}_{x_{j},y_{j}}$, the new path $\tilde{\gamma}$ will be an
element of $\Lambda^{N^{2}}$ if the following condition is satisfied,

$\left|\textup{Enclose}(\tilde{\gamma}_{x_{j},y_{j}}\cup[0,x_{j}]\cup[0,y_{j}])\right|-\left|\mathbf{T}_{0,x_{j},y_{j}}\right|\geq\left(\tfrac{1}{2}(20)^{3}K_{1}^{2}\chi^{3}+20K_{1}\varphi(\chi)\chi\right)N\log
N.$ (4.21)

####  4.2.4 The right shape

Now that we have a sufficient condition for a resampled path to be accepted,
we need to exhibit an event that will ensure this sufficient condition. We
make good use of the event $\mathsf{LogGAC}(x,y,\eta)$ introduced in
Definition 4.2. As we are about to see, the realisation of this event will
ensure that the condition of Corollary 4.13 will be satisfied. Nevertheless,
we also need our resamplings to have a large local roughness and that is what
motivates Definition 4.14.

######  Definition 4.14 (Significant inward deviation).

Let $\eta>0$ and $x,y\in\mathbb{N}^{2}$ with $\arg(x)>\arg(y)$. Let
$\gamma\in\Lambda$ be a path which passes by $x$ and $y$. We say that $\gamma$
realizes the event $\mathsf{LogSID}(x,y,\eta)$ (meaning “logarithmic
significant inward deviation”) if there exists $z\in\gamma_{x,y}$ such that

$\mathrm{d}\left(z,\partial\text{conv}([0,x]\cup[0,y]\cup\gamma_{x,y})\right)\geq\eta||x-y||^{\frac{1}{2}}\left(\log||x-y||\right)^{\frac{1}{2}},$
(4.22)

where $\partial\textup{conv}$ is the border of the convex hull.

We are going to resample in the sectors $\mathbf{A}_{j}(N)$ introduced above.
For a path $\gamma\in\Lambda^{N^{2}}$, call