MnLargeSymbols’164 MnLargeSymbols’171 mathx”30 mathx”38

# An upper bound on geodesic length in 2D critical first-passage percolation

Erik Bates
Department of Mathematics
North Carolina State University
SAS Hall
2311 Stinson Drive
Raleigh, NC 27695-8205 USA
<EMAIL_ADDRESS>,  David Harper
Department of Mathematics
Georgia Institute of Technology
Skiles Classroom Building
686 Cherry St NW
Atlanta, GA 30332 USA
<EMAIL_ADDRESS>,  Xiao Shen
Department of Mathematics
University of Utah
John A. Widtsoe Building
155 S 1400 E
Salt Lake City, Utah 84112 USA
<EMAIL_ADDRESS>and  Evan Sorensen
Department of Mathematics
Columbia University
Room 614, MC 4432
2990 Broadway
New York, NY 10027 USA
<EMAIL_ADDRESS>
###### Abstract.

We consider i.i.d. first-passage percolation (FPP) on the two-dimensional
square lattice, in the critical case where edge-weights take the value zero
with probability $1/2$. Critical FPP is unique in that the Euclidean lengths
of geodesics are superlinear—rather than linear—in the distance between their
endpoints. This fact was speculated by Kesten in 1986 but not confirmed until
2019 by Damron and Tang, who showed a lower bound on geodesic length that is
polynomial with degree strictly greater than $1$. In this paper we establish
the first non-trivial upper bound. Namely, we prove that for a large class of
critical edge-weight distributions, the shortest geodesic from the origin to a
box of radius $R$ uses at most $R^{2+\varepsilon}\pi_{3}(R)$ edges with high
probability, for any $\varepsilon>0$. Here $\pi_{3}(R)$ is the polychromatic
3-arm probability from classical Bernoulli percolation; upon inserting its
conjectural asymptotic, our bound converts to $R^{4/3+\varepsilon}$. In any
case, it is known that $\pi_{3}(R)\lesssim R^{-\delta}$ for some $\delta>0$,
and so our bound gives an exponent strictly less than $2$. In the special case
of Bernoulli($1/2$) edge-weights, we replace the additional factor of
$R^{\varepsilon}$ with a constant and give an expectation bound.

###### Key words and phrases:

critical first-passage percolation, geodesic length, square lattice

###### 2020 Mathematics Subject Classification:

60K35, 60K37, 82B27, 82B43

E.B. was partially supported by NSF grants DMS-1902734 and DMS-2246616

D.H. was partially supported by NSF grant DMS-2054559.

X.S. was partially support from the Wylie Research Fund at the University of
Utah.

E.S. was partially supported by the Fernholz foundation. This work was partly
performed while E.S. was a PhD student at the University of Wisconsin–Madison,
where he was partially supported by Timo Seppäläinen under NSF grants
DMS-1854619 and DMS-2152362.

###### Contents

  1. 1 Introduction
  2. 2 Topological preliminaries: circuits and associated regions
  3. 3 Proofs for Bernoulli edge-weights
  4. 4 Proofs for general edge-weights
  5. 5 Kesten’s separation results and consequences
  6. 6 Topological details concerning circuits and paths
  7. 7 Proof of Proposition 2.5 (construction of geodesic)
  8. 8 Acknowledgments

##  1\. Introduction

###  1.1. The model of critical first-passage percolation (FPP)

Let $E(\mathbb{Z}^{2})$ denote the edge set of the square lattice
$\mathbb{Z}^{2}$. Consider a family of i.i.d. random variables $(t_{e})_{e\in
E(\mathbb{Z}^{2})}$ defined on some probability space
$(\Omega,\mathcal{F},\mathbb{P})$ such that

$\displaystyle\begin{split}\mathbb{P}(t_{e}<0)=0\quad\text{and}\quad\mathbb{P}(t_{e}=0)=1/2.\end{split}$
(1.1)

We say that $t_{e}$ is the weight of edge $e$. For each pair
$x,y\in\mathbb{Z}^{2}$, let $\mathcal{P}(x,y)$ denote the collection of all
self-avoiding nearest-neighbor paths starting at $x$ and ending at $y$. The
passage time between $x$ and $y$ is the random quantity

$\displaystyle\begin{split}T(x,y)=\inf_{\gamma\in\mathcal{P}(x,y)}T(\gamma),\quad\text{where}\quad
T(\gamma)=\sum_{e\in\gamma}t_{e}.\end{split}$ (1.2)

When $x=y$, we allow the empty path so that $T(x,x)=0$. The map
$T(\cdot\,,\cdot)$ is thus a pseudometric on $\mathbb{Z}^{2}$, and it
naturally extends to sets: for $\mathcal{A},\mathcal{B}\subset\mathbb{Z}^{2}$,
we define

$\displaystyle
T(\mathcal{A},\mathcal{B})=\inf_{x\in\mathcal{A},\,y\in\mathcal{B}}T(x,y).$

A path $\gamma$ is said to be a geodesic between $\mathcal{A}$ and
$\mathcal{B}$ if $\gamma$ starts at a vertex in $\mathcal{A}$, ends at a
vertex in $\mathcal{B}$, and achieves the minimal passage time
$T(\gamma)=T(\mathcal{A},\mathcal{B})$. It is known that with probability one,
geodesics exist between every pair of points [28, Cor. 1.3]. Therefore,
geodesics exist between any two finite sets $\mathcal{A}$ and $\mathcal{B}$.

This model can be projected to classical Bernoulli percolation by declaring
that all edges $e$ with $t_{e}=0$ are open, while those with $t_{e}>0$ are
closed. The assumption (1.1) means that the resulting projection is critical
percolation; in particular, there is no infinite connected cluster of open
edges. Consequently, for $x$ and $y$ far apart, a geodesic between $x$ and $y$
will typically use a large number of open edges without penalty, but will also
need to traverse a small number of closed edges to go between distinct open
clusters. On the criticality assumption (1.1), Kesten [14, Sec. 9.24]
conjectured that the length of a geodesic (i.e. the total number of edges it
contains) between $0$ and $x$ grows superlinearly in $\|x\|$. This conjecture
was verified by Damron and Tang [7]. To state their result precisely, we let
$\mathcal{N}_{0,x}$ denote the minimum length of a geodesic between $0$ and
$x$.

######  Theorem A.

[7, Thm. 1] Assume (1.1). Then there exist $c>0$ and $\beta>1$ such that

$\displaystyle\begin{split}\mathbb{P}(\mathcal{N}_{0,x}\leq\|x\|^{\beta}_{1})\leq(1/c)\exp(-\|x\|_{1}^{c})\quad\text{for
all $x\in\mathbb{Z}^{2}$}.\end{split}$ (1.3)

It should be emphasized that this result is special to the critical case, and
at least superficially to $d=2$. Indeed, if $\mathbb{P}(t_{e}=0)\neq 1/2$,
then $\mathcal{N}_{0,x}$ grows linearly in $\|x\|$ and is known in some cases
to even satisfy a law of large numbers if $x$ is brought to infinity along a
fixed direction (see [2, Sec. 1.5] and references therein, including [1, Thm.
4.9] for the subcritical case, and [29, Thm. 4] for supercritical). What
remains unsettled by Theorem A is the exact growth rate of $\mathcal{N}_{0,x}$
in critical FPP, as there is no matching upper bound. In fact, until now, no
upper bound whatsoever has been established.

###  1.2. Main results for Bernoulli weights

The most well-studied case of critical FPP is that of Bernoulli weights:

$\displaystyle\begin{split}\mathbb{P}(t_{e}=0)=\mathbb{P}(t_{e}=1)=\frac{1}{2}.\end{split}$
(1.4)

Our results are strongest in this case. The estimates we provide for geodesic
length are given in terms of arm events, which are of fundamental interest in
the study of Bernoulli percolation and not immediately connected to FPP. We
now recall the relevant definitions.

Define the dual lattice
$\widehat{\mathbb{Z}}^{2}=\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2})$. For
clarity, we will often refer to $\mathbb{Z}^{2}$ as the primal lattice. The
shift by $(\frac{1}{2},\frac{1}{2})$ means that for each primal edge $e\in
E(\mathbb{Z}^{2})$, there is a unique dual edge $\hat{e}\in
E(\widehat{\mathbb{Z}}^{2})$ that bisects it. We say that the endpoints of
$\hat{e}$ are the dual neighbors of $e$, and similarly, the endpoints of $e$
are the dual neighbors of $\hat{e}$. We also define the dual neighbors of a
vertex $v$, which are the four points on the dual lattice closest to $v$. Once
the edges of $\mathbb{Z}^{2}$ are given open or closed status, their dual
edges are given the same status. That is, we declare $\hat{e}$ to be open if
$e$ is open, or closed if $e$ is closed. We can then speak of open or closed
paths; all open paths we consider are on the primal lattice, while all closed
paths are on the dual lattice. Furthermore, it will often be useful to
identify self-avoiding paths with the simple curves their edges trace out in
the plane. For instance, this identification makes defining disjointness very
intuitive: two paths are disjoint if their associated curves are
disjoint.111If the two paths are on the same lattice, then disjointness is
equivalent to sharing no vertices; if on different lattices, it is equivalent
to sharing no dual pair of edges (i.e. never crossing).

Consider the box $B_{R}=[-R,R]^{2}\cap\mathbb{Z}^{2}$. The boundary of $B_{R}$
is written $\partial B_{R}=B_{R}\setminus B_{R-1}$. Let $\pi_{3}(R)$ denote
the probability of the following event depicted in Figure 1:

$e$$\partial B_{R}$ Figure 1. The polychromatic 3-arm event at edge
$e=\\{(0,0),(1,0)\\}$: two open primal paths (shown solid) and one closed dual
path (shown dashed).

there exist two primal paths and one dual path that are disjoint and satisfy
the following conditions:

  * •

The two primal paths are open, start at $(0,0)$ and $(1,0)$ respectively, and
both end at $\partial B_{R}$.

  * •

The dual path is closed, starts at either $(\frac{1}{2},\frac{1}{2})$ or
$(\frac{1}{2},-\frac{1}{2})$, and eventually reaches a dual neighbor of a
point inside $\partial B_{R}$.

In the percolation literature, this is called a 3-arm event, and $\pi_{3}(R)$
is the 3-arm probability at distance $R$. Our bounds are given in terms of
this quantity.

Let $\mathcal{N}_{R}$ denote the length of the shortest geodesic from $0$ to
$\partial B_{R}$.

######  Theorem 1.1.

Assume (1.4). Then there exists a constant $C$ such that for all $R\geq 1$,

$\displaystyle\begin{split}\mathbb{E}[\mathcal{N}_{R}]\leq
CR^{2}\pi_{3}(R).\end{split}$ (1.5)

Our next result considers a more general setting. For finite sets
$\mathcal{A},\mathcal{B}\subset\mathbb{Z}^{2}$, let
$\mathcal{N}_{\mathcal{A},\mathcal{B}}$ denote the minimum length of a
geodesic from $\mathcal{A}$ to $\mathcal{B}$. When $\mathcal{A}$ and
$\mathcal{B}$ are single points, the following gives a result for point-to-
point geodesics.

######  Theorem 1.2.

Assume (1.4). Let $\mathcal{A}$ and $\mathcal{B}$ be disjoint finite connected
sets of vertices, and let $d=\operatorname{dist}(\mathcal{A},\mathcal{B})$.
There exist constants $C,c>0$, independent of $d,|\mathcal{A}|$, and
$|\mathcal{B}|$ such that

$\displaystyle\begin{split}\mathbb{P}(\mathcal{N}_{\mathcal{A},\mathcal{B}}\geq\lambda
d^{2}\pi_{3}(d))\leq
C(|\mathcal{A}|+|\mathcal{B}|)^{3}\lambda^{-c}\quad\text{for all
$\lambda>0$}.\end{split}$ (1.6)

The exact decay rate of $\pi_{3}(R)$ as $R\to\infty$ has not been established
on the square lattice, but on the triangular lattice, it is known that
$\pi_{3}(R)=R^{-2/3+o(1)}$ [24, Thm. 4] (see also [21, Thm. 21]). It is widely
believed that the same asymptotic holds on the square lattice, although we are
not aware of any rigorous estimates other than those inferred from bounds for
the one-arm and five-arm exponents; see Remark 3.6.

######  Remark 1.3.

In the case $|\mathcal{A}|=|\mathcal{B}|=1$, Theorem 1.2 is analogous to [3,
Cor. 2] concerning chemical distance in critical percolation. We do not obtain
a precise value of $c$ in the proof, but we explain here why $c$ cannot exceed
the value $2$ when $\mathcal{A}=\\{(0,0)\\}$ and $\mathcal{B}=\\{(1,0)\\}$.
Indeed, it was shown in [3, Prop. 3] that

$\mathbb{E}\big{[}(\mathcal{N}_{(0,0),(1,0)})^{2}\>\big{|}\>(0,0)\leftrightarrow(1,0)\big{]}=\infty,$

where $(0,0)\leftrightarrow(1,0)$ denotes the event that the two points are
connected by an open path. In particular,
$\mathbb{E}\big{[}(\mathcal{N}_{(0,0),(1,0)})^{2}\big{]}=\infty$. But if $c$
in (1.6) were greater than $2$, then $\mathcal{N}_{(0,0),(1,0)}$ would have a
finite second moment.

###  1.3. Main results for general edge-weights

To obtain bounds for geodesic length beyond the Bernoulli case, we require
either of two possible assumptions on the distribution function $F$ of the
edge-weights. These are (1.8) and (1.9) stated below. These conditions involve
quantities $(p_{R})_{R\geq 1}$ which are standard in near-critical
percolation. We now recall their definition.

Consider i.i.d. random variables $(U_{e})_{e\in E(\mathbb{Z}^{2})}$, each
uniformly distributed on $[0,1]$. For a specific choice of distribution
function $F$, we can realize the edge-weights as $t_{e}=F^{-1}(U_{e})$, where

$\displaystyle F^{-1}(u)=\inf\\{t\in\mathbb{R}:\,F(t)\geq u\\}.$

We say a self-avoiding path $\gamma$ is $p$-open if, for each edge
$e\in\gamma$, we have $U_{e}\leq p$. With $p_{\mathrm{c}}=1/2$, the open paths
we have defined previously are the same as $p_{\mathrm{c}}$-open paths defined
here. For positive integers $n,m$ and $p\in(p_{\mathrm{c}},1]$, let

$\sigma(n,m,p)=\mathbb{P}(\text{there is a }p\text{-open left-right crossing
of }[0,n]\times[0,m]),$

where “left-right crossing of $[0,n]\times[0,m]$” means a path in
$[0,n]\times[0,m]$ that starts in $\\{0\\}\times[0,m]$ and ends in
$\\{n\\}\times[0,m]$. For $\varepsilon>0$ and $p>p_{\mathrm{c}},$ define

$L(p,\varepsilon)=\min\\{R\geq 1:\sigma(R,R,p)>1-\varepsilon\\}.$

This $L(p,\varepsilon)$ is called the (finite-size scaling) correlation
length. It is shown in [15, Thm. 4] that there exists $\varepsilon_{1}>0$ such
that for all $0<\varepsilon,\varepsilon^{\prime}\leq\varepsilon_{1}$, one has
$L(p,\varepsilon)\asymp L(p,\varepsilon^{\prime})$ as $p\searrow
p_{\mathrm{c}}$. We therefore write $L(p)=L(p,\varepsilon_{1})$ with this
fixed $\varepsilon_{1}$, as is customary. For $R\geq 1$, define

$p_{R}=\inf\\{p>p_{\mathrm{c}}:L(p)\leq R\\}.$ (1.7)

The two maps $R\mapsto p_{R}$ and $p\mapsto L(p)$ should be thought of as
inverse to each other.

######  Theorem 1.4.

Assume that the edge-weight distribution function $F$ satisfies (1.1) and one
of the following two conditions:

$\displaystyle\limsup_{n\to\infty}F^{-1}(p_{2^{n+1}})/F^{-1}(p_{2^{n}})<1,$
(1.8)
$\displaystyle\text{or}\quad\liminf_{n\to\infty}F^{-1}(p_{2^{2n}})/F^{-1}(p_{2^{n}})>0.$
(1.9)

Then for any $\varepsilon>0$, there exist constants $C,c,s>0$ such that for
all $\lambda\geq 1$ and $R\geq 1$,

$\displaystyle\begin{split}\mathbb{P}(\mathcal{N}_{R}\geq\lambda
R^{2+\varepsilon}\pi_{3}(R))\leq Ce^{-c(\log\lambda R)^{s}}.\end{split}$
(1.10)

Since $p_{R}\searrow\frac{1}{2}$ as $R\to\infty$ (for instance, see (4.6)),
the conditions (1.8) and (1.9) control the behavior of $F$ near its atom at
$0$. Similar conditions have been used in [6] for characterizing the
asymptotic behavior of $T(0,\partial B_{R})$ as $R\to\infty$, but what is
striking here is that (1.8) and (1.9) represent opposite (indeed, mutually
exclusive) regimes for the behavior of $F$ near $0$. Intuitively, (1.8)
suggests that as $\varepsilon\searrow 0$, the value of
$F^{-1}(1/2+\varepsilon)$ does not decrease too rapidly towards zero.
Meanwhile, (1.9) indicates that the graph of $F$ becomes very flat to the
right of $0$, so that $F^{-1}(1/2+\varepsilon)$ rapidly vanishes as
$\varepsilon\searrow 0$. Somewhat surprisingly, either of these opposite
possibilities is enough to prove the same upper bound (1.10).

In either case, the strategy is to control the number of nonzero edges along
the geodesics. Under condition (1.8), we control the number of nonzero edges
in two ways. When the maximum edge weight along the geodesics is less than or
equal to $F^{-1}(1/2+\varepsilon)$, we employ an argument based on the work of
Kiss [17, 18]. In particular, we cite a result from [7] that uses this
machinery. Otherwise, we show that it is rare for the maximum edge weight
along a geodesic to be greater than $F^{-1}(1/2+\varepsilon)$. It is worth
noting that we need this estimate to hold for all large boxes, which
translates to all small values of $\varepsilon$. If $F^{-1}(1/2+\varepsilon)$
decays too quickly, the tail estimate for the maximum edge weight breaks down.
This is where condition (1.8) comes into play: it controls the decay rate of
$F^{-1}(1/2+\varepsilon)$ as $\varepsilon\searrow 0$.

On the other hand, the condition (1.9) gives a different decomposition. It
ensures that edges with small positive values are infrequent across the entire
box $B_{R}$. Consequently, there cannot be many of these edges along the
geodesic. Additionally, edges with larger values do not appear on the geodesic
too often, as their presence would again result in a higher first-passage
value.

The following collection of examples demonstrates that conditions (1.8) and
(1.9) allow for a broad class of edge-weight distributions. Figure 2 provides
an illustration.

######  Example 1.5.

In each example below, it is assumed $F(t)=0$ for $t<0$ and
$F(0)=p_{\mathrm{c}}$. We specify how $F(t)$ behaves for small positive $t$.

  1. (a)

Suppose there are constants $C,\alpha,h>0$ such that
$F(t)=p_{\mathrm{c}}+Ct^{\alpha}$ for all $t\in[0,h]$. Then $F$ satisfies
(1.8).

  2. (b)

Any distribution whose support has a gap at $0$ (i.e. $F(h)=p_{\mathrm{c}}$
for some $h>0$) satisfies (1.9) because the ratio is $1$ for all $n$. In
particular, the Bernoulli distribution (1.4) satisfies (1.9).

  3. (c)

Suppose there are constants $C,\alpha,h>0$ such that
$F(t)=p_{\mathrm{c}}+Ce^{-t^{-\alpha}}$ for all $t\in(0,h]$. Then $F$
satisfies (1.9).

  4. (d)

Not all distributions must satisfy one of the conditions. For example, since
$p_{R}\searrow p_{\mathrm{c}}$ as $R\to\infty$, there exists a distribution
$F$ with $F(0)=p_{\mathrm{c}}$ and $F^{-1}(p_{2^{n}})=e^{-\sqrt{n}}$ for every
$n\geq 1$. Such a distribution satisfies neither (1.8) nor (1.9).

The proof that these examples have the stated properties is given in Section
4.1.

$\frac{1}{2}$$F(t)$$t$$\frac{1}{2}$$F(t)$$t$ Figure 2. An illustration of
conditions (1.8) and (1.9). Left: Three distribution functions that behave
near $t=0$ as $1/2+t^{1/10}$, $1/2+t$, and $1/2+t^{10}$. All three
distributions satisfy (1.8). Right: A distribution function that is constant
on $[0,h]$ for some $h>0$. Any such distribution satisfies (1.9).

###  1.4. Three-arm heuristic and organization of the paper

The appearance of $\pi_{3}(R)$ in (1.5), (1.6), and (1.10) can be intuitively
explained by the following heuristic. From every edge $e$ in a geodesic
$\gamma$ between two sets, the geodesic itself provides two paths: one to the
starting set and another to the ending set. These subpaths of $\gamma$ consist
mostly of open edges, making it plausible that they resemble open arms. By
duality, there is also a closed dual path from the starting set to the ending
set that consists mostly of closed edges. We show that if $\gamma$ is chosen
“as close as possible” to $\zeta$, then for each edge $e$ in $\gamma$, there
must exist a closed path from (a dual neighbor of) $e$ to $\zeta$. Joining
this closed path with $\zeta$ itself, we obtain a path resembling a closed
arm. Hence $e$ resembles an edge admitting a 3-arm event.

Carrying out this argument has two main challenges. The first is to overcome
the obvious issue that all three “arms” described above have some defect edges
of the wrong type. This is dealt with in two essentially independent ways. In
the Bernoulli case, Section 3 employs a patching argument that stitches
together arm events on different scales in exchange for only constant factors
in front of probabilities. The main technical devices are found in the proof
of Lemma 3.10.

For more general edge-weight distributions, Section 4 provides a quite
different approach. Instead of avoiding the defects by patching as before, we
allow our three arms to actually use the defects. Each defect incurs a small
probabilistic penalty thanks to a result of [21] (this is the reason for the
additional factor of $R^{\varepsilon}$ in (1.10)). In order for these
penalties to not accumulate too much, we must control the number and location
of closed edges used by a geodesic (Sections 4.3 and 4.5), which in turn
requires some understanding of the passage time across annuli (Section 4.2).
These intermediate results are interesting in their own right, as they shed
light on the structure of geodesics in critical FPP.

The second main challenge is to supply all of these arguments with rigorous
topological constructions. The paths involved in arm events are arrived at by
particular interactions of the geodesic with open and closed circuits. Proving
these interactions do occur requires a careful treatment of the relevant
topological objects. So as to not distract from the probabilistic arguments,
we state the requisite definitions in Section 2 and postpone the finer
topological details until Sections 5, 6, and 7.

Section 5 recalls a result from [13] regarding percolation on planar graphs,
and then collects various consequences for bond percolation on
$\mathbb{Z}^{2}$. For instance, a key fact is that a bounded cluster of open
edges is enclosed by a closed dual circuit (Lemma 5.6). This and other results
in Section 5 are well-known but not conveniently quotable from the literature,
hence the inclusion of this section for completeness. Section 6 provides some
general facts about Jordan curves. Finally, Section 7 constructs the desired
geodesic. The outcome is summarized in Proposition 2.5, which gives a circuit-
based description of this distinguished geodesic in any critical FPP model on
$\mathbb{Z}^{2}$. This and other topological results we develop are quite
general and could be useful for future studies. It should be noted that our
construction performs several modifications of the percolation environment (in
order to use the separation result following from [13]) that are similar to
techniques used in [16] on the triangular lattice. In that setting,
topological considerations are somewhat simpler because there is no auxiliary
dual lattice. We work on the square lattice to highlight the general
applicability of our approach, as our methods work just as well on the
triangular lattice.

###  1.5. Related literature and methodology

The study of the shortest geodesic in critical FPP is similar to the concept
of chemical distance. The chemical distance
$\operatorname{dist}_{\mathrm{c}}(x,y)$ between $x,y\in\mathbb{Z}^{2}$ is the
minimum number of edges in an open path starting at $x$ and ending at $y$. If
no such open path exists, the chemical distance is considered to be infinite.

A natural question is how the chemical distance scales with the Euclidean
distance. For instance, if we condition on the event $\\{x\leftrightarrow
y\\}$, then how does
$\mathbb{E}[\operatorname{dist}_{\mathrm{c}}(x,y)\>|\>x\leftrightarrow y]$
depend on $\|x-y\|$? It is generally believed that there is some $\beta>1$
such that

$\displaystyle\mathbb{E}[\operatorname{dist}_{\mathrm{c}}(x,y)\>|\>x\leftrightarrow
y]\asymp\|x-y\|^{\beta}.$

Establishing the existence of $\beta$, let alone computing its value, remains
a very challenging open problem. In fact, it was posited in [23] that even on
the triangular lattice, the value of $\beta$ is not obtainable through SLE
methods. Currently, there are not even many non-rigorous arguments for the
value of $\beta$. Nevertheless, numerical results [10, 11, 30] have suggested
that $\beta\approx 1.13$.

Another version of the chemical distance question to ask about the minimal
length of an open left-right crossing of the box $B_{R}$, conditional on the
event $\mathsf{C}_{R}$ of such a crossing. It turns out that a more amenable
quantity is the length $L_{R}$ of the lowest such crossing. This is because
the lowest crossing consists exclusively of $3$-arm edges. Using this fact,
Morrow and Zhang [20] showed (for critical site percolation on the triangular
lattice) that

$\displaystyle\begin{split}\mathbb{E}[L_{R}\>|\>\mathsf{C}_{R}]=R^{4/3+o(1)}=R^{2+o(1)}\pi_{3}(R).\end{split}$
(1.11)

Since the length of the lowest crossing is an upper bound on the length of the
shortest crossing, this result effectively provides upper bound on the
exponent $\beta$, namely $\beta\leq 4/3$. Our arguments in the Bernoulli case
(1.4) can be viewed as a generalization of this approach, where “lowest” is
replaced by “closest to a certain closed dual path”.

More recently, the upper bound was improved to $\beta<4/3$ in [4, 5], by
creating shortcuts along the lowest crossing. This use of shortcuts has also
been applied to give the same strict inequality for radial chemical distance
[25], improving on [3].

###  1.6. Open problems

We enumerate some open problems suggested by our work.

  1. (1)

Can the result of Theorem 1.4 be generalized to apply to any critical FPP
edge-weight distribution?

  2. (2)

Does the rate of growth of geodesic length in critical FPP depend on the edge-
weight distribution?

  3. (3)

Find the exact order of growth of geodesic length in critical FPP. Does there
exist $s>0$ so that $\mathcal{N}_{R}\asymp R^{1+s+o(1)}$? The existence of
such an exponent $s$ governing the growth of geodesic length is unknown even
on the triangular lattice, even though the arm exponents are known there.

  4. (4)

Our results give length bounds for a specific choice of geodesic that is
constructed in Proposition 2.5. Is $R^{2}\pi_{3}(R)$ optimal for this
geodesic, as in (1.11)? With a different choice of geodesic, can the ideas of
[4, 5, 25] be adapted to FPP in order to replace the bound
$R^{2+\varepsilon}\pi_{3}(R)$ with $R^{2-\delta}\pi_{3}(R)$ for some small
$\delta>0$?

  5. (5)

Related to Remark 1.3, [3] asks whether the expected chemical distance between
two points is finite. In our notation, this is the problem of determining
whether, in the setting (1.4), we have
$\mathbb{E}\big{[}\mathcal{N}_{(0,0),(0,1)}\>\big{|}\>(0,0)\leftrightarrow(1,0)\big{]}<\infty$.

##  2\. Topological preliminaries: circuits and associated regions

Critical FPP is intimately connected to the structure of open and closed
circuits.

######  Definition 2.1.

A circuit is a path of length at least $4$ that starts and ends at the same
vertex but is otherwise self-avoiding. An open circuit is a circuit consisting
of open primal edges. A closed circuit is a circuit consisting of closed dual
edges.

A circuit $\mathcal{C}$ can be naturally identified with the Jordan curve its
edges trace out in the plane; in particular, we can speak of its interior and
exterior, and we will denote these two disjoint sets by
$\mathsf{int}(\mathcal{C})$ and $\mathsf{ext}(\mathcal{C})$.

######  Definition 2.2.

We say a circuit $\mathcal{C}$ encloses a set of vertices $\mathcal{A}$ (i.e.
either $\mathcal{A}\subseteq\mathbb{Z}^{2}$ or
$A\subset\widehat{\mathbb{Z}}^{2}$) if
$\mathcal{A}\subseteq\mathsf{int}(\mathcal{C})$. We say a circuit
$\mathcal{C}$ encloses another circuit $\mathcal{C}^{\prime}$ if
$\mathsf{int}(\mathcal{C}^{\prime})\subseteq\mathsf{int}(\mathcal{C})$.

We note that each circuit can also be considered as a collection of vertices.
From this perspective, the two notions of enclosure are not the same, but for
technical topological reasons, it is essential for the that we establish the
two definitions in the present paper. In all applications, it should be clear
from context whether we are enclosing a circuit or enclosing a set of
vertices. Notice that if the two circuits live on the same lattice, enclosure
still allows for them to intersect. So we will often impose edge-disjointness,
meaning the two circuits have no common edges. A stronger condition is vertex-
disjointness, which means the two circuits share not vertices. Under the
criticality assumption (1.1), it is well-known that the following event, which
we shall call $\Omega_{\infty}$, occurs with probability one.

######  Definition 2.3.

Let $\Omega_{\infty}$ be the full probability event on which, for every $n\geq
1$, there exist both an open circuit and a closed circuit containing $B_{n}$
in their interiors.

In the context of planar FPP, circuits are useful objects to consider for the
simple reason that paths must cross them to go from their interior to their
exterior. However, when we speak of paths going from one circuit to another,
there is potential for confusion depending on which lattice—primal or
dual—each object belongs to. Instead of exhausting the reader with
clarifications every time, we simply rely on the conventions specified in
Definition 2.4 to maintain precision.

######  Definition 2.4.

A primal path $\gamma$ is open if all its edges are open.

  * •

If $A\subset\mathbb{Z}^{2}$, then we say $\gamma$ starts (ends) at
$\mathcal{A}$ if its first (last) vertex is an element of $\mathcal{A}$.

  * •

If $\mathcal{E}$ is a collection of primal edges, then we say $\gamma$ starts
(ends) at $\mathcal{E}$ if its first (last) vertex is an endpoint of some
element of $\mathcal{E}$.

  * •

If $\widehat{\mathcal{E}}$ is a collection of dual edges, then we say $\gamma$
starts (ends) at $\widehat{\mathcal{E}}$ if its first (last) vertex is a dual
neighbor of some element of $\widehat{\mathcal{E}}$.

A dual path $\zeta$ is closed if all its edges are closed.

  * •

If $\mathcal{A}\subset\mathbb{Z}^{2}$, then we say $\zeta$ starts (ends) at
$\mathcal{A}$ if its first (last) vertex is equal to
$x\pm(\frac{1}{2},\frac{1}{2})$ or $x\pm(\frac{1}{2},-\frac{1}{2})$ for some
$x\in\mathcal{A}$.

  * •

If $\mathcal{E}$ is a collection of primal edges, then we say $\gamma$ starts
(ends) at $\mathcal{E}$ if its first (last) vertex is a dual neighbor of some
element of $\mathcal{E}$.

  * •

If $\widehat{\mathcal{E}}$ is a collection of dual edges, then we say $\gamma$
starts (ends) at $\widehat{\mathcal{E}}$ if its first (last) vertex is an
endpoint of some element of $\widehat{\mathcal{E}}$.

To streamline the exposition, certain topological facts regarding circuits and
paths are presented in Sections 5 and 6 near the end of the present paper.

###  2.1. Constructing the geodesic

In the present paper, we are concerned with bounding the length of the
shortest geodesic. This is achieved by obtaining an upper bound for the length
of a specific choice of geodesic that is carefully constructed. Proposition
2.5 below states the properties of the construction that we need. Rigorously
constructing this geodesic requires technical topological care, which is done
in Section 7. Here, we outline how this geodesic is constructed.

Let $\mathcal{A}$ and $\mathcal{B}$ be finite, connected, disjoint sets of
vertices in $\mathbb{Z}^{2}$. We first construct a sequence of edge-disjoint
circuits, each of which either encloses $\mathcal{A}$ or $\mathcal{B}$. Let
$\mathcal{I}_{1}$ be the innermost open circuit enclosing $\mathcal{A}$ such
that $\mathcal{B}$ is contained in $\mathsf{ext}(\mathcal{I}_{1})$ (if such a
circuit exists). Next, let $\mathcal{I}_{2}$ be the innermost open circuit
enclosing and edge disjoint from $\mathcal{I}_{1}$ such that $\mathcal{B}$ is
contained in $\mathsf{ext}(\mathcal{I}_{2})$. Continue this way, obtaining a
finite sequence $\mathcal{I}_{1},\ldots,\mathcal{I}_{L}$ of nested open
circuits, until we can no longer create any more such circuits. Then, let
$\mathcal{I}_{L+1}$ be the outermost open circuit, edge-disjoint from
$\mathcal{I}_{L}$ that encloses $\mathcal{B}$ and keeps
$\mathsf{int}(\mathcal{I}_{L})$ in its exterior. From there, let
$\mathcal{I}_{L+2}$ be the outermost open circuit enclosing $\mathcal{B}$ that
is enclosed by and edge disjoint from $\mathcal{I}_{L+1}$. Continue this way,
obtaining a nested sequence of open circuits
$\mathcal{I}_{L+1},\mathcal{I}_{L+2},\ldots,\mathcal{I}_{P}$, each enclosing
$\mathcal{B}$. Section 7 proves that this sequence is well-defined on the
full-probability event $\Omega_{\infty}$.

With this sequence constructed, there exists a dual path $\zeta$, from
$\mathcal{A}$ to $\mathcal{B}$, that consists of exactly $P$ open edges. These
open edges are seen exactly when the path crosses each of the open circuits
$\mathcal{I}_{1},\ldots,\mathcal{I}_{P}$. We start with any geodesic $\gamma$
between $\mathcal{A}$ and $\mathcal{B}$. We then argue that for any closed
edge $e$ along the geodesic, the dual edge $e^{\star}$ belongs to a closed
circuit that either encloses $\mathcal{A}$ and keeps $\mathcal{B}$ in its
exterior or encloses $\mathcal{B}$ and keeps $\mathcal{A}$ in its exterior. By
rerouting the paths $\gamma$ and $\zeta$ around the open and closed circuits
if needed, we can choose $\zeta$ and $\gamma$ to be disjoint. Between any two
closed edges on $\gamma$, we may choose an alternate path for $\gamma$
consisting of open edges. We choose $\gamma$ to be the closest geodesic (in
the appropriate sense) to $\gamma$. By this choice, for each open edge
$e\in\gamma$, there exists a closed dual path from $e$ to $\zeta$. Then, the
edge $e$ satisfies a 3-arm event: one closed arm to $\zeta$ and two open arms
(by following the geodesic in each direction to the next closed edges).
Quantitatively controlling the size of this arm event is done in the next
section.

The properties of this construction are summarized in the following
proposition, whose proof may be found in Section 7.

######  Proposition 2.5.

Let $\mathcal{A}$ and $\mathcal{B}$ be finite disjoint connected subsets of
$\mathbb{Z}^{2}$. On the event $\Omega_{\infty}$, there exists a (possibly
empty) sequence of edge-disjoint open circuits
$\mathcal{I}_{1},\ldots,\mathcal{I}_{L},\mathcal{I}_{L+1},\ldots,\mathcal{I}_{P}$
satisfying the following:

  1. (i)

$\mathcal{A}\subseteq\mathsf{int}(\mathcal{I}_{1})\subseteq\mathsf{int}(\mathcal{I}_{2})\subseteq\cdots\subseteq\mathsf{int}(\mathcal{I}_{L})\subseteq\mathsf{int}(\mathcal{I}_{L})\cup\mathcal{I}_{L}\subseteq
B^{C}$.

  2. (ii)

$\mathsf{int}(\mathcal{I}_{L})\cap\mathsf{int}(\mathcal{I}_{L+1})=\varnothing$.

  3. (iii)

$A^{C}\supseteq\mathsf{int}(\mathcal{I}_{L+1})\cup\mathcal{I}_{L+1}\supseteq\mathsf{int}(\mathcal{I}_{L+1})\supseteq\mathsf{int}(\mathcal{I}_{L+2})\supseteq\cdots\supseteq\mathsf{int}(\mathcal{I}_{P})\supseteq
B$.

  4. (iv)

For $j\in\\{1,\ldots,P-2\\}$, the circuits $\mathcal{I}_{j}$ and
$\mathcal{I}_{j+2}$ are vertex-disjoint.

  5. (v)

For $j\in\\{1,\ldots,P\\}$ and every $e\in\mathcal{I}_{j}$, there exists a
dual path $\zeta_{e}$ to $\mathcal{A}$ that has exactly $j-1$ open edges–one
crossing each of the circuits $\mathcal{I}_{1},\ldots,\mathcal{I}_{j-1}$.

Furthermore, there exists a geodesic $\gamma$ from $\mathcal{A}$ to
$\mathcal{B}$ and a disjoint dual path $\zeta$ from $\mathcal{A}$ to
$\mathcal{B}$ satisfying the following properties (here we note that the paths
$\zeta_{e}$ in Item (v) are not necessarily disjoint from $\gamma$):

  1. (vi)

$\zeta$ has exactly $P$ open edges–one crossing each of the circuits
$\mathcal{I}_{1},\ldots,\mathcal{I}_{L},\mathcal{I}_{L+1},\ldots,\mathcal{I}_{P}$.

  2. (vii)

For each circuit $\mathcal{I}_{j}$, let $x$ and $y$ be the first and last
vertices from $\gamma$ on that circuit. Then, the portion of $\gamma$ between
$x$ and $y$ lies entirely on $\mathcal{I}_{j}$. If $\mathcal{I}_{j}$ and
$\mathcal{I}_{j+1}$ are not vertex-disjoint, then the path $\gamma$ crosses
from $\mathcal{I}_{j}$ to $\mathcal{I}_{j+1}$ via a common vertex, and
$\gamma$ contains no edges lying strictly between the circuits.

  3. (viii)

If $\gamma_{R}$ is the geodesic corresponding to $\mathcal{A}=\\{0\\}$ and
$\mathcal{B}_{R}=\partial B_{R}$ for $R\geq 0$, then the sequence $\gamma_{R}$
can be chosen so that the portion of $\\{\gamma_{R}\\}_{R\in\mathbb{Z}}$
between successive open circuits does not depend on $R$ (while the portion
between the last circuit in $\partial B_{R}$ and the boundary does depend on
$R$).

  4. (ix)

From every open edge $e\in\gamma$ with
$e\notin\mathcal{I}_{1}\cup\cdots\cup\mathcal{I}_{L}\cup\mathcal{I}_{L+1}\cup\cdots\cup\mathcal{I}_{P}$,
there exists a closed path from $e$ to $\zeta$.

  5. (x)

The dual of each closed edge along $\gamma$ belongs to a closed circuit
$\mathcal{U}$ that either contains $\mathcal{A}$ in its interior and
$\mathcal{B}$ in its exterior, or vice versa. The circuit $\zeta$ does not
contain the dual of any other edges along $\gamma$.

  6. (xi)

With $\\{0,1\\}$-valued edge-weights, the closed circuits $\mathcal{U}$ from
Item (x) can be chosen to form a edge-disjoint collection
$\mathcal{U}_{1},\ldots,\mathcal{U}_{V}$. The union of the circuits
$\mathcal{I}_{1},\ldots,\mathcal{I}_{P}$ and the circuits
$\mathcal{U}_{1},\ldots,\mathcal{U}_{V}$ forms a sequence
$\mathcal{C}_{1},\ldots,\mathcal{C}_{K}$, which is ordered so that, for some
index $W\in\\{0,\ldots,K\\}$,
$\mathsf{int}(\mathcal{C}_{W})\cap\mathsf{int}(\mathcal{C}_{W+1})=\varnothing$
(with the convention
$\mathsf{int}(\mathcal{C}_{0})=\mathsf{int}(\mathcal{C}_{K+1})=\varnothing)$,
and we have the following inclusions

$\mathcal{A}\subseteq\mathsf{int}(\mathcal{C}_{1})\subseteq\cdots\subseteq\mathsf{int}(\mathcal{C}_{W}),\quad\text{and}\quad\mathsf{int}(\mathcal{C}_{W+1})\supseteq\cdots\mathsf{int}(\mathcal{C}_{K})\supseteq
B.$

Figure 3 shows an example of two circuits $\mathcal{I}_{i},\mathcal{I}_{i+1}$
that are not vertex-disjoint. The only way that this can happen is if the two
circuits touch at a corner. Since each vertex is only seen in four edges on
$\mathbb{Z}^{2}$, $\mathcal{I}_{i}$ and $\mathcal{I}_{i+2}$ must always be
vertex-disjoint, as stated in Item (ii).

Figure 3. Two primal circuits (one in solid, the other dashed) which are edge-
disjoint but not vertex-disjoint, and such that one encloses the other.

##  3\. Proofs for Bernoulli edge-weights

In this Section, we prove Theorems 1.1 and 1.2. So we assume the edge-weights
have the Bernoulli($\frac{1}{2}$) distribution from (1.4). In this case, all
closed edges have weight $1$, and so the passage time along any path is simply
the number of closed edges it includes. Hence the passage time between two
sets $\mathcal{A},\mathcal{B}\subset\mathbb{Z}^{2}$ is the minimal number of
closed edges that are encountered by a path starting at $\mathcal{A}$ and
ending at $\mathcal{B}$. By Proposition 2.5(x), that number of closed edges is
also the number of edge-disjoint closed circuits
$\mathcal{U}_{1},\ldots,\mathcal{U}_{V}$ that separate $\mathcal{A}$ and
$\mathcal{B}$.

Throughout our arguments, the symbols $c$ and $C$ will denote positive
constants that are chosen sufficiently small or sufficiently large,
respectively. Their values may change from line to line, but they will never
depend on the radii on arm events.

###  3.1. Arm events along the modified geodesic.

We start by defining the open and closed arms which form the arm events. For
an edge $e$, Let $x_{e}$ denote the end point of $e$ which has the smaller
norm, and define $B_{b}(e)=x_{e}+B_{b}$. For integers $a<b$, define

$A[a,b)(e)=\\{x\in\mathbb{R}^{2}:a\leq|x-x_{e}|_{\infty}<b\\}.$

Fix two boxes $B_{a}$ and $B_{b}$ where $a,b$ are integers with $1\leq a\leq
b$. Recall the definition of “start” and “end” from Definition 2.4, and we
define

  * •

a (primal) open arm between $\partial B_{a}$ and $\partial B_{b}$ is a primal
open path inside $A[a,b)$ (except for the ending point) that starts on
$\partial B_{a}$ and ends on $\partial B_{b}$, and

  * •

a (dual) closed arm between $\partial B_{a}$ and $\partial B_{b}$ is a dual
closed path inside $A[a,b)$ that starts at $\partial B_{a}$ and ends at
$\partial B_{b}$. Note $\partial B_{a}$ and $\partial B_{b}$ are both on the
primal lattice, so we invoke Definition 2.4 here.

For $k\in\mathbb{Z}_{>0}$, a $k$-arm event event between $\partial B_{a}$ and
$\partial B_{b}$ is the event that there exist $k$ open or closed arms between
$\partial B_{a}$ and $\partial B_{b}$. If the arms are all open or all closed,
we call this monochromatic $k$-arm event. Otherwise, it will be called a
polychromatic $k$-arm event. We will also use polychromatic 3-arm events
starting from a single edge instead of a box. Fix the edge
$e=\\{(0,0),(1,0)\\}$, recall a $3$-arm event starting from $e$ up to
$\partial B_{a}$ means there exist two open arms that start at $(0,0)$ and
$(1,0)$ and end at $\partial B_{b}$, and a closed arm inside $B_{b}$ starting
at $(\tfrac{1}{2},\tfrac{1}{2})$ or $(\tfrac{1}{2},-\tfrac{1}{2})$ and ending
at $\partial B_{b}$. Note the above definitions immediately extend to
$x+B_{a},x+B_{b}$, $x+(1,0)$, and $x+(0,1)$ for any $x\in\mathbb{Z}^{2}$.

Next, let us introduce notation for the probabilities of arm events. For
$a<b$, let $\pi_{k}^{\prime}(a,b)$ and $\pi_{k}^{\prime\prime}(a,b)$ denote
the probability of a $k$-arm event between $\partial B_{a}$ and $\partial
B_{b}$ where all the arms are open or closed, respectively. The probability of
the polychromatic $k$-arm event between $\partial B_{a}$ and $B_{b}$ with
$k-1$ open arms and $1$ closed arm will be denoted as $\pi_{k}(a,b)$. We only
define this single probability for polychromatic $k$-arm events because the
probabilities for different polychromatic color sequences are comparable up to
absolute constants [22]. Lastly, the probability of a polychromatic 3-arm
event described above from the edge $e$ to $\partial B_{b}$ described above
will be denoted as $\pi_{3}(b)$.

We start by stating three propositions for the arm events. The first
proposition is known as quasi-multiplicativity, which is widely used in near-
critical percolation.

######  Proposition 3.1.

[21, Prop. 17] Fix $j\geq 1$ and let $n_{0}=n_{0}(j)$ be the smallest integer
such that $\pi_{j}(n_{0},n_{0}+1)>0$. Then there exists $C=C(j)>0$ such that
for all $n_{0}(j)\leq n_{1}<n_{2}<n_{3}$,

$\displaystyle\begin{split}\pi_{j}(n_{1},n_{3})\leq\pi_{j}(n_{1},n_{2})\pi_{j}(n_{2},n_{3})\leq
C\pi_{j}(n_{1},n_{3}).\end{split}$ (3.1a) In the case $j=3$ we have, for
$1\leq n_{1}\leq n_{2}$,
$\displaystyle\begin{split}\pi_{3}(n_{2})\leq\pi_{3}(n_{1})\pi_{3}(n_{1},n_{2})\leq
C\pi_{3}(n_{2}).\end{split}$ (3.1b)

######  Remark 3.2.

It follows from the Russo–Seymour–Welsh (RSW) theorem [13, Chap. 6] that for
any fixed constant $\alpha>1$, there is some constant $c=c(\alpha,j)>0$ such
that $\pi_{j}(n,\lceil\alpha n\rceil)\geq c$ for all $n\geq 1$. Combining this
observation with quasi-multiplicativity, one obtains

$\displaystyle\begin{split}\pi_{j}(1,n)\leq\pi_{j}(1,n)\frac{\pi_{j}(n,\lceil\alpha
n\rceil)}{c}\readlist*\mylist{quasi_{j}}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}C\pi_{j}(1,\lceil\alpha
n\rceil).\end{split}$ (3.2)

The second result is an upper bound on the 1-arm probability. It follows from
the RSW theorem.

######  Proposition 3.3.

[13, Lem. 8.5] There exists $c>0$ such that

$\displaystyle\begin{split}\pi_{1}(a,b)\leq(b/a)^{-c}\quad\text{for all $b\geq
a\geq 1$.}\end{split}$ (3.3)

Meanwhile, the polychromatic 5-arm exponent is known.

######  Proposition 3.4.

[21, Thm. 24(3)] There exist constants $c,C>0$ such that

$\displaystyle\begin{split}c(b/a)^{-2}\leq\pi_{5}(a,b)\leq
C(b/a)^{-2}\quad\text{for all $b\geq a\geq 1$}.\end{split}$ (3.4)

######  Remark 3.5.

It should be noted that [21] works on the triangular lattice instead of the
square lattice. On the triangular lattice, the polychromatic arm exponents are
known exactly: $\pi_{1}(1,R)=R^{-5/48+o(1)}$ [19] while
$\pi_{j}(1,R)=R^{-(j^{2}-1)/12+o(1)}$ for $j\geq 2$ [24]. But unlike the
derivations of the other arm exponents on the triangular lattice that use
conformal invariance, the proof of [21, Thm. 24(3)] works on the square
lattice as well; see the discussion above [21, Thm. 24]. Alternatively, one
can use the argument outlined in [27, Chap. 1] together with the equivalence
of polychromatic arm probabilities [22].

######  Remark 3.6.

Propositions 3.3 and 3.4 imply that for $j\in\\{2,3,4\\}$, there exists $c>0$
such that

$c(b/a)^{-(2-c)}\leq\pi_{j}(a,b)\leq(b/a)^{-c}\quad\text{for all $b\geq a\geq
1$}.$

Indeed, the upper bound is immediate from (3.3) since
$\pi_{j}(a,b)\leq\pi_{1}(a,b)$. For the lower bound, first use the Van den
Berg–Kesten (BK) inequality inequality to determine that
$\pi_{5}(a,b)\leq\pi_{1}(a,b)\pi_{4}(a,b)$, or equivalently
$\pi_{4}(a,b)\geq\pi_{5}(a,b)/\pi_{1}(a,b)$. Then use (3.3) and (3.4) to
conclude $\pi_{4}(a,b)\geq C(b/a)^{-(2-c)}$. The same lower bound follows for
$j=2$ and $j=3$ since then $\pi_{j}(a,b)\geq\pi_{4}(a,b)$.

In the case $j=4$, the upper bound can be improved to $(b/a)^{-(1+c)}$ for
some $c>0$ [26], but we will not need this.

######  Lemma 3.7.

There exists $C>0$ such that for all $L\in\mathbb{Z}_{>0}$,

$\sum_{i=1}^{L}L\pi_{3}(i)\leq CL^{2}\pi_{3}(L).$

###### Proof.

By quasi-multiplicativity (3.1b),

$\sum_{i=1}^{L}L\pi_{3}(i)\leq CL\pi_{3}(L)\sum_{i=1}^{L}\pi_{3}(i,L)^{-1}.$

By [4, Lemma 3.1] (in the first inequality below), there exists
$\alpha\in(0,1)$ such that

$\sum_{i=1}^{L}\pi_{3}(i,L)^{-1}\leq C\sum_{i=1}^{L}(i/L)^{\alpha-1}\leq
C\frac{\int_{1}^{L}x^{\alpha-1}dx}{L^{\alpha-1}}\leq CL.\qed$

Our final preparatory result gives an upper bound for the probability of a
sequence of arm events by a single polychromatic 3-arm event.

######  Proposition 3.8.

Fix any $L\in\mathbb{Z}_{>0}$. Let $k_{1},\ldots,k_{L}$ be any sequence of
integers greater than or equal to $4$. There exists a constant $C>0$,
depending only on $L$, such that for all $M\in\mathbb{Z}_{>0}$,

$\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{j_{2}=j_{1}+1}^{\lfloor\log_{2}M\rfloor}\dots\sum_{j_{L}=j_{L-1}+1}^{\lfloor\log_{2}M\rfloor}\pi_{3}(2^{j_{1}})\pi_{k_{1}}(2^{j_{1}+1},2^{j_{2}})\dots\pi_{k_{L}}(2^{j_{L}+1},M)\leq
C\pi_{3}(M).$

Furthermore, if $k_{1},\dots,k_{L-1}\geq 4$ and $k_{L}=3$, then for each
$0<\varepsilon<1$, there exists a positive constant $C_{\varepsilon}$

$\displaystyle\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{j_{2}=j_{1}+1}^{\lfloor\log_{2}M\rfloor}\dots\sum_{j_{L}=j_{L-1}+1}^{\lfloor\log_{2}M\rfloor}\pi_{3}(2^{j_{1}})\pi_{k_{1}}(2^{j_{1}+1},2^{j_{2}})\dots\pi_{k_{L-1}}(2^{j_{L-1}+1},2^{j_{L}})\pi_{3}(2^{j_{L}+1},M)\Big{(}\frac{2^{j_{L}+1}}{M}\Big{)}^{\varepsilon}$
$\displaystyle\leq C_{\varepsilon}\pi_{3}(M).$

###### Proof.

We prove the first statement, as the second statement essentially follows from
the same proof. By the BK inequality, $\pi_{4}(k)\leq\pi_{3}(k)\pi_{1}(k).$
Then, using quasi-multiplicativity (3.1) and Proposition 3.3, we have

$\displaystyle\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{j_{2}=j_{1}+1}^{\lfloor\log_{2}M\rfloor}\dots\sum_{j_{L}=j_{L-1}+1}^{\lfloor\log_{2}M\rfloor}\pi_{3}(2^{j}_{1})\pi_{k_{1}}(2^{j_{1}+1},2^{j_{2}})\cdots\pi_{k_{L}}(2^{j_{L}+1},M)$
$\displaystyle\leq\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{j_{2}=j_{1}+1}^{\lfloor\log_{2}M\rfloor}\dots\sum_{j_{L}=j_{L-1}+1}^{\lfloor\log_{2}M\rfloor}\pi_{3}(2^{j}_{1})\pi_{4}(2^{j_{1}+1},2^{j_{2}})\cdots\pi_{4}(2^{j_{L}+1},M)$
$\displaystyle\leq\pi_{3}(M)\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{j_{2}=j_{1}+1}^{\lfloor\log_{2}M\rfloor}\cdots\sum_{j_{L}=j_{L-1}+1}^{\lfloor\log_{2}M\rfloor}\pi_{1}(2^{j_{1}+1},2^{j_{2}})\cdots\pi_{1}(2^{j_{L}+1},M)$
$\displaystyle\leq
C\pi_{3}(M)\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{j_{2}=j_{1}+1}^{\lfloor\log_{2}M\rfloor}\cdots\sum_{j_{L}=j_{L-1}+1}^{\lfloor\log_{2}M\rfloor}\Big{(}\frac{2^{j_{1}+1}}{2^{j_{2}}}\Big{)}^{\varepsilon}\dots\Big{(}\frac{2^{j_{L}+1}}{M}\Big{)}^{\varepsilon}$
$\displaystyle\leq
C2^{L}\pi_{3}(M)\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{j_{2}=j_{1}+1}^{\lfloor\log_{2}M\rfloor}\cdots\sum_{j_{L}=j_{L-1}+1}^{\lfloor\log_{2}M\rfloor}\Big{(}\frac{2^{j_{1}+1}}{M}\Big{)}^{\varepsilon}$
$\displaystyle\leq
C2^{L}\pi_{3}(M)\sum_{j_{1}=0}^{\lfloor\log_{2}M\rfloor}(\lfloor\log_{2}M\rfloor-
j_{1}+1)^{L-1}\Big{(}\frac{2^{j_{1}+1}}{M}\Big{)}^{\varepsilon}\quad$
$\displaystyle\leq
C2^{L}\pi_{3}(M)\int_{0}^{\lfloor\log_{2}M\rfloor}(\lfloor\log_{2}M\rfloor-x+1)^{L-1}\Big{(}\frac{2^{x+1}}{M}\Big{)}^{\varepsilon}\,dx\quad(\text{then,
let $x=\log_{2}M-z-1$})$
$\displaystyle=C2^{L}\pi_{3}(M)\int_{-1}^{\lfloor\log_{2}M\rfloor-1}(z+2)^{L-1}\Big{(}\frac{1}{2^{z}}\Big{)}^{\varepsilon}\,dz$
$\displaystyle\leq
C2^{L}\pi_{3}(M)\int_{-1}^{\infty}(z+2)^{L-1}\Big{(}\frac{1}{2^{z}}\Big{)}^{\varepsilon}\,dz\leq
C_{L}\pi_{3}(M).\qed$

We state one last needed result from [25]. Recall that $\pi^{\prime}$
indicates an arm event where all arms are open, and $\pi^{\prime\prime}$
indicates an arm event where all arms are closed.

######  Proposition 3.9.

[25, Lem. 2.3] On the square lattice $\mathbb{Z}^{2}$, there exists
$\varepsilon>0$ and a large positive integer $r_{0}$ such that for any $1\leq
R_{1}<R_{2}$,

$\pi^{\prime}_{r_{0}}(R_{1},R_{2})\leq\pi_{3}(R_{1},R_{2})\Big{(}\frac{R_{1}}{R_{2}}\Big{)}^{\varepsilon}.$

By duality, it also holds that

$\pi^{\prime\prime}_{r_{0}}(R_{1},R_{2})\leq\pi_{3}(R_{1},R_{2})\Big{(}\frac{R_{1}}{R_{2}}\Big{)}^{\varepsilon}.$

Notice that the 3-arm event in (1.5) and (1.6) requires arms that reach all
the way to distance $R$. But the arms in Proposition 2.5 only extend as far as
the nearby circuits. To actually furnish the quantity $\pi_{3}(R)$ from these
shorter arms, we define a larger family of arm events between annuli, and then
make a patching argument. The goal is to establish the following theorem.

######  Theorem 3.10.

Assume (1.4). Let $\mathcal{A}$ and $\mathcal{B}$ be finite, connected,
disjoint subsets of $\mathbb{Z}^{2}$. Let $\gamma$ be the geodesic between
$\mathcal{A}$ and $\mathcal{B}$ defined in Proposition 2.5. There exists a
positive constant $C$ such that for each edge $e\in E(\mathbb{Z}^{2})$,

$\mathbb{P}(e\in\gamma)\leq C\pi_{3}(M)$

where $M=\min\\{d(e,\mathcal{A}),d(e,\mathcal{B})\\}.$

To prove this inequality, we split the event $\\{e\in\gamma\\}$ into various
disjoint events listed below in Lemma 3.11. First define the event

$D_{0}=\\{\text{the set of circuits $\mathcal{C}_{1},\ldots,\mathcal{C}_{K}$
in Proposition \ref{prop:geod_cons}\ref{itm:Csequence} is nonempty}\\}.$

######  Lemma 3.11.

Assume (1.4). Let $M=\min\\{d(e,\mathcal{A}),d(e,\mathcal{B})\\}.$ There
exists an absolute positive constant $C$ (independent of
$\operatorname{dist}(\mathcal{A},\mathcal{B})$,$|\mathcal{A}|$,$|\mathcal{B}|$)
such that

$\displaystyle\;\mathbb{P}(\\{e\in\gamma\\}\cap D_{0}^{\mathrm{c}})\leq
C\pi_{3}(M)$ (3.5) $\displaystyle\;\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$
is between two open circuits $\mathcal{C}_{k}$, $\mathcal{C}_{k+1}$}\\}\cap
D_{0})\leq C\pi_{3}(M)$ (3.6)
$\displaystyle\;\mathbb{P}(\\{e\in\gamma\\}\cap\\{e\in\cup_{k=1}^{K}\mathcal{C}_{k})\\}\cap\\{\tau_{e}=0\\}\cap
D_{0})\leq C\pi_{3}(M)$ (3.7)
$\displaystyle\begin{rcases*}\mathbb{P}(\\{e\in\gamma\\}\cap\\{\tau_{e}=1\\}\cap
D_{0})\\\ \mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is between two closed
circuits $\mathcal{C}_{k}$, $\mathcal{C}_{k+1}$}\\}\cap D_{0})\\\
\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is between two circuits
$\mathcal{C}_{k}$, $\mathcal{C}_{k+1}$, one open, one closed}\\}\cap D_{0})\\\
\end{rcases*}\leq C\pi_{3}(M)$ (3.8)
$\displaystyle\begin{rcases*}\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is
between $\mathcal{A}$, $\mathcal{C}_{1}$}\\}\cap D_{0})\\\
\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is between $\mathcal{C}_{K}$,
$\mathcal{B}$}\\}\cap D_{0})\end{rcases*}\leq C\pi_{3}(M)$ (3.9)

In the remainder of this subsection, we prove Lemma 3.11.

###### Proof of Lemma 3.11.

By quasi-multiplicativity (3.1b), we have $\pi_{3}(M)\asymp\pi_{3}(1,M)$, and

$\pi_{3}(1,2^{\lfloor\log_{2}M\rfloor})\leq C\pi_{3}(1,M).$

Thus, it suffices to bound the events in our lemma by
$\pi_{3}(1,2^{\lfloor\log_{2}M\rfloor})$ instead of $\pi_{3}(M)$.

We can start by observing that (3.5) is a direct consequence of Proposition
2.5: On the event where $e\in\gamma$ and the sequence
$\mathcal{C}_{1},\ldots,\mathcal{C}_{K}$ is empty, Proposition 2.5(x) implies
that we can obtain two open arms to distance $2^{\lfloor\log_{2}M\rfloor}$ by
following the geodesic from $e$ in both directions until reaching
$\mathcal{A}$ and $\mathcal{B}$, respectively. On the other hand, the closed
arm to distance $2^{\lfloor\log_{2}M\rfloor}$ can be obtained using Item (ix)
of the same proposition. Specifically, we take a closed dual path from $e$ to
$\zeta$ and then follow the closed dual path $\zeta$ to either $\mathcal{A}$
or $\mathcal{B}$.

$\mathcal{A}$$e$$\zeta$$\mathcal{B}$$\mathcal{C}_{k}$$\mathcal{C}_{k+1}$
Figure 4. An illustration of the case when $e\in\gamma$ is between two open
circuits $\mathcal{C}_{k}$ and $\mathcal{C}_{k+1}$ (light/thick). There exists
a dual path $\zeta$ (dark/thin) between $\mathcal{A}$ and $\mathcal{B}$. The
open dual edges in $\zeta$ are contained in the collection of annuli
$\\{A_{i_{j}}\\}$, which are shown in light gray.

Next, we show (3.6). For an illustration of the argument, see Figure 4. Take
an open edge $e\in\gamma$ lying between two consecutive open circuits
$\mathcal{C}_{k},\mathcal{C}_{k+1}$. Proposition 2.5(vii) implies that
$\mathcal{C}_{k}$ and $\mathcal{C}_{k+1}$ are vertex-disjoint. Two arms to
distance $2^{\lfloor\log_{2}M\rfloor}$ are obtained by following $\gamma$ in
different directions until reaching $\mathcal{C}_{k},\mathcal{C}_{k+1}$,
respectively, then following those circuits around the origin. Proposition
2.5(xi) guarantees that all closed edges on $\gamma$ cross one of the closed
circuits in the sequence $\mathcal{C}_{1},\ldots,\mathcal{C}_{K}$, so since
$\mathcal{C}_{k}$ and $\mathcal{C}_{k+1}$ are open, these two arms to distance
$2^{\lfloor\log_{2}M\rfloor}$ must be open.

Next, let $\zeta$ be the dual path from $\mathcal{A}$ to $\mathcal{B}$
constructed in Proposition 2.5. By Proposition 2.5(ix), there exists a closed
dual path which connects $\zeta$ and $e$. Furthermore, the edges along $\zeta$
are mostly closed; the only open edges are places where $\zeta$ crosses the
open circuits $\mathcal{I}_{1},\dots\mathcal{I}_{P}$ among the circuits
$\mathcal{C}_{1},\dots,\mathcal{C}_{K}$ defined in Proposition 2.5.

Now, consider a sequence of annuli around $e$, labelled as

$\displaystyle A_{i}^{(e)}$ $\displaystyle=A[2^{i},2^{i+1})(e)\text{ where
$i=0,\dots,\lfloor\log_{2}M\rfloor-1$},$ (3.10)

with the convention that a subset of $\mathbb{R}^{2}$ contains an edge if it
contains the midpoint of the edge. We now extract a (possibly empty)
subsequence $\\{i_{j}\\}_{j=1}^{J}$ such that each of these annuli contains at
least one dual open edge along $\zeta$. In addition, by ignoring the last few
terms of this subsequence, we may require the random variable $J$ to be no
greater than $100r_{0}$, where $r_{0}$ is the large positive integer from
Lemma 3.9.

First, suppose $J=0$, meaning the collection $\\{i_{j}\\}$ is empty. Then,
there is a polychromatic 3-arm event from $e$ up to distance
$2^{\lfloor\log_{2}M\rfloor}$. Recall the two open arms exist because $e$ is
connected to $\mathcal{C}_{k}$ and $\mathcal{C}_{k+1}$ via the geodesic
$\gamma$. The closed dual arm is obtained by following the closed dual path
from $e$ to $\zeta$, then following $\zeta$ towards $\mathcal{A}$. Thus,

$\displaystyle\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is between open
circuits $\mathcal{C}_{k}$, $\mathcal{C}_{k+1}$
}\\}\cap\\{J=0\\})\leq\pi_{3}(2^{\lfloor\log M\rfloor}).$ (3.11)

Next, suppose that $J=1$. In this case, we have a polychromatic 3-arm event
from $\partial B_{1}{(e)}$ to $\partial B_{2^{{i}_{1}}}(e)$, where the
configuration of these three arms is the same as the case when $J=0$, except
that the arms only reach until the dual arm meets an open edge inside
$A_{i_{1}}^{(e)}$. Let us consider the arm events outside of $\partial
B_{2^{{i}_{1}+1}}(e)$. By assumption, $\zeta$ contains an open edge inside
$B_{2^{{i}_{1}+1}}(e)$ at some vertex. That open edge is contained in an open
circuit $\mathcal{I}_{j}$ for some $j$. Then, between $\partial
B_{2^{{i}_{1}+1}}(e)$ and $\partial B_{2^{\lfloor\log_{2}M\rfloor}}(e)$, there
are two open paths formed by the open circuit(s) which meets $\zeta$ in
$B_{2^{{i}_{1}+1}}(e)$, and two closed paths from $\zeta$. Using asymptotic
equivalence of polychromatic arm events ([22]), there exists a constant $C$
(changing from line to line) so that

$\displaystyle\quad\;\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is between
open circuits $\mathcal{C}_{k}$, $\mathcal{C}_{k+1}$ }\\}\cap\\{J=1\\})$
(3.12) $\displaystyle\leq
C\sum_{{i}_{1}=0}^{\lfloor\log_{2}M\rfloor}\pi_{3}(1,2^{{i}_{1}})\pi_{4}(2^{{i}_{1}+1},2^{\lfloor\log_{2}M\rfloor})$
$\displaystyle\leq
C\sum_{{i}_{1}=1}^{\lfloor\log_{2}M\rfloor}\pi_{3}(1,2^{{i}_{1}})\pi_{4}(2^{{i}_{1}},2^{\lfloor\log_{2}M\rfloor})\leq
C\pi_{3}(1,2^{\lfloor\log_{2}M\rfloor}),$

where the last two inequalities follow from quasi-multiplicativity (3.1a) and
Proposition 3.8, respectively.

Now, suppose $J\geq 2$. Again, we have a polychromatic 3-arm event from
$B_{1}{(e)}$ to $\partial B_{2^{{i}_{1}}}(e)$, just as in the case when $J=1$.
We now describe arm events between the successive annuli. Fixing $j$ such that
$1\leq j\leq j+1\leq{J}$, we show there are at least $\max\\{j,4\\}$
polychromatic arms between $\partial B_{2^{{i}_{j}+1}}(e)$ and $\partial
B_{2^{{i}_{j+1}}}(e)$. To see this, by the definition of $M$ in Theorem 3.10
and the construction of the subsequence $\\{i_{j}\\}$ below (3.10),
$B_{2^{{i}_{j}+1}}(e)$ intersects at least $j$ edge-disjoint open circuits.
And each pair of circuits gives at least two open arms to distance $M$ since
$\mathcal{I}_{i}$ and $\mathcal{I}_{i+2}$ are always disjoint (Proposition
2.5(iv)). The remaining two closed arms are obtained from $\zeta$ going from
$\partial B_{2^{{i}_{j}+1}}(e)$ toward $\mathcal{A}$ and $\mathcal{B}$. They
form two closed dual paths that connect to $\partial B_{2^{{i}_{j+1}}}(e)$.
This uses again the fact that $\zeta$ must intersect $B_{2^{{i}_{j}+1}}(e)$ by
the definition of the subsequence$\\{i_{j}\\}$.

To summarize, we have shown that on the event

$\\{e\in\gamma\\}\cap\\{\text{$e$ is between open circuits $\mathcal{C}_{k}$,
$\mathcal{C}_{k+1}$ }\\}\cap\\{J\geq 2\\},$

the following holds:

there is a polychromatic 3-arm event between $B_{1}{(e)}$ and $\partial
B_{2^{i_{1}}}(e)$, and there are at least $\max\\{j,4\\}$ polychromatic arms
between $\partial B_{2^{{i}_{j}+1}}(e)$ (3.13) and $\partial
B_{2^{{i}_{j+1}}}(e)$ for $1\leq j\leq J$.

Finally, we split the situation into two subcases: whether $J\leq 10r_{0}$ or
otherwise. If $J\leq 10r_{0}$, there cannot be any open dual edges in the
portion of $\zeta$ between $B_{2^{{i}_{J}+1}}(e)$ and
$B_{2^{\lfloor\log_{2}M\rfloor}}(e)$. Hence, there must be two closed dual
paths from $B_{2^{{i}_{J}+1}}(e)$ to $B_{2^{\lfloor\log_{2}M\rfloor}}(e)$,
obtained by following $\zeta$ toward $\mathcal{A}$ and $\mathcal{B}$. At the
same time, there must also be two open paths from $B_{2^{{i}_{J}+1}}(e)$ to
$B_{2^{\lfloor\log_{2}M\rfloor}}(e)$ from by an open circuit as $J\geq 2$.
Hence, we have the following, where the first inequality comes from (3.13),
and the second inequality comes from Proposition 3.8,

$\displaystyle\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is between open
$\mathcal{C}_{k}$, $\mathcal{C}_{k+1}$ }\\}\cap\\{2\leq J\leq 10r_{0}\\})$
(3.14)
$\displaystyle\leq\sum_{{i}_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{{i}_{2}={i}_{1}+1}^{\lfloor\log_{2}M\rfloor}\hbox
to10.00002pt{.\hss.\hss.}\sum_{{i}_{{J}}={i}_{{J}-1+1}}^{\lfloor\log_{2}M\rfloor}\pi_{3}(1,2^{{i}_{1}})\pi_{4}(2^{{i}_{1}+1},2^{{i}_{2}})\hbox
to10.00002pt{.\hss.\hss.}\pi_{4}(2^{{i}_{{J}-1}+1},2^{{i}_{{J}}})\pi_{4}(2^{{i}_{{J}}+1},2^{\lfloor\log_{2}M\rfloor})$
$\displaystyle\leq C\pi_{3}(1,2^{\lfloor\log_{2}M\rfloor}).$

In the other case when $10r_{0}\leq J\leq 100r_{0}$, the number of open
circuits intersecting the box $B_{2^{{i}_{J}+1}}(e)$ is at least $10r_{0}$.
Thus, we have

$\displaystyle\mathbb{P}(\\{e\in\gamma\\}\cap\\{\text{$e$ is between open
$\mathcal{C}_{k}$, $\mathcal{C}_{k+1}$ }\\}\cap\\{10r_{0}\leq J\leq
100r_{0}\\})$ (3.15)
$\displaystyle\leq\sum_{{i}_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{i_{2}=i_{1}+1}^{\lfloor\log_{2}M\rfloor}\cdots\sum_{{i}_{{J}}={i}_{{J}-1}+1}^{\lfloor\log_{2}M\rfloor}\pi_{3}(1,2^{i_{1}})\pi_{4}(2^{i_{1}+1},2^{i_{2}})\hbox
to10.00002pt{.\hss.\hss.}\pi_{4}(2^{{i}_{{J}-1}+1},2^{{i}_{{J}}})\pi^{\prime}_{r_{0}}(2^{{i}_{{J}}+1},M)$
$\displaystyle\leq\sum_{{i}_{1}=0}^{\lfloor\log_{2}M\rfloor}\sum_{i_{2}=i_{1}+1}^{\lfloor\log_{2}M\rfloor}\cdots\sum_{{i}_{{J}}={i}_{{J}-1}+1}^{\lfloor\log_{2}M\rfloor}\pi_{3}(1,2^{{i}_{1}})\pi_{4}(2^{i_{1}+1},2^{i_{2}})\hbox
to10.00002pt{.\hss.\hss.}\pi_{4}(2^{{i}_{{J}-1}+1},2^{{i}_{{J}}})\pi_{3}(2^{{i}_{{J}}+1},M)\Big{(}\frac{2^{{i}_{{J}}+1}}{M}\Big{)}^{\varepsilon}$
$\displaystyle\leq C\pi_{3}(1,M)$

where the second inequality follows from Proposition 3.9, and the third
inequality follows from Proposition 3.8. Combining (3.11),(3.12),(3.14), and
(3.15) proves (3.6).

Next, we show (3.7). Suppose $e\in\gamma\cap\mathcal{C}_{k}$ for some
$k\in\\{1,\ldots,K\\}$. Since $\gamma$ lives on the primal lattice,
$\mathcal{C}_{k}$ is open. By Proposition 2.5(v), there exists a dual path
$\zeta_{e}$ between $e$ and $\mathcal{A}$ which consists of closed edges,
except for the places where it crosses the open circuits among
$\mathcal{C}_{1},\dots,\mathcal{C}_{K}$.

If $k=1$, or $k>1$ and $\mathcal{C}_{k-1}$ is closed, then there is a
polychromatic 3-arm event around $e$ to distance $M$. The two open arms come
from following $\mathcal{C}_{k}$ in both directions, and the closed arm is
obtained by following $\zeta_{e}$ to either $\mathcal{A}$, or to
$\mathcal{C}_{k-1}$ and then around either $\mathcal{A}$ or $\mathcal{B}$.

Otherwise, suppose $k>2$, and $\mathcal{C}_{k-1}$ is open. We will further
split the estimate into two cases, depending on $\mathcal{C}_{k+1}$:

  1. (1)

$\mathcal{C}_{k+1}$ does not exist, or it is open, or

  2. (2)

$\mathcal{C}_{k+1}$ is closed.

The arguments for these cases are similar to the proof of (3.6). However,
unlike $\zeta$ which connects $\mathcal{A}$ and $\mathcal{B}$, $\zeta_{e}$
only goes from $e$ to the set $\mathcal{A}$. However, the combination of
$\zeta_{e}$ and the portion of $\gamma$ between $e$ and $\mathcal{B}$ will
form a connection between $\mathcal{A}$ and $\mathcal{B}$. This will
essentially play the role of $\zeta$ in the previous estimate (3.6).

In case (1), let us again define the sequence of annuli $\\{A_{i}^{(e)}\\}$ as
in (3.10). Then, we extract a subsequence $\\{i_{j}\\}_{j=1}^{J}$ such that
each of these annuli contains at least one dual open edge along $\zeta_{e}$.
Again, we truncate the random variable $J$ to be no greater than $100r_{0}$.

If $J=0$, then we get a polychromatic 3-arm event from $B_{1}(e)$ to distance
$2^{\lfloor\log_{2}M\rfloor}$–two open arms from following $\mathcal{C}_{k}$
and a closed dual arm from following $\zeta_{e}$ to
$B_{2^{\lfloor\log_{2}M\rfloor}}(e)$. Next, we will assume $J\geq 1$. By
construction, there is a polychromatic 3-arm event from $B_{1}(e)$ to
$\partial B_{2^{i_{1}}}(e)$, constructed from the open circuit
$\mathcal{C}_{k}$ and the closed dual arm, $\zeta_{e}$. Next, we show that
there must be a polychromatic 4-arm arm event between $\partial
B_{2^{i_{1}+1}}(e)$ and $\partial B_{2^{i_{2}}}(e)$ (or $\partial
B_{2^{\lfloor\log_{2}{M}\rfloor}}(e)$ if $J=1$). First, because of
$\zeta_{e}$, there must be a closed arm. If we follow $\gamma$ from $e$ toward
$\mathcal{B}$, it will either encounter $\mathcal{C}_{k+1}$ ,which is assumed
to be open, or $\mathcal{B}$ if $\mathcal{C}_{k+1}$ does not exist. This gives
us another open arm. Lastly, because $B_{2^{i_{1}+1}}(e)$ contains an open
dual edge along $\zeta_{e}$, $\partial B_{2^{i_{1}+1}}(e)$ must intersect an
open circuit $\mathcal{C}_{i}$ for some $1\leq i\leq k-1$, and this gives us
two more open arms. We note that $\mathcal{C}_{i}$ in the previous sentence
and the segment of $\gamma$ must be disjoint by Proposition 2.5(vii), thus
they indeed give three vertex-disjoint open arms. Finally after this, for each
$2\leq j\leq J-1$, there will always be at least $\max\\{4,j\\}$ number of
polychromatic arms between $\partial B_{2^{i_{j}+1}}(e)$ and $\partial
B_{2^{i_{j+1}}}(e)$. The $4$ appears inside the maximum simply because the
three open arms from the previous step are actually to distance $M$, hence
paired with $\zeta_{e}$, there must be at least 4 arms. The $j$ here is
because there must be at least $j$ edge-disjoint open circuits intersecting
the box $B_{2^{i_{j}+1}}(e)$. With these, the rest of the arguments can be
read off starting at the argument after (3.13).

In case (2), compared to case (1), when choosing our subsequence
$\\{i_{j}\\}_{j=1}^{J}$, in addition to the annuli which cover the open edges
along $\zeta_{e}$, we also require them to cover the edge where $\gamma$
crosses $\mathcal{C}_{k+1}$, and we will denote this edge as $\widetilde{e}$.
Again, let us look at the arm event between $\partial B_{2^{i_{1}+1}}(e)$ and
$\partial B_{2^{i_{2}}}(e)$.

First, if $\widetilde{e}\in A_{i_{1}}$, then there are three closed arms and 2
open arms between $\partial B_{2^{i_{1}+1}}(e)$ and $\partial
B_{2^{i_{2}}}(e)$. The closed arms are from $\zeta_{e}$ and
$\mathcal{C}_{k+1}$, and the open arms are from $\mathcal{C}_{k}$. In addition
for $2\leq j\leq J-1$, between $\partial B_{2^{i_{j}+1}}(e)$ and $\partial
B_{2^{i_{j+1}}}(e)$, there must be at least $\max\\{j,4\\}$ number of
polychromatic arms. This gives the same statement as in case (1), and the rest
of the argument directly follows s before.

Otherwise if $\widetilde{e}\in A_{i_{j}}$ for some $j\geq 2$. Then, we have 1
closed arm, and 3 open arms between $\partial B_{2^{i_{1}+1}}(e)$ and
$\partial B_{2^{i_{2}}}(e)$, which follows from the same argument as in case
1. And again, between $\partial B_{2^{i_{j}+1}}(e)$ and $\partial
B_{2^{i_{j+1}}}(e)$, there must be at least $\max\\{j,4\\}$ number of
polychromatic arms. Because $B_{2^{i_{j}+1}}(e)$ must intersect at least $j$
edge-disjoint open circuits or $j-1$ edge-disjoint open circuits and a closed
circuit. This is again similar to (3.13), and the exact same argument would
follow as over there.

Next, we will show (3.8). The argument is again similar to (3.6). The first
difference is that $\gamma$ will now play the role of $\zeta$ from the proof
of (3.6). Let us again fix a sequence of annuli around $e$ just like in
(3.10). We extract a (possibly empty) subsequence $\\{i_{j}\\}_{j=1}^{J}$ such
that each of these annuli contains at least one closed edge along $\gamma$,
and again truncate $J$ to be no larger than $100r_{0}$.

The second difference is how we obtain our polychromatic 3-arm event from
$B_{1}(e)$ to $\partial B_{2^{i_{1}}}(e)$, or from $B_{1}(e)$ to $\partial
B_{2^{\lfloor\log_{2}M\rfloor}}(e)$ if $J=0$. In this case, the two open paths
are from $\gamma$. To see the closed path, if the edge $e$ is on a closed
circuit, then it is clear. Otherwise, by Proposition 2.5, $e$ is connected to
$\zeta$ by a closed path, then, we can follow $\zeta$ to $\partial
B_{2^{i_{1}}}(e)$, or to $\partial B_{2^{\lfloor\log_{2}M\rfloor}}(e)$ if
$J=0$.

We note that if $J=0$, we have the polychromatic 3-arm events described in the
paragraph above. If $J\geq 1$, the polychromatic 3-arm event from $B_{1}(e)$
to $\partial B_{2^{i_{1}}}(e)$ is also described in the previous paragraph.
And all the arm events outside of the box $B_{2^{i_{1}+1}}(e)$ are exactly the
same as in (3.6), except the words “open” and “closed” are exchanged (because
$\gamma$ now plays the role of $\zeta$ in (3.6)). This means we again have
(3.13), and the rest of the argument follows as the argument after (3.13). We
omit the details here.

For (3.9), without the loss of generality, we will assume that $e$ is between
$\mathcal{A}$ and $\mathcal{C}_{1}$, and the exact same argument can be
applied to the other case. First, note that there exists at least one open
path and one closed path from $e$ to distance $M$. The open path is because
$e$ is connected to $\mathcal{A}$ by $\gamma$. The closed path exists because
by Proposition 2.5(ix), there is a closed dual path from $e$ to $\zeta$, which
is connected to $\mathcal{A}$ by a closed path. Now, if $\mathcal{C}_{1}$ is
open, then we are done since this gives another open arm from $e$ reaching
distance $M$. Otherwise, suppose $\mathcal{C}_{1}$ is closed, then $\gamma$
must cross $\mathcal{C}_{1}$ at some edge $\widetilde{e}$. Recall the annuli
around $e$ defined in (3.10). Suppose $\widetilde{e}\not\in\cup A_{i}$, then
there will be a third open arm from $B_{1}(e)$ to distance
$2^{\lfloor\log_{2}M\rfloor}$, by following $\gamma$ toward $\mathcal{C}_{1}$.
By quasi-multiplicativity (3.1b), we are done. Otherwise if $\widetilde{e}\in
A_{i^{*}}$ for some random index $i^{*}\in[1,\lfloor\log_{2}M\rfloor]$, there
will be a polychromatic 3-arm event from $B_{1}(e)$ to $\partial
B_{2^{i_{*}}}(e)$ and a 4-arm polychromatic event from $\partial
B_{2^{i_{*}+1}}(e)$ to $\partial B_{2^{\lfloor\log_{2}M\rfloor}}(e)$. The
additional two arms are coming from $\mathcal{C}_{1}$, which must intersect
$B_{2^{\lfloor\log_{2}M\rfloor}}(e)$. Then, the same calculation from (3.12)
gives our desired estimate. ∎

We finish this section by proving Theorems 1.1 and 1.2.

###### Proof of Theorem 1.1.

Let $\gamma$ be the geodesic constructed in Proposition 2.5, where
$\mathcal{A}=\\{(0,0)\\}$, and $\mathcal{B}=\partial B_{R}$. Let $|\gamma|$ be
the number of edges in $\gamma$. Let $E(B_{R})$ be the set of edges with at
least one endpoint in the interior of $B_{R}$. Then, by Theorem 3.10 (in the
penultimate inequality), followed by Lemma 3.7,

$\displaystyle\quad\;\mathbb{E}[\mathcal{N}_{R}]\leq\mathbb{E}[|\gamma|]=\sum_{e\in
E(B_{R})}\mathbb{P}(e\in\gamma)$
$\displaystyle=\sum_{k=0}^{R}\sum_{\begin{subarray}{c}e\in E(B_{R})\\\
\operatorname{dist}(e,\\{0\\})\vee\operatorname{dist}(e,\partial
B_{R})=k\end{subarray}}\mathbb{P}(e\in\gamma)\leq
C\sum_{k=1}^{R}R\pi_{3}(k)\leq CR^{2}\pi_{3}(R).$

In the penultimate equality, we simply subsumed the $k=0$ term into the $k=1$
term, bounding $1$ by a constant times $\pi_{3}(1)$. ∎

###### Proof of Theorem 1.2.

Let $\mathfrak{b}$ be a box of side length $2(d+|\mathcal{A}|+|\mathcal{B}|)$
which encloses both $\mathcal{A}$ and $\mathcal{B}$, and let $x$ be its center
so that $\mathfrak{b}=B_{d+|\mathcal{A}|+|\mathcal{B}|}(x)$. Then, for $k\geq
0$, let $\mathfrak{b}_{k}=B_{2^{k}(d+|\mathcal{A}|+|\mathcal{B}|)}(x)$. For
$k\geq 1$, let $D_{k}$ be the event on which

  1. (i)

There exists an open circuit $\mathcal{C}$ in
$\mathfrak{b}_{2k-1}\setminus\mathfrak{b}_{2k-2}$,  and

  2. (ii)

There exists a closed dual circuit $\mathcal{D}$ in
$\mathfrak{b}_{2k}\setminus\mathfrak{b}_{2k-1}$.

The events $D_{k}$ are pairwise independent, and the RSW theorem implies that
there exists a constant $c>0$, independent of $d,|\mathcal{A}|$, and
$|\mathcal{B}|$ so that

$\mathbb{P}(D_{k})\geq 1-e^{-c}\qquad\forall k\geq 1.$ (3.16)

Next, we argue that on the event $D_{k}$, all geodesics from $\mathcal{A}$ to
$\mathcal{B}$ (and in particular, the geodesic in Proposition 2.5) are
contained in $\mathfrak{b}_{2k}$. To do this, let $\mathcal{C}$ and
$\mathcal{D}$ be a choice of the open and closed circuits guaranteed in the
event $D_{k}$, respectively. Let $\gamma$ be a geodesic from $\mathcal{A}$ to
$\mathcal{B}$. If $\gamma$ contains an edge $e$ whose relative interior lies
in $\mathsf{ext}(\mathcal{C})$, it must be open. Otherwise, since $\gamma$
both starts and ends in $\mathsf{int}(\mathcal{C})$, the edge $e$ lies along a
portion of $\gamma$ lying in $\mathsf{ext}(\mathcal{C})$ and connecting two
points of $\mathcal{C}$. An alternate path between the two points is obtained
by following the open circuit $\mathcal{C}$ between those points. This path
bypasses the closed edge $e$, so $\gamma$ cannot be a geodesic. Therefore, on
the event $D_{k}$, $\gamma$ cannot move outside $\mathfrak{b}_{2k}$, or else
it must cross $\mathcal{D}$ and therefore take a closed edge whose relative
interior lies in $\mathsf{ext}(\mathcal{C})$.

For $\omega\in\Omega$, let $T(\omega)=\min\\{k:\omega\in D_{k}\\}$. Then, for
each $\lambda>0$ and $n\geq 1$,

$\mathbb{P}(\mathcal{N}_{\mathcal{A},\mathcal{B}}\geq\lambda
d^{2}\pi_{3}(d))\leq\mathbb{P}(\mathcal{N}_{\mathcal{A},\mathcal{B}}\geq\lambda
d^{2}\pi_{3}(d),T\leq n)+\mathbb{P}(T>n).$ (3.17)

By (3.16), and independence of the events $\\{D_{k}\\}_{k\geq 1}$, for any
$n\geq 1$,

$\mathbb{P}(T>n)\leq\mathbb{P}\Bigl{(}\bigcap_{k=1}^{n}D_{k}^{c}\Bigr{)}=\prod_{k=1}^{n}(1-\mathbb{P}(D_{k}))\leq
e^{-cn}.$ (3.18)

On the other hand, letting $\gamma$ be the geodesic from $\mathcal{A}$ to
$\mathcal{B}$ in our construction and using Chebyshev’s inequality,

$\quad\;\mathbb{P}(\mathcal{N}_{\mathcal{A},\mathcal{B}}\geq\lambda
d^{2}\pi_{3}(d),T\leq
n)\leq\mathbb{P}(\mathcal{N}_{\mathcal{A},\mathcal{B}}\geq\lambda
d^{2}\pi_{3}(d),\gamma\subseteq\mathfrak{b}_{2n})\leq\frac{\mathbb{E}[\mathcal{N}_{\mathcal{A},\mathcal{B}}\mathds{1}(\gamma\subseteq\mathfrak{b}_{2n})]}{\lambda
d^{2}\pi_{3}(d)}.$ (3.19)

Let $\mathcal{S}(\mathfrak{b}_{k})$ be the set of edges in $\mathcal{B}_{k}$
that do not connect two vertices of $\mathcal{A}$ or two vertices of
$\mathcal{B}$. Using Lemma 3.11,

$\displaystyle\quad\;\mathbb{E}[\mathcal{N}_{\mathcal{A},\mathcal{B}}\mathds{1}(\gamma\subseteq\mathfrak{b}_{2n})]=\sum_{e\in\mathcal{S}(\mathfrak{b}_{2n})}\mathbb{P}(e\in\gamma)\leq
C\sum_{e\in\mathcal{S}(\mathfrak{b}_{2n})}\pi_{3}(\min(\operatorname{dist}(e,\mathcal{A}),\operatorname{dist}(e,\mathcal{B})))$
$\displaystyle=\sum_{k=1}^{4^{n+1}(d+|\mathcal{A}|+|\mathcal{B}|)}\sum_{e\in\mathcal{S}(\mathfrak{b}_{2n}):\operatorname{dist}(e,\mathcal{A})\wedge\operatorname{dist}(e,\mathcal{B})=k}\pi_{3}(k)\leq\sum_{k=1}^{4^{n+1}(d+|\mathcal{A}|+|\mathcal{B}|)}C(|\mathcal{A}|+|\mathcal{B}|)k\pi_{3}(k)$
$\displaystyle\leq
C(|\mathcal{A}|+|\mathcal{B}|)(4^{n}(d+|\mathcal{A}|+|\mathcal{B}|))^{2}\pi_{3}(4^{n+1}(d+|\mathcal{A}|+|\mathcal{B}|))\leq
C(4^{n}d)^{2}(|\mathcal{A}|+|\mathcal{B}|)^{3}\pi_{3}(d),$

where $C>0$ is a constant independent of $d,|\mathcal{A}|$, and
$|\mathcal{B}|$, and the second-to-last inequality follows by Lemma 3.7. In
the last equality, we have used $(d+|\mathcal{A}|+|\mathcal{B}|)\leq
2d(|\mathcal{A}|+|\mathcal{B}|)$, which holds because $d\geq 1$ and
$|\mathcal{A}|+|\mathcal{B}|\geq 2$. Then, using (3.17),(3.18), and (3.19),

$\mathbb{P}(\mathcal{N}_{\mathcal{A},\mathcal{B}}\geq\lambda
d^{2}\pi_{3}(d))\leq
e^{-cn}+\frac{C(4^{n}d)^{2}(|\mathcal{A}|+|\mathcal{B}|)^{3}\pi_{3}(d)}{\lambda
d^{2}\pi_{3}(d)}\leq
e^{-cn}+C(|\mathcal{A}|+|\mathcal{B}|)^{3}\frac{16^{n}}{\lambda}.$

The proof is complete upon setting
$n=\lfloor\frac{1}{2}\log_{16}\lambda\rfloor$. ∎

##  4\. Proofs for general edge-weights

Throughout this section, we assume the edge-weight distribution function $F$
satisfies (1.1) and additionally one of following conditions from Theorem 1.4:

$\displaystyle\limsup_{n\to\infty}F^{-1}(p_{2^{n+1}})/F^{-1}(p_{2^{n}})$
$\displaystyle<1,$ (4.1)
$\displaystyle\liminf_{n\to\infty}F^{-1}(p_{2^{2n}})/F^{-1}(p_{2^{n}})$
$\displaystyle>0,$ (4.2)

where $p_{R}$ is defined in (1.7).

###  4.1. Correlation length and verification of examples

The lemma below provides various properties of the correlation length
$L(\cdot)$ defined in Section 1.3.

######  Lemma 4.1.

The following statements hold.

  1. (a)

There exists $c>0$ such that

$cR\leq L(p_{R})\leq R\quad\text{for all $R\geq 1$.}$ (4.3)

  2. (b)

(Kesten’s scaling relation) There exists $c>0$ such that

$c\leq L(p)^{2}\pi_{4}(1,L(p))(p-p_{\mathrm{c}})\leq 1/c\quad\text{for all
$p>p_{\mathrm{c}}$.}$ (4.4)

  3. (c)

There exists $c>0$ such that

$c\Bigl{(}\frac{R}{r}\Bigr{)}^{-(2-c)}\leq\frac{p_{R}-p_{\mathrm{c}}}{p_{r}-p_{\mathrm{c}}}\leq(1/c)\Bigl{(}\frac{R}{r}\Bigr{)}^{-c}\quad\text{for
all $R\geq r\geq 1$}.$ (4.5)

  4. (d)

There exists $c>0$ such that

$cR^{-(2-c)}\leq p_{R}-p_{\mathrm{c}}\leq(1/c)R^{-c}\quad\text{for all $R\geq
1$}.$ (4.6)

###### Proof.

Part (a) follows from [12, eq. (2.10)].

Part (b) appears as [15, Prop. 34].

Part (c): Making using of the previous parts, we have

$\displaystyle\begin{split}\frac{p_{R}-p_{\mathrm{c}}}{p_{r}-p_{\mathrm{c}}}\readlist*\mylist{eq:scaling_{r}elation}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}C\frac{L(p_{r})^{2}\pi_{4}(1,L(p_{r}))}{L(p_{R})^{2}\pi_{4}(1,L(p_{R}))}\readlist*\mylist{eq:L_{p}{}_{n}}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}C\frac{r^{2}\pi_{4}(1,cr)}{(cR)^{2}\pi_{4}(1,R)}\readlist*\mylist{quasi_{c}}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}C\frac{r^{2}\pi_{4}(1,r)}{R^{2}\pi_{4}(1,R)}.\end{split}$
(4.7)

By analogous reasoning, we also have

$\displaystyle\begin{split}\frac{p_{R}-p_{\mathrm{c}}}{p_{r}-p_{\mathrm{c}}}\geq
c\frac{r^{2}\pi_{4}(1,r)}{R^{2}\pi_{4}(1,R)}.\end{split}$ (4.8)

Now study the ratio of $\pi_{4}$ quantities:

$\displaystyle\frac{1}{\pi_{1}(r,R)}\leq\frac{1}{\pi_{4}(r,R)}\readlist*\mylist{quasi_{j}}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}\frac{\pi_{4}(1,r)}{\pi_{4}(1,R)}\readlist*\mylist{quasi_{j}}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}\frac{C}{\pi_{4}(r,R)}\leq\frac{C\pi_{1}(r,R)}{\pi_{5}(r,R)},$

where the final inequality is due to the BK inequality. Inserting these
estimates into (4.7) and (4.8) yields

$\displaystyle
c\frac{r^{2}}{R^{2}\pi_{1}(r,R)}\leq\frac{p_{R}-p_{\mathrm{c}}}{p_{r}-p_{\mathrm{c}}}\leq
C\frac{r^{2}\pi_{1}(r,R)}{R^{2}\pi_{5}(r,R)}.$

Finally, use the bounds provided by Propositions 3.3 and 3.4:

$\displaystyle\frac{r^{2}}{R^{2}\pi_{1}(r,R)}\readlist*\mylist{ncb3}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\geq}}\frac{r^{2}}{R^{2}(r/R)^{c}}=\Big{(}\frac{R}{r}\Big{)}^{-(2-c)}$
while
$\displaystyle\frac{r^{2}\pi_{1}(r,R)}{R^{2}\pi_{5}(r,R)}\readlist*\mylist{dm8i}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}C\frac{r^{2}(R/r)^{-c}}{R^{2}((R/r)^{-2})}\leq(1/c)\Big{(}\frac{R}{r}\Big{)}^{-c}.$

The two previous displays together yield (4.5).

Part (d) follows from part (c) upon setting $r=1$. ∎

For completeness, we next check that the distributions in Example 1.5 have the
claimed properties.

###### Proof of Example 1.5.

Part (a): $F(t)=p_{\mathrm{c}}+Ct^{\alpha}$ for $t\in[0,h]$, where
$C,\alpha,h>0$. For $p>p_{\mathrm{c}}$ sufficiently close to $p_{\mathrm{c}}$,
we have $F^{-1}(p)=\bigl{(}(p-p_{\mathrm{c}})/{C}\bigr{)}^{1/r}$. So by (4.5),
there exists $c>0$ such that

$F^{-1}(p_{2^{n+1}})/F^{-1}(p_{2^{n}})\leq(1/c)2^{-c/r}<1\quad\text{for all
$n$ sufficiently large.}$

Hence (4.1) holds.

Part (b): $F(t)=p_{\mathrm{c}}$ for some $h>0$. Choose $h^{\prime}>h$ such
that $F(h^{\prime})>p_{\mathrm{c}}$. Then for all $p>p_{\mathrm{c}}$
sufficiently close to $p_{\mathrm{c}}$, we have $h\leq F^{-1}(p)\leq
h^{\prime}$. Hence (4.2) holds.

Part (c): $F(t)=p_{\mathrm{c}}+Ce^{-t^{-\alpha}}$ for $t\in(0,h]$, where
$C,\alpha,h>0$. Then for $p>p_{\mathrm{c}}$ sufficiently close to
$p_{\mathrm{c}}$, we have

$F^{-1}(p)=\Big{(}\frac{1}{\log(C/(p-p_{\mathrm{c}}))}\Big{)}^{1/\alpha}.$

In particular,

$\displaystyle\frac{F^{-1}(p_{2^{2n}})}{F^{-1}(p_{2^{n}})}=\Big{(}\frac{\log
C-\log(p_{2^{n}}-p_{\mathrm{c}})}{\log
C-\log(p_{2^{2n}}-p_{\mathrm{c}})}\Big{)}^{1/\alpha}.$

Since $p_{R}\searrow p_{\mathrm{c}}$ as $R\to\infty$, we have

$\displaystyle\liminf_{n\to\infty}\frac{F^{-1}(p_{2^{2n}})}{F^{-1}(p_{2^{n}})}=\Big{(}\liminf_{n\to\infty}\frac{\log(p_{2^{n}}-p_{\mathrm{c}})}{\log(p_{2^{2n}}-p_{\mathrm{c}})}\Big{)}^{1/\alpha}.$

So define $\beta_{n}$ implicitly by
$p_{2^{n}}-p_{\mathrm{c}}=(p_{2^{2n}}-p_{\mathrm{c}})^{\beta_{n}}$, which
makes

$\displaystyle\frac{\log(p_{2^{n}}-p_{\mathrm{c}})}{\log(p_{2^{2n}}-p_{\mathrm{c}})}=\beta_{n}.$

Therefore, in order to establish (4.2), it suffices to show that

$\displaystyle\liminf_{n\to\infty}\beta_{n}>0.$

For this observe that

$\displaystyle\big{(}c(2^{2n})^{-(2-c)})^{\beta_{n}}\readlist*\mylist{pn-
pcbd}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}(p_{2^{2n}}-p_{\mathrm{c}})^{\beta_{n}}=p_{2^{n}}-p_{\mathrm{c}}\readlist*\mylist{pn-
pcbd}\stackrel{{\scriptstyle\mbox{\footnotesize\foreachitem\x\in\mylist[]{,\eqref{\x}}}}}{{\leq}}(1/c)(2^{n})^{-c}.$

Isolating just the first and last expressions, we see that

$\displaystyle\beta_{n}(\log_{2}c-2n(2-c))$
$\displaystyle\leq\log_{2}(1/c)-nc$
$\displaystyle\implies\quad\liminf_{n\to\infty}\beta_{n}$
$\displaystyle\geq\frac{c}{2(2-c)}>0.$

Part (d): $F$ is such that $F^{-1}(p_{2^{n}})=e^{-\sqrt{n}}$ for each $n\geq
1$. We then have

$\displaystyle\frac{F^{-1}(p_{2^{n+1}})}{F^{-1}(p_{2^{n}})}=\frac{e^{-\sqrt{n+1}}}{e^{-\sqrt{n}}}\to
1\quad\text{as $n\to\infty$},$

as well as

$\displaystyle\frac{F^{-1}(p_{2^{2n}})}{F^{-1}(p_{2^{n}})}=\frac{e^{-\sqrt{2n}}}{e^{-\sqrt{n}}}\to
0\quad\text{as $n\to\infty$}.$

Therefore, neither (4.1) nor (4.2) holds. ∎

###  4.2. Bounds for passage times of geodesics across annuli

We start this subsection by citing the following result.

######  Lemma 4.2.

[7, Cor. 2.3] Let $F$ be the distribution function of a nonnegative random
variable with $F(0)=\frac{1}{2}$. Given an integer $K\geq 2$, there exists
$C>0$ such that, for all $n$ and $p$ with $L(p)\leq 2^{n}$,

$\mathbb{P}\Big{(}T_{(K)}(n)\geq\lambda
F^{-1}(p)\Bigl{(}\frac{2^{n}}{L(p)}\Bigr{)}^{2}\Big{)}\leq
e^{-C\lambda}+\exp\Bigl{(}-C\frac{2^{n}}{L(p)}\Bigr{)},\quad\text{ for
}\lambda\geq 0,$

where $T_{(K)}(n)$ is the minimal passage time between the left and right
sides of $[-K2^{n},K2^{n}]\times[-2^{n},2^{n}]$, among all paths that remain
in this rectangle.

In the next result, we use Lemma 4.2 to obtain a result about maximal passage
times of geodesics across annuli. Before making a definition, we make the
following technical convention. For $k\geq 1$, let $E_{k}$ be the set of edges
with both endpoints in the set $B_{2^{k+1}}\setminus B_{2^{k}}\cup\partial
B_{2^{k}}$. We also set $E_{0}$ to be the edges with both endpoints in
$B_{2}$. Given a path $\gamma$ connecting the origin to $\partial
B_{2^{n+1}}$, for $k\in\\{0,\ldots,n\\}$, we set

$T_{k}(\gamma)=\begin{cases}\sum_{e\in\gamma\cap(E_{0}\cup E_{1})}t_{e}&k=0\\\
\sum_{e\in\gamma\cap(E_{k-1}\cup E_{k}\cup E_{k+1})}t_{e}&1\leq k<n\\\
\sum_{e\in\gamma\cap(E_{n-1}\cup E_{n})}t_{e}&1\leq k=n.\end{cases}$

Then, for $0\leq k\leq n$, define

$T_{k,n}^{\max}=\sup_{\gamma\in\operatorname{Geo}(0,\partial
B_{2^{n+1}})}T_{k}(\gamma).$ (4.9)

######  Lemma 4.3.

Let $F$ be the distribution function of a nonnegative random variable with
$F(0)=\frac{1}{2}$. There exists $c>0$ such that for all $0\leq k\leq n$ and
$p>p_{\mathrm{c}}$ with $L(p)\leq 2^{k}$,

$\mathbb{P}\Big{(}T_{k,n}^{\max}\geq\lambda\Big{(}\dfrac{2^{k}}{L(p)}\Big{)}^{2}F^{-1}(p)\Big{)}\leq
e^{-c\lambda}+\frac{1}{c}\exp\Big{(}-c\dfrac{2^{k}}{L(p)}\Big{)}.$

$T_{k,n}^{\max}$$\mathsf{T}_{1}$$\mathsf{T}_{2}$$\mathsf{T}_{3}$$\mathsf{T}_{4}$$\mathsf{T}_{5}$$\mathsf{T}_{9}$$\mathsf{T}_{6}$$\mathsf{T}_{7}$$\mathsf{T}_{8}$
Figure 5. Because the weights are all non-negative, the sum of passage values
of $\mathsf{T}_{1},+\dots+\mathsf{T}_{9}$ is greater than or equal to
$T^{\max}_{k,n}$, which is shown as the thick dotted line along the geodesic
(thin dotted line).

###### Proof.

We may assume that $k\geq 2$ because $L(p)\geq 1$, so for $k=0,1$, the
statement can be made trivial by making $c$ small. To prove the bound, we will
replace $T^{\max}_{k,n}$ by the sum of nine first-passage values crossing nine
different rectangles. For an illustration, see Figure 5.

To describe the setup, we break the annulus $B_{2^{k-1}}\setminus
B_{2^{k-2}}\cup\partial B_{2^{k-2}}$ into four pieces (which are not mutually
disjoint). These pieces below correspond to the top, right, bottom, and left
of the annulus, numbered in that order.

  1. (1)

$[-2^{k-1},2^{k-1}]\times[2^{k-2},2^{k-1}]$

  2. (2)

$[2^{k-2},2^{k-1}]\times[-2^{k-1},2^{k-1}]$

  3. (3)

$[-2^{k-1},2^{k-1}]\times[-2^{k-1},-2^{k-2}]$

  4. (4)

$[-2^{k-1},-2^{k-2}]\times[-2^{k-1},2^{k-1}]$.

Let $\mathsf{T}_{1}$ be the minimal passage time of all paths crossing from
the left boundary of $[-2^{k-1},2^{k-1}]\times[2^{k-2},2^{k-1}]$ to the right
boundary and staying in the box. Let $\mathsf{T}_{2}$ be the minimal passage
time of all paths crossing from the top edge of
$[2^{k-2},2^{k-1}]\times[-2^{k-1},2^{k-1}]$ to the bottom and staying in the
box. Let $\mathsf{T}_{3}$ and $\mathsf{T}_{4}$ be defined similarly for the
bottom and left parts of the annulus. If we consider paths that achieve these
minima, we note that parts of these four paths can be concatenated to form a
circuit in the annulus $B_{2^{k-1}}\setminus B_{2^{k-2}}\cup\partial
B_{2^{k-2}}$, which has passage time less no greater than
$\sum_{i=1}^{4}\mathsf{T}_{i}$. We call this circuit $\mathcal{C}_{1}$.

Next, we partition $B_{2^{k+3}}\setminus B_{2^{k+2}}\cup\partial B_{2^{k+2}}$
into four, nondisjoint pieces and let $\mathsf{T}_{5},\ldots,\mathsf{T}_{8}$
be defined similarly. In the cases $k=n$ or $k=n-1$, we disregard these four
variables and set their values to $0$. Again, the minimal paths form a circuit
in the annulus $B_{2^{k+3}}\setminus B_{2^{k+2}}\cup\partial B_{2^{k+2}}$,
which has passage time no greater than $\sum_{i=1}^{4}\mathsf{T}_{i+4}$. We
call this circuit $\mathcal{C}_{2}$. Let $\mathsf{T}_{9}$ be the minimal
passage time of all paths crossing from the left side of the box
$[2^{k-2},2^{k+3}]\times[-2^{k-2},2^{k-2}]$ to the right, and staying in the
box. In the case $k=n-1$, we replace this box with
$[2^{n-3},2^{n+1}]\times[-2^{n-3},2^{n-3}]$, an in the case $k=n$, we replace
this box with $[2^{n-2},2^{n+1}]\times[-2^{n-2},2^{n-2}]$. Choose a minimal
path achieving passage time $\mathsf{T}_{9}$ and call it $\gamma^{\prime}$. We
now claim that

$T_{k,n}^{\max}\leq\sum_{i=1}^{9}\mathsf{T}_{i}.$ (4.10)

Suppose, by way of contradiction, that there exists a geodesic
$\gamma:0\to\partial B_{2^{n+1}}$ with
$T_{k}(\gamma)>\sum_{i=1}^{9}\mathsf{T}_{i}$. We claim that we can find a new
path $\tilde{\gamma}:0\rightarrow\partial B_{2^{n+1}}$ with passage time
strictly smaller than $T(\gamma)$, contradicting the optimally of $\gamma$.
The path $\gamma$ must always cross $\mathcal{C}_{1}$ and must also cross
$\mathcal{C}_{2}$ (when $k<n-1)$ as it passes from $0$ to $\partial
B_{2^{n+1}}$. Let $x\in\mathcal{C}_{1}$ denote the first vertex that $\gamma$
shares in common with $\mathcal{C}_{1}$ when traversing along the vertices of
$\gamma$, beginning from the origin. Accordingly, let $\gamma^{x}:0\rightarrow
x$ denote this segment of $\gamma$. Similarly, let $y\in C_{2}$ denote the
last vertex that $\gamma$ shares in common with $C_{2}$ and let
$\gamma^{y}:y\rightarrow\partial B_{2^{n+1}}$ denote the corresponding segment
of $\gamma$ (In the case $k=n-1$ or $k=n$, $y$ is the last vertex of
$\gamma$–the unique vertex on $\partial B_{2^{n+1}}$). Let $\gamma_{x,y}$
denote the segment of $\gamma$ connecting $x$ and $y$. Let $w$ be the first
point of $\gamma$ on $\partial B_{2^{k-1}}$, and let $z$ be the last point of
$\gamma$ on $\partial B_{2^{k+2}}$ (in the case $k=n-1$ of $k=n$, $z$ is the
last point of $\gamma$). Let $\gamma_{w,z}$ denote the portion of $\gamma$
connecting $w$ and $z$. Then, $T_{k}(\gamma)\leq T(\gamma_{w,z})$ because all
edges on $\gamma$ that also lie in $E_{k}$ must be contained in the portion of
$\gamma$ between $w$ and $z$. We claim now that
$\gamma_{w,z}\subseteq\gamma_{x,y}$ so that $T_{k}(\gamma)\leq
T(\gamma_{x,y})$. Equivalently, $x$ preceedes (or is equal to) $w$, and $z$
preceedes (or is equal to) $y$ in the path $\gamma$ as it travels from $0$ to
$\partial B_{2^{n+1}}$. This follows because $\mathcal{C}_{1}\subseteq
B_{2^{k-1}}\setminus B_{2^{k-2}}\cup\partial B_{2^{k-2}}$, so Lemma 6.1
implies that the Jordan curve defined by $\partial B_{2^{k-1}}$ encloses
$\mathcal{C}_{1}$. Since these two curves both enclose $0$, the path $\gamma$
cannot meet $B_{2^{k-1}}$ before it meets $\mathcal{C}_{1}$. Similarly,
$\mathcal{C}_{2}$ encloses the Jordan curve defined by $\partial B_{2^{k+2}}$,
so $y$ cannot preceede $z$.

Next, define the path $\gamma^{\prime\prime}$ from $x$ to $y$ as follows (int
the case $k\geq n-1)$, $\gamma^{\prime\prime}$ will be a path from $x$ to
$\partial B_{2^{n+1}}$. Follow the path $\mathcal{C}_{1}$ from $x$ until
meeting the path $\gamma^{\prime}$. Then, folow $\gamma^{\prime}$ until
reaching the circuit $\mathcal{C}_{2}$ (in the case $k<n-1)$ or until reaching
$\partial B_{2^{n+1}}$ (in the case $k\geq n-1$). In the case $k<n-1$, we
then, follow $\mathcal{C}_{2}$ until reaching $y$. It follows that
$\gamma^{\prime\prime}$ is a self-avoiding path consisting of edges in
$\mathcal{C}_{1},\mathcal{C}_{2}$, and $\gamma^{\prime}$, so by our
assumption,

$T(\gamma^{\prime\prime})\leq\sum_{i=1}^{9}\mathsf{T}_{i}<T_{k}(\gamma).$

Define $\tilde{\gamma}$ to be the concatenation of
$\gamma_{x},\gamma^{\prime\prime}$, and $\gamma_{y}$ (ignoring the
$\gamma_{y}$ part in the case $k\geq n-1$). Then,

$T(\tilde{\gamma})=T(\gamma_{x})+T(\gamma^{\prime\prime})+T(\gamma_{y})<T(\gamma_{x})+T_{k}(\gamma)+T(\gamma_{y})\leq
T(\gamma_{x})+T(\gamma_{x,y})+T(\gamma_{y})=T(\gamma),$

giving the desired contradiction, and thus proving (4.10). Then, by a simple
union bound,

$\mathbb{P}\Big{(}T_{k,n}^{\max}\geq\lambda\Big{(}\dfrac{2^{k}}{L(p)}\Big{)}^{2}F^{-1}(p)\Big{)}\leq\sum_{i=1}^{9}\Big{(}\mathsf{T}_{i}\geq\frac{\lambda}{9}\Big{(}\dfrac{2^{k}}{L(p)}\Big{)}^{2}F^{-1}(p)\Big{)},$

and the result now follows from Lemma 4.2. ∎

######  Corollary 4.4.

Let $F$ be the distribution funciton of a nonnegative random variable with
$F(0)=\frac{1}{2}$. There exists $c,C>0$ so that for for all $0\leq k\leq n$
and all $1\leq j\leq 2^{k}$,

$\mathbb{P}\Big{(}T_{k,n}^{\max}\geq j^{3}F^{-1}(p_{\lfloor
2^{k}/j\rfloor})\Big{)}\leq Ce^{-cj}.$

###### Proof.

The result can be obtained by first making the substitutions $p=p_{\lfloor
2^{k}/j\rfloor}$ and $\lambda=j$ into the inequality given as the result of
Lemma 4.3. Then one can use the bounds $cR\leq L(p_{R})\leq R$ from (4.3) to
obtain the desired result. ∎

######  Lemma 4.5.

Assume (4.1). There exists $r,a,c_{1},c_{2},C_{1}>0$ so that for all $j,k$
satisfying $2\leq j\leq c_{1}2^{{c_{2}}k}$ we have

$\mathbb{P}\Big{(}T_{k,n}^{\max}\geq F^{-1}(p_{\lfloor
2^{k}/j^{r}\rfloor})\Big{)}\leq C_{1}e^{-aj}.$

###### Proof.

To prove the result we show that there exists $r,c_{1},c_{2}>0$ so that for
all $k$ satisfying $2\leq j\leq c_{1}2^{c_{2}k}$ we have

$F^{-1}(p_{\lfloor 2^{k}/j^{r}\rfloor})\geq j^{3}F^{-1}(p_{\lfloor
2^{k}/j\rfloor})$

Then the result follows from Corollary 4.4. In fact, our choice of constants
will be so that $c_{1},c_{2}<1$ and $c_{2}r=1$ so that the condition $j\leq
c_{1}2^{c_{2}k}$ implies $j^{r}\leq 2^{k}$, and $p_{\lfloor
2^{k}/j^{r}\rfloor}$ is well-defined.

By assumption (4.1), there exists $L>1$ and $N>0$ so that, for any $m\geq N$,
we have $F^{-1}(p_{2^{m-1}})/F^{-1}(p_{2^{m}})\geq L$. Then for non-negative
integers $m$ and $\ell$ satisfying $m-\ell\geq N$, iterating this inequality
yields

$F^{-1}(p_{2^{m-\ell}})/F^{-1}(p_{2^{m}})\geq L^{\ell}.$ (4.11)

We fix a large $\alpha>0$ to be specified later, and choose the constants
$c_{1},c_{2}$ by

$c_{2}=\frac{1}{1+\alpha},\qquad-\frac{1}{c_{2}}\log_{2}c_{1}=N.$

With these choices of constants, whenever $j\leq c_{1}2^{c_{2}k}$, we have

$k\geq\frac{\log_{2}j-\log_{2}c_{1}}{c_{2}}=\log_{2}j+\log_{2}j^{\alpha}+N\geq\lfloor\log_{2}j\rfloor+\lfloor\log_{2}j^{\alpha}\rfloor+N.$

Now, apply (4.11) with $m=k-\lfloor\log_{2}j\rfloor$ and
$\ell=\lfloor\log_{2}j^{\alpha}\rfloor$, along with the simple estimate that
$x/2\leq\lfloor x\rfloor\leq x$ and Equation (4.3), which allows us to remove
the floor function below. This yields

$F^{-1}(p_{\lfloor 2^{k}/j^{1+\alpha}\rfloor})\geq
j^{\frac{\log_{2}L}{2}\alpha}F^{-1}(p_{\lfloor 2^{k}/j\rfloor}).$

Now, choose $\alpha$ so that $j^{\frac{\log_{2}L}{2}\alpha}\geq j^{3}$ for all
$j\geq 2$. The result follows with $r=1+\alpha$. ∎

We now prove a result of a similar type as Lemma 4.5, this time assuming
(4.2).

######  Lemma 4.6.

Assume (4.2). There exist constants $c,a,C,u>0$ so that, for all $n\geq 0$,
$0\leq k\leq n$, and $1\leq\lambda\leq c8^{k}$,

$\mathbb{P}(T_{k,n}^{\max}\geq\lambda F^{-1}(p_{2^{k}}))\leq
Ce^{-a\lambda^{u}}.$

###### Proof.

By the assumption (4.2) as well as the fact that $F$ is nondecreasing, there
exists $\rho>0$ and $N\geq 1$ so that, for all $k\geq N$,

$\dfrac{F^{-1}(p_{2^{k+1}})}{F^{-1}(p_{2^{k}})}\geq\dfrac{F^{-1}(p_{2^{2k}})}{F^{-1}(p_{2^{k}})}\geq\rho.$

Then, as long as $1\leq\lambda\leq 2^{k-N}$, we have
$k-\lfloor\log_{2}\lambda\rfloor\geq N$, and

$F^{-1}(p_{\lfloor 2^{k}/\lambda\rfloor})/F^{-1}(p_{2^{k}})\leq
F^{-1}(p_{2^{k-\lfloor\log_{2}\lambda\rfloor}})/F^{-1}(p_{2^{k}})\leq
1/\rho^{\lfloor\log_{2}\lambda\rfloor}\leq\widetilde{C}\lambda^{s}$ (4.12)

for constants $\widetilde{C},s>0$. Then, using (4.12) followed by Corollary
4.4,

$\displaystyle\mathbb{P}\Big{(}T_{k,n}^{\max}\geq\lambda^{3+s}F^{-1}(p_{2^{k}})\Big{)}\leq\mathbb{P}\Big{(}T_{k,n}^{\max}\geq(\widetilde{C}^{-1/3}\lambda)^{3}F^{-1}(p_{\lfloor
2^{k}/\lambda\rfloor})\Big{)}\leq Ce^{-a\lambda},$

for constants $C,a>0$. Replacing $\lambda$ with $\lambda^{1/(3+s)}$, we have

$\mathbb{P}(T_{k,n}^{\max}\geq\lambda F^{-1}(p_{2^{k}}))\leq
Ce^{-a\lambda^{1/(3+s)}},$

and this holds so long as $1\leq\lambda^{1/(3+s)}\leq 2^{k-N}$, which holds
when $1\leq\lambda\leq 2^{-N(3+s)}8^{k}$. ∎

###  4.3. Bounding the number of closed edges of a geodesic between annuli

For $n\geq 1$, define

$R(n)=B_{2^{n+1}}\setminus B_{2^{n}}\cup\partial
B_{2^{n}},\qquad\text{and}\qquad S(n)=B_{2^{n+2}}\setminus
B_{2^{n-1}}\cup\partial B_{2^{n-1}}.$

We also define

$S^{\prime}(n)=B_{2^{n+1}}\setminus B_{2^{n-1}}\cup\partial B_{2^{n-1}}.$

For and edge $e\in E(\mathbb{Z}^{2})$ with both endpoints in $R(n)$ and
$p>p_{\mathrm{c}}$, let $A_{n}(p,e)$ be the event that all of the following
occur:

  1. (1)

$U_{e}\in(p_{\mathrm{c}},p]$,

  2. (2)

there are two vertex-disjoint $p$-open paths from $e$ to $\partial
B_{2^{n-1}}(e)$ that remain in $S(n)$, and

  3. (3)

there are two vertex-disjoint $p_{\mathrm{c}}$-closed dual paths from
$e^{\star}$ to $\partial B_{2^{n}}(e)$ (disjoint from the $p$-open paths
above).

We also define events $A_{n}^{\prime}(p,e)$ similarly, the only differences
being that there is only one vertex-disjoint $p$-open path, and that path,
along with the two $p_{\mathrm{c}}$-closed paths, are requried to remain
inside $B_{2^{n+1}}$. Define

$N_{n}(p)=\sum_{e\subset R(n)}\textbf{1}_{A_{n}(p,e)}\qquad\text{and}\qquad
N_{n}^{\prime}(p)=\sum_{e\subset R(n)}\textbf{1}_{A_{n}^{\prime}(p,e)}$

We borrow the following result from [7, Lem. 2.2], whose original idea goes
back to the work of Kiss [17]. The only substantive difference is our arm
events are defined inside annuli while their arm events are defined inside of
rectangles. We also note that this lemma is a general result for critical two-
dimensional percolation (i.e. not depending on the edge weights $t_{e}$)
because it only makes reference to the uniform random variables $U_{e}$.

######  Lemma 4.7.

[7, Lem. 2.2] There exists $C>0$ such that for all $n\geq 1$ and
$p>p_{\mathrm{c}}$ with $L(p)\leq 2^{n}$, and all $\lambda\geq 0$,

$\mathbb{P}\Big{(}N_{n}(p)\geq\lambda\Big{(}\dfrac{2^{n}}{L(p)}\Big{)}^{2}\Big{)}\leq
e^{-C\lambda},\quad\text{and}\quad\mathbb{P}\Big{(}N_{n}^{\prime}(p)\geq\lambda\Big{(}\dfrac{2^{n}}{L(p)}\Big{)}^{2}\Big{)}\leq
e^{-C\lambda}.$

As before, for $k\geq 0$, let $E_{k}$ be the set of edges with both endpoints
in the set $B_{2^{k+1}}\setminus B_{2^{k}}\cup\partial B_{2^{k}}$, with the
convention that $E_{0}$ is the set of edges with both endpoints in $B_{2}$. We
will let $\gamma_{n}$ be the geodesic between $\mathcal{A}=\\{0\\}$ and
$\mathcal{B}=\partial B_{2^{n+1}}$ as constructed in Proposition 2.5. For
$0\leq k\leq n$, let

$D_{k,n}=\\#\\{e\in\gamma_{n}\cap E_{k}:t_{e}>0\\}$ (4.13)

Furthermore, for $1\leq k\leq n$, let

$M_{k,n}=\begin{cases}\max\\{U_{e}:e\in\gamma_{n}\cap(E_{k-1}\cup E_{k}\cup
E_{k+1})\\}&k<n\\\ \max\\{U_{e}:e\in\gamma_{n}\cap(E_{n-1}\cup
E_{n})\\}&k=n.\end{cases}$ (4.14)

We now prove the following corollary to Lemma 4.7.

######  Corollary 4.8.

Assume (1.1). Then there exists a constant $c^{\prime}>0$ (independent of the
choice of edge-weight distribution) so that, for all $n\in\mathbb{Z}_{\geq
0}$, $1\leq k\leq n$, and $\lambda\geq 0$ and $1\leq j\leq 2^{k}$,

$\mathbb{P}(D_{k,n}\geq\lambda,M_{k,n}\leq p_{\lfloor 2^{k}/j\rfloor})\leq
e^{-c^{\prime}\lambda/j^{2}}.$

###### Proof.

The statement is trivial for $\lambda=0$, so since $D_{k,n}$ is an integer, we
take $\lambda\geq 1$ without loss of generality. For this, we show that

$\\{D_{k,n}\geq\lambda,M_{k,n}\leq p_{\lfloor
2^{k}/j\rfloor}\\}\subseteq\begin{cases}\mathbb{\\{}N_{k}(p_{\lfloor
2^{k}/j\rfloor})\geq\lambda\\}&k<n\\\ \mathbb{\\{}N_{k}^{\prime}(p_{\lfloor
2^{k}/j\rfloor})\geq\lambda\\}&k=n.\end{cases}$ (4.15)

Before proving (4.15), we show here why this is sufficient to prove the lemma.
Using (4.3), which tells us that $L(p_{R})\geq cR$ for some constant $c>0$, we
have

$\displaystyle\mathbb{P}(N_{k}(p_{\lfloor 2^{k}/j\rfloor})\geq\lambda)$
$\displaystyle=\mathbb{P}\Big{(}N_{k}(p_{\lfloor
2^{k}/j\rfloor})\geq\Big{(}\frac{L(p_{\lfloor
2^{k}/j\rfloor})}{2^{k}}\Big{)}^{2}\lambda\Big{(}\frac{2^{k}}{L(p_{\lfloor
2^{k}/j\rfloor})}\Big{)}^{2}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}N_{k}(p_{\lfloor
2^{k}/j\rfloor})\geq\frac{c\lambda}{j^{2}}\Big{(}\frac{2^{k}}{L(p_{\lfloor
2^{k}/j\rfloor})}\Big{)}^{2}\Big{)},$

and the same holds with $N_{k}$ replaced with $N_{k}^{\prime}$. The lemma then
follows directly from Lemma 4.7. With this observation, we return to proving
(4.15).

We first consider the case $k<n$. Assume that $D_{k,n}\geq\lambda$ and
$M_{k,n}\leq p_{\lfloor 2^{k}/j\rfloor}$. For each edge $e\in\gamma_{n}\cap
E_{k}$ with $t_{e}>0$, we verify that the event $A_{k}(p,e)$ occurs with
$p=p_{\lfloor 2^{k}/j\rfloor}$. It must be the case that $U_{e}\leq p$ by the
assumption $M_{k,n}\leq p_{\lfloor 2^{k}/j\rfloor}$. This assumption also
guarantees there are two disjoint $p$-open paths formed from the two portions
of the geodesic $\gamma_{n}$ traveling from the endpoints of $e$ to the first
time they meet $\partial B_{2^{k-1}}\cup\partial B_{2^{k+2}}$. These paths
each travel a distance at least $2^{k-1}$ and therefore must meet $\partial
B_{2^{k-1}}(e)$. Since $t_{e}>0$, Proposition 2.5(x) implies that $e^{\star}$
belongs to a dual closed circuit that encloses the origin, and the closed
circuit does not intersect $\gamma_{n}$ at a location other than $e^{\star}$.
Hence, there exist two $p_{\mathrm{c}}$-closed dual arms to distance at least
$2^{k}$, disjoint from the two $p$-open arms, and $A_{n}(p,e)$ occurs. Thus,
since there are at least $\lambda$ of these edges $e$,

$N_{k}(p_{\lfloor 2^{k}/j\rfloor})=\sum_{e\subseteq
R(n)}\mathbf{1}_{A_{n}(p_{\lfloor 2^{k}/j\rfloor},e)}\geq\lambda.$

The case $k=n$ is similar, this time working with the events
$A_{n}^{\prime}(p,e)$ instead of $A_{n}(p,e)$. Since the geodesic $\gamma_{n}$
is near the boundary $\partial B_{2^{n+1}}$ for edges $e\in E_{n}$, we
consider the single $p$-open path starting backward from $e$ and traveling
until the first time it reaches $\partial B_{2^{n-1}}$. This path must stay
inside $B_{2^{n+1}}$. The two $p_{\mathrm{c}}$-closed paths are again formed
by the closed circuit containing $e^{\star}$, and the requirement that they
stay inside $B_{2^{n+1}}$ is met by the construction of the circuit in
Proposition 2.5(x). ∎

Before the next proposition, we make the following elementary observation.

######  Lemma 4.9.

Let $Np<1$. Then, if $\chi$ has the binomial distribution with parameters
$N\in\mathbb{N}$ and $p\in(0,1)$,

$\mathbb{P}(\chi\geq\lambda)\leq e\times(Np)^{\lambda}$

###### Proof.

This is a straightforward calculation.

$\mathbb{P}(\chi\geq\lambda)=\sum_{k=\lceil\lambda\rceil}^{n}{N\choose
k}p^{k}(1-p)^{N-k}\leq\sum_{k=\lceil\lambda\rceil}^{n}\frac{N^{k}}{k!}p^{k}\leq
e\times(Np)^{\lceil\lambda\rceil}\leq e\times(Np)^{\lambda}.\qed$

######  Proposition 4.10.

Assume either (4.1) or (4.2). Then, there exists $C,c,s>0$ so that

$\mathbb{P}(D_{k,n}\geq\lambda)\leq Ce^{-c\lambda^{s}}$

for all $\lambda\geq 0$ and $0\leq k\leq n$.

###### Proof.

We start by noting that $D_{k,n}$ is trivially upper bounded by the size of
$E_{k}$, which is bounded by $C^{\prime}4^{k}$ for some constant $C^{\prime}$.
Combined with the fact that $D_{k,n}$ is an integer, we may take
$1\leq\lambda\leq C^{\prime}4^{k}$. By bounding the total number of edges by a
deterministic $O(1)$ term, in the first $K_{0}$, without loss of generality,
we may prove the result for all $k\geq K_{0}$ for some $K_{0}\geq 1$ to be
determined later. We handle the cases of assuming (4.1) and (4.2) separately,
with the number $K_{0}$ being chosen differently in each case.

Case 1: Assuming (4.1)

Let $r,a,c_{1},c_{2},C_{1}$ be chosen as in Lemma 4.5, and fix
$\eta\in(0,c_{2})$. Observe that

$\displaystyle\mathbb{P}\Big{(}D_{k,n}\geq\lambda\Big{)}$
$\displaystyle=\mathbb{P}\Big{(}D_{k,n}\geq\lambda,M_{k,n}\leq p_{\lfloor
2^{k}/2^{r}\rfloor}\Big{)}$ (4.16) $\displaystyle+\sum_{j=2}^{\lfloor 2^{\eta
k}\rfloor}\mathbb{P}\Big{(}D_{k,n}\geq\lambda,p_{\lfloor
2^{k}/j^{r}\rfloor}<M_{k,n}\leq p_{\lfloor 2^{k}/(j+1)^{r}\rfloor}\Big{)}$
$\displaystyle+\mathbb{P}\Big{(}D_{k,n}\geq\lambda,M_{k,n}>p_{\lfloor
2^{k}/(\lfloor 2^{\eta k}\rfloor+1)^{r}\rfloor}\Big{)}$

We bound each of the three terms in (4.16). By Corollary 4.8, there exists
$c>0$ so that

$\mathbb{P}\Big{(}D_{k,n}\geq\lambda,M_{k,n}\leq p_{\lfloor
2^{k}/2^{r}\rfloor}\Big{)}\leq e^{-c\lambda}.$

Since $\eta<c_{2}$, there exists $K_{0}\geq 0$ so that $(\lfloor 2^{\eta
k}\rfloor+1)\leq c_{1}2^{c_{2}k}$ for all $k\geq K_{0}$. Next, recalling the
definition of $T_{k,n}^{\max}$ (4.9), if $M_{k,n}>p$, then $T_{k,n}^{\max}\geq
F^{-1}(p)$. Then, by Lemma 4.5, for such $k$,

$\displaystyle\mathbb{P}\Big{(}D_{k,n}\geq\lambda,M_{k,n}>p_{\lfloor
2^{k}/(\lfloor 2^{\eta k}\rfloor+1)^{r}\rfloor}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}M_{k,n}>p_{\lfloor 2^{k}/(\lfloor 2^{\eta
k}\rfloor+1)^{r}\rfloor}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}T_{k,n}^{\max}\geq F^{-1}(p_{\lfloor
2^{k}/(\lfloor 2^{\eta k}\rfloor+1)^{r}\rfloor})\Big{)}$ $\displaystyle\leq
C_{1}e^{-a(\lfloor 2^{\eta k}\rfloor+1)}\leq Ce^{-c\lambda^{u}},$

where the last inequality holds because we have assumed that $\lambda\leq
C^{\prime}4^{k}$ at the beginning of the proof, and $u$ is a sufficiently
small fixed number. It remains to bound the middle term on the right-hand side
of (4.16). Using Hölder’s inequality,

$\displaystyle\quad\;\mathbb{P}\Big{(}D_{k,n}\geq\lambda,p_{\lfloor
2^{k}/j^{r}\rfloor}<M_{k,n}\leq p_{\lfloor 2^{k}/(j+1)^{r}\rfloor}\Big{)}$
(4.17) $\displaystyle\leq\mathbb{P}\Big{(}D_{k,n}\geq\lambda,M_{k,n}\leq
p_{\lfloor
2^{k}/(j+1)^{r}\rfloor}\Big{)}^{1/2}\mathbb{P}\Big{(}M_{k,n}>p_{\lfloor
2^{k}/j^{r}\rfloor}\Big{)}^{1/2}.$

By Corollary 4.8, we may choose $c>0$ so that

$\mathbb{P}\Big{(}D_{k,n}\geq\lambda,M_{k,n}\leq p_{\lfloor
2^{k}/(j+1)^{r}\rfloor}\Big{)}\leq e^{-c\lambda/(j+1)^{2r}}.$ (4.18)

On the other hand, because $j\leq\lfloor 2^{\eta k}\rfloor<c_{1}2^{c_{2}k}$,
Lemma 4.5 implies

$\mathbb{P}\Big{(}M_{k,n}>p_{\lfloor
2^{k}/j^{r}\rfloor}\Big{)}\leq\mathbb{P}\Big{(}T_{k,n}^{\max}\geq
F^{-1}(p_{\lfloor 2^{k}/j^{r}\rfloor})\Big{)}\leq Ce^{-cj}.$ (4.19)

We now consider two cases. In the first case, suppose that
$\lfloor\lambda^{\frac{1}{2r+1}}\rfloor\leq\lfloor 2^{\eta k}\rfloor$. Then,
using (4.17)–(4.19), and adjusting the constants $C,c$ as needed due to the
square root, we obtain

$\displaystyle\quad\;\sum_{j=2}^{\lfloor 2^{\eta
k}\rfloor}\mathbb{P}\Big{(}D_{k,n}\geq\lambda,p_{\lfloor
2^{k}/j^{r}\rfloor}<M_{k,n}\leq p_{\lfloor 2^{k}/(j+1)^{r}\rfloor}\Big{)}\leq
C\sum_{j=2}^{\lfloor 2^{\eta k}\rfloor}e^{-c\lambda/(j+1)^{2r}}e^{-cj}$
$\displaystyle\leq
C\sum_{j=2}^{\lfloor\lambda^{\frac{1}{2r+1}}\rfloor-1}e^{-c\lambda/(j+1)^{2r}}e^{-cj}+C\sum_{j=\lfloor\lambda^{\frac{1}{2r+1}}\rfloor}^{\lfloor
2^{\eta k}\rfloor}e^{-c\lambda/(j+1)^{2r}}e^{-cj}.$ (4.20)

In the first sum on the right in (4.20), approximate each summand by
substituting largest value of $j$. In the second sum, discard the first factor
because it is bounded above by $1$. Then, we obtain the following bound for
(4.20) by adjusting the constants as needed:

$\displaystyle
Ce^{-c\lambda^{\frac{1}{2r+1}}}\sum_{j=2}^{\lfloor\lambda^{\frac{1}{2r+1}}\rfloor-1}e^{-cj}+C\sum_{j=\lfloor\lambda^{\frac{1}{2r+1}}\rfloor}^{\lfloor
2^{\eta k}\rfloor}e^{-cj}\leq Ce^{-c\lambda^{\frac{1}{2r+1}}}.$

This gives the desired bound. In the second case where
$\lfloor\lambda^{\frac{1}{2r+1}}\rfloor>\lfloor 2^{\eta k}\rfloor$, we
directly have that

$C\sum_{j=2}^{\lfloor 2^{\eta k}\rfloor}e^{-c\lambda/(j+1)^{2r}}e^{-cj}\leq
Ce^{-c\lambda/(\lfloor 2^{\eta k}\rfloor+1)^{2r}}\leq
Ce^{-c\lambda/(\lfloor\lambda^{\frac{1}{2r+1}}\rfloor+1)^{2r}}\leq
Ce^{-\frac{c}{2^{2r}}\lambda^{\frac{1}{2r+1}}},$

where the last inequality holds because
$\lfloor\lambda^{\frac{1}{2r+1}}\rfloor+1\leq 2\lambda^{\frac{1}{2r+1}}$ for
all $\lambda\geq 1$.

Case 2: Assuming (4.2): Let $\beta$ be a fixed nonnegative integer, to be
chosen later. Recall the coupling from uniform random variables $U_{e}$, where
$t_{e}=F^{-1}(U_{e})$. Then, $t_{e}>0$ if and only if $U_{e}>p_{\mathrm{c}}$.
Define

$\displaystyle D_{k,n}^{1}$ $\displaystyle=\\#\\{e\in\gamma_{n}\cap
E_{k}:U_{e}\in(p_{\mathrm{c}},p_{2^{\beta k}})\\},$ $\displaystyle
D_{k,n}^{2}$ $\displaystyle=\\#\\{e\in\gamma_{n}\cap E_{k}:U_{e}\geq
p_{2^{\beta k}}\\}.$

Then,

$\mathbb{P}(D_{k,n}\geq\lambda)\leq\mathbb{P}(D_{k,n}^{1}\geq\lambda/2)+\mathbb{P}(D_{k,n}^{2}\geq\lambda/2).$
(4.21)

We handle the two terms on the right-hand side of (4.21) separately. For the
first term, we trivially bound $D_{k,n}^{1}$ by total the number of edges
$e\in E_{k}$ with $U_{e}\in(p_{\mathrm{c}},p_{2^{\beta k}})$. The random
number $D_{k,n}^{1}$ has the binomial distribution with parameters $N\leq
C_{1}4^{k}$ and $p=p_{2^{\beta k}}-p_{c}\leq C_{2}2^{-\alpha\beta k}$ (4.6),
and constants $C_{1},C_{2},\alpha>0$. Fix $\delta>0$, and choose the integer
$\beta\geq 2$ sufficiently large so that $1-\alpha\beta\leq-\delta$. By Lemma
4.9, there exists a constant $C>0$ so that

$\mathbb{P}(D_{k,n}^{1}\geq\lambda/2)\leq Ce^{-\delta k\lambda/2}\leq
Ce^{-\delta\lambda/2},$

as long as we choose $k\geq K_{0}\geq 1$. This gives an appropriate bound on
the first term in (4.21).

For the second term, we first argue that the assumption (4.2) implies that for
any integer $\beta\geq 1$,

$\liminf_{n\to\infty}\frac{F^{-1}(p_{2^{\beta n}})}{F^{-1}(p_{2^{n}})}>0.$
(4.22)

From monotonicity of $F$, if (4.22) holds for some $\beta$, then it holds for
all $\alpha<\beta$. Thus, it suffices to prove the statement for
$\beta=2^{j}$. Note that

$\liminf_{n\to\infty}\frac{F^{-1}(p_{2^{2^{j}n}})}{F^{-1}(p_{2^{n}})}=\liminf_{n\to\infty}\prod_{i=1}^{j}\frac{F^{-1}(p_{2^{2^{i}n}})}{F^{-1}(p_{2^{2^{i-1}n}})}=\Big{(}\liminf_{n\to\infty}\frac{F^{-1}(p_{2^{2n}})}{F^{-1}(p_{2^{n}})}\Big{)}^{j}>0.$

Hence, there exists a constant $C_{\beta}>0$ so that

$F^{-1}(p_{2^{\beta k}})\geq C_{\beta}F^{-1}(p_{2^{k}})$ (4.23)

for all $k\geq 1$. Next, we make the definition

$Z_{k,n}=\max_{\gamma\in\operatorname{Geo}(0,\partial
B_{2^{n+1}})}\\#\\{e\in\gamma\cap E_{k}:F^{-1}(U_{e})\geq F^{-1}(p_{2^{\beta
k}})\\},$

and observe that $D_{k,n}^{2}\leq Z_{k,n}$. Observe further that
$F^{-1}(p_{2^{\beta k}})Z_{k,n}\leq T_{k,n}^{\max}$. Then, by (4.23) and Lemma
4.6, whenever $C_{\beta}\lambda/2\leq c^{\prime}8^{k}$ for some fixed constant
$c^{\prime}$,

$\displaystyle\mathbb{P}(D_{k,n}^{2}\geq\lambda/2)$
$\displaystyle\leq\mathbb{P}(Z_{k,n}\geq\lambda/2)\leq\mathbb{P}(T_{k,n}^{\max}\geq\lambda
F^{-1}(p_{2^{\beta k}})/2)$
$\displaystyle\leq\mathbb{P}(T_{k,n}^{\max}\geq(C_{\beta}\lambda/2)F^{-1}(p_{2^{k}}))\leq
Ce^{-c\lambda^{s}}.$

This gives an upper bound for the second term in (4.21). If we choose $K_{0}$
so that for all $k\geq K_{0}$, $\frac{C_{\beta}C^{\prime}}{2}4^{k}\leq
c^{\prime}8^{k}$, then the condition $C_{\beta}\lambda/2\leq c^{\prime}8^{k}$
is guaranteed by the assumption $\lambda\leq C^{\prime}4^{k}$. ∎

###  4.4. Spacings within and between circuits

Recall the sets $E_{k}$ of edges having both endpoints in
$B_{2^{k+1}}\setminus B_{2^{k}}\cup\partial B_{2^{k}}$ for $k\geq 1$, and we
define $E_{0}$ to be the set of edges with both endpoints in $B_{2}$. Next,
for a circuit $\mathcal{C}$, define

$\mathfrak{i}(\mathcal{C})=\min\\{k\geq
0:E_{k}\cap\mathcal{C}\not=\emptyset\\}$
$\mathfrak{o}(\mathcal{C})=\max\\{k\geq
0:E_{k}\cap\mathcal{C}\not=\emptyset\\}$

In each of the following lemmas, we let
$\mathcal{I}_{1},\mathcal{I}_{2},\ldots$ be the sequence of successive
disjoint innermost collection of open circuits enclosing the origin. On the
event $\Omega_{\infty}$, this sequence is infinite. The rigorous construction
of such sequences follows just as in Section 7, except that we only consider
circuits surrounding $\\{0\\}$, without a second reference set. Alternatively,
one can construct this set of circuits directly using Proposition 2.5 by
considering the two sets $\mathcal{A}=\\{0\\}$ and $\mathcal{B}=\partial
B_{R}$, constructing the set of successive edge-disjoint innermost open
circuits surrounding $0$ but keeping $\partial B_{R}$ in the exterior, then
letting $R\to\infty$ (note that there can be no circuits with $\partial B_{R}$
in the interior with $0$ in the exterior). In other words, for $R\geq 1$, let

$Y_{R}=\sup\\{j\geq 1:\mathcal{I}_{j}\subseteq B_{R}\\}.$ (4.24)

Then, since we chose each circuit to be the successive innermost circuit,
$\mathcal{I}_{1},\ldots,\mathcal{I}_{Y_{R}}$ is the sequence from Proposition
2.5, with $\mathcal{A}=\\{0\\}$, and $\mathcal{B}=\partial B_{R}$. The results
in this subsection make no assumptions on the edge-weight distribution. In
fact, they are general statements about critical percolation on
$\mathbb{Z}^{2}$, without reference to a distribution function $F$.

######  Lemma 4.11.

For $j\geq 1$, $\mathfrak{o}(\mathcal{I}_{j})$ (and therefore also
$\mathfrak{i}(\mathcal{I}_{j})$) is stochastically dominated by the sum of $j$
i.i.d. nondegenerate geometric random variables supported on
$\\{1,2,\ldots\\}$. In particular, for each $k\geq 1$, there exsits a constant
$C=C(k)$ so that
$\mathbb{E}[\mathfrak{i}(\mathcal{I}_{j})^{k}]\leq\mathbb{E}[\mathfrak{o}(\mathcal{I}_{j})^{k}]\leq
Cj^{k}$ for all $k\geq 1$.

###### Proof.

Consider the sequence of disjoint annuli $A_{k}=B_{2^{k+1}}\setminus
B_{2^{k}}$ for $k\geq 1$. the events

$\Omega_{k}=\\{\exists\text{ an open circuit, all of whose vertices lie in
}A_{k}\\}$

are mutually independent, and the probability of this seqeunce of events is
bounded away from $0$ and $1$ by the RSW theorem.

If $\Omega_{k}$ occurs for some $k\geq 1$, choose exactly one circuit
$\mathcal{C}$ in some measurable fashion, and note that
$\mathfrak{o}(\mathcal{C})=k$. Let $\mathcal{C}_{1},\mathcal{C}_{2},\ldots$ be
the infinite sequence of circuits constructed in this manner. Note that each
$\mathcal{C}_{j+1}$ encloses and is vertex-disjoint (and therefore also edge-
disjoint) from $\mathcal{C}_{j}$. Since
$\mathcal{I}_{1},\mathcal{I}_{2},\ldots$ was defined as the sequence of
successive innermost open circuits, it follows by induction that each
$\mathcal{I}_{j}$ is enclosed by $\mathcal{C}_{j}$, so
$\mathfrak{o}(\mathcal{C}_{j})\geq\mathfrak{o}(\mathcal{I}_{j})$.

Now, the circuit $\mathcal{C}_{j}$ lies entirely in a single annulus
$B_{2^{k+1}}\setminus B_{2^{k}}$ for some $k$, and the values
$\mathfrak{o}(\mathcal{C}_{j})$ is equal to that $k$. Consider the
$\\{0,1\\}$-valued sequence $\\{\eta_{k}\\}_{k\geq 1}$ defined by $\eta_{k}=1$
if $\Omega_{k}$ occurs, and $\eta_{k}=0$ otherwise. Then,
$\mathfrak{o}(\mathcal{C}_{1})=\inf\\{k\geq 1:\eta_{k}=1\\}$, and
$\mathfrak{o}(\mathcal{C}_{j})=\inf\\{k>\mathfrak{o}(\mathcal{C}_{j}):\eta_{k}=1\\}$.

Letting $p=\inf_{j}\mathbb{P}(\Omega_{j})>0$, the independence of the events
$\Omega_{j}$ implies that $\mathfrak{o}(\mathcal{C}_{j})$ is stochastically
dominated by the sum of $j$ i.i.d. geometric random variables with success
probability $p$. The bound on the moments follows immediately. ∎

######  Lemma 4.12.

There exists $C,c>0$ so that, for all $j\geq 1$

$\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j})-\mathfrak{i}(\mathcal{I}_{j})\geq
t)\leq Cje^{-ct}.$

###### Proof.

When $\mathfrak{i}(\mathcal{I}_{j})=k$, and $\mathfrak{o}(\mathcal{I}_{j})\geq
t+k$, there exists an open path from $\partial B_{2^{k+1}}$ to $\partial
B_{2^{k+t}}$ formed by following a portion of $\mathcal{I}_{j}$. We use
$\\{\mathcal{A}\leftrightarrow\mathcal{B}\\}$ to denote the event that
$\mathcal{A}$ is connected to $\mathcal{B}$ by an open path. Then,

$\displaystyle\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j})-\mathfrak{i}(\mathcal{I}_{j})\geq
t)$
$\displaystyle=\sum_{k=0}^{\infty}\mathbb{P}(\\{\mathfrak{o}(\mathcal{I}_{j})-\mathfrak{i}(\mathcal{I}_{j})\geq
t\\}\cap\\{\mathfrak{i}(\mathcal{I}_{j})=k\\})$
$\displaystyle\leq\sum_{k=0}^{\infty}\mathbb{P}(\\{\partial
B_{2^{k+1}}\leftrightarrow\partial
B_{2^{k+t}}\\}\cap\\{\mathfrak{i}(\mathcal{I}_{j})\geq k\\}).$

Observe that $\\{\mathfrak{i}(\mathcal{I}_{j})\geq k\\}$ is a decreasing event
(in the sense that adding more open edges cannot change the event from not
occuring to occuring), while each $\\{\partial
B_{2^{k+1}}\leftrightarrow\partial B_{2^{k+t}}\\}$ is an increasing event. So
by the FKG inequality, followed by a bound on the $1$-arm probability
(Proposition 3.3), we have

$\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j})-\mathfrak{i}(\mathcal{I}_{j})\geq
t)\leq\sum_{k=0}^{\infty}\mathbb{P}(\partial
B_{2^{k+1}}\leftrightarrow\partial
B_{2^{k+t}})\mathbb{P}(\mathfrak{i}(\mathcal{I}_{j})\geq k)\leq
Ce^{-ct}\mathbb{E}\mathfrak{i}(\mathcal{I}_{j}).$

The result now follows from Lemma 4.11.∎

######  Lemma 4.13.

There exists $C,c>0$ so that, for all $j\geq 1$,

$\mathbb{P}(\mathfrak{i}(\mathcal{I}_{j+1})-\mathfrak{o}(\mathcal{I}_{j})\geq
t)\leq Cje^{-ct}.$

###### Proof.

Let $\\{\mathcal{A}\overset{\text{cd}}{\leftrightarrow}\mathcal{B}\\}$ denote
the event that there exists a closed dual connection from the set
$\mathcal{A}$ to the set $\mathcal{B}$. By Proposition 2.5(vi), for each
$j\geq 1$, there exists a closed dual connection from $\mathcal{I}_{j}$ to
$\mathcal{I}_{j+1}$. Hence, using a bound on the critical 1-arm probability
(Proposition 3.3),

$\displaystyle\sum_{k=0}^{\infty}\mathbb{P}(\\{\mathfrak{i}(\mathcal{I}_{j+1})-\mathfrak{o}(\mathcal{I}_{j})\geq
t\\}\cap\\{\mathfrak{o}(\mathcal{I}_{j})=k\\})\leq\sum_{k=0}^{\infty}\mathbb{P}(\\{\partial
B_{2^{k+1}}\overset{\text{cd}}{\leftrightarrow}\partial
B_{2^{k+t}}\\}\cap\\{\mathfrak{o}(\mathcal{I}_{j})=k\\})$
$\displaystyle\leq\sum_{k=0}^{\infty}\mathbb{P}(\partial
B_{2^{k+1}}\overset{\text{cd}}{\leftrightarrow}\partial
B_{2^{k+t}})^{1/2}\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j})=k)^{1/2}\leq
Ce^{-ct}\sum_{k=0}^{\infty}\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j})=k)^{1/2}$
$\displaystyle\leq
Ce^{-ct}\Big{(}\sum_{k=0}^{\infty}(k+1)^{2}\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j})=k)\Big{)}^{1/2}\Big{(}\sum_{k=0}^{\infty}\dfrac{1}{(k+1)^{2}}\Big{)}^{1/2}\leq
Ce^{-ct}\sqrt{\mathbb{E}\mathfrak{o}(\mathcal{I}_{j})^{2}}.$

The result now follows by Lemma 4.11. ∎

######  Lemma 4.14.

There exists $C,c>0$ so that, for all $j\geq 0$,

$\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j+1})-\mathfrak{i}(\mathcal{I}_{j})\geq
t)\leq Cje^{-ct}.$

###### Proof.

This is a Corollary of Lemmas 4.12 and 4.13:

$\displaystyle\quad\;\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j+1})-\mathfrak{i}(\mathcal{I}_{j})\geq
t)$
$\displaystyle\leq\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j+1})-\mathfrak{i}(\mathcal{I}_{j+1})+\mathfrak{i}(\mathcal{I}_{j+1})-\mathfrak{o}(\mathcal{I}_{j})+\mathfrak{o}(\mathcal{I}_{j})-\mathfrak{i}(\mathcal{I}_{j})\geq
t)$
$\displaystyle\leq\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j+1})-\mathfrak{i}(\mathcal{I}_{j+1})\geq
t/3)+\mathbb{P}(\mathfrak{i}(\mathcal{I}_{j+1})-\mathfrak{o}(\mathcal{I}_{j})\geq
t/3)+\mathbb{P}(\mathfrak{o}(\mathcal{I}_{j})-\mathfrak{i}(\mathcal{I}_{j})\geq
t/3).\qed$

The next result estimates the distance between the last circuit inside the box
and the boundary of the box of radius $2^{n+1}$.

######  Lemma 4.15.

Recall the notation $Y_{R}$ from (4.24). Then, there exist constants $C,c>0$
so that

$\displaystyle\mathbb{P}(n-\mathfrak{o}(\mathcal{I}_{Y_{2^{n+1}}})\geq t)\leq
Ce^{-ct}.$

###### Proof.

On the event $n-\mathfrak{o}(\mathcal{I}_{Y_{2^{n+1}}})\geq t$, Proposition
2.5(vi) implies that there must be a closed crossing from from $B_{2^{n-t+1}}$
to $\partial B_{2^{n+1}}$ and so the bound follows from a bound on the $1$-arm
probability from Proposition 3.3. ∎

######  Lemma 4.16.

There exists $C,c>0$ so that, for all $n\geq 1$,

$\mathbb{P}(n-\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})\geq t)\leq
Cn^{5/2}e^{-ct}.$

###### Proof.

$\displaystyle\mathbb{P}(n-\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})\geq t)$
$\displaystyle=\mathbb{P}(n-\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})-\mathfrak{o}(\mathcal{I}_{Y_{2^{n+1}}})+\mathfrak{o}(\mathcal{I}_{Y_{2^{n+1}}})\geq
t)$ $\displaystyle\leq\mathbb{P}(n-\mathfrak{o}(\mathcal{I}_{Y_{2^{n+1}}})\geq
t/2)+\mathbb{P}(\mathfrak{o}(\mathcal{I}_{Y_{2^{n+1}}})-\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})\geq
t/2)$

The first term on the right-hand side is bounded by $Ce^{-ct}$ by Lemma 4.15.
Using Lemma 4.12 and Hölder’s inequality,

$\displaystyle\quad\;\mathbb{P}(\mathfrak{o}(\mathcal{I}_{Y_{2^{n+1}}})-\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})\geq
t/2)=\sum_{k}\mathbb{P}(\mathfrak{o}(\mathcal{I}_{k})-\mathfrak{i}(\mathcal{I}_{k})\geq
t/2,Y_{2^{n+1}}=k)$ $\displaystyle\leq
C\sum_{k}\sqrt{k}e^{-ct}\sqrt{\mathbb{P}(Y_{2^{n+1}}=k)}=C\sum_{k}\frac{\sqrt{k}}{k^{2}}e^{-ct}k^{2}\sqrt{\mathbb{P}(Y_{2^{n+1}}=k)}$
$\displaystyle\leq
Ce^{-ct}\Big{(}\sum_{k}k^{4}\mathbb{P}(Y_{2^{n+1}}=k)\Big{)}^{1/2}=Ce^{-ct}(\mathbb{E}Y_{2^{n+1}}^{4})^{1/2}.$

We conclude the proof by showing that $\mathbb{E}Y_{2^{n+1}}^{4}\leq Cn^{5}$.
Decompose $Y_{2^{n+1}}$ as $\sum_{i=0}^{n}V_{i}$, where $V_{i}$ is the number
of circuits $\mathcal{I}_{j}$ in the sequence
$\mathcal{I}_{1},\ldots,\mathcal{I}_{Y_{2^{n+1}}}$ with
$\mathfrak{o}(\mathcal{I}_{j})=i$. There is a deterministic constant $C$ so
that $(\sum_{i=0}^{n}x_{i})^{4}\leq Cn^{4}\sum_{i=1}^{n}x_{i}^{4}$ for all
$n\geq 1$ and nonnegative numbers $x_{0},\ldots,x_{n}$. Then,

$\mathbb{E}Y_{2^{n+1}}^{4}=\mathbb{E}\Bigl{(}\sum_{i=0}^{n}V_{i}\Bigr{)}^{4}\leq
Cn^{4}\sum_{i=0}^{n}\mathbb{E}V_{i}^{4}\leq Cn^{5},$

where the last inequality uses the fact that $\mathbb{E}[V_{i}^{4}]$ is a
bounded sequence, which we now prove. Define the event

$\displaystyle A_{i}$ $\displaystyle=\\{\exists\text{ a closed circuit whose
veritices lie in $B_{2^{i}}\setminus B_{2^{i-1}}$}$
$\displaystyle\qquad\text{and $\exists$ a closed dual left-right crossing of
$[2^{i-1},2^{i+2}]\times[-2^{i-1},2^{i-1}]$ }\\}$

By the FKG inequality and the RSW theorem, $\mathbb{P}(A_{i})\geq\varepsilon$
for some constant $\varepsilon>0$. On the event $A_{i}$, there exists no open
circuit $\mathcal{C}$ with $\mathfrak{o}(\mathcal{C})=i$. To see this, note
that the closed circuit in $A_{k}$, which we call $\mathcal{D}$, must be
enclosed by $\mathcal{C}$ by Lemma 6.9. The closed dual left-right crossing of
$[2^{i-1},2^{i+2}]\times[-2^{i-1},2^{i-1}]$ must cross from the interior of
$\mathcal{C}$ to the exterior and must therefore cross the open circuit
$\mathcal{C}$, a contradiction. Hence

$\mathbb{P}(\exists\text{ an open circuit $\mathcal{C}$ with
$\mathfrak{o}(\mathcal{C})=i$ })\leq\mathbb{P}(A_{i}^{c})\leq 1-\varepsilon.$

Then, using the BK inequality, for all $i\geq 0$,

$\displaystyle\mathbb{P}(V_{i}\geq k)=\mathbb{P}(\text{$\exists k$ edge-
disjoint open circuits $\mathcal{C}$ with $\mathfrak{o}(\mathcal{C})=i$
})\leq(1-\varepsilon)^{k},$ (4.25)

from which the uniform upper bound for $\mathbb{E}[V_{i}^{4}]$ follows. ∎

###  4.5. Controlling the number of closed edges between circuits

We recall here again that $\gamma_{n}$ is the geodesic from
$\mathcal{A}=\\{0\\}$ to $\mathcal{B}=\partial B_{2^{n+1}}$ from Proposition
2.5.

For any $j\geq 1$, define

$X_{j}=\\#\\{e\in\gamma_{n}\cap
E(\text{int}(\mathcal{I}_{j+1})\cap\text{ext}(\mathcal{I}_{j})):t_{e}>0\\},$
(4.26)

where $E(\text{int}(\mathcal{I}_{j+1})\cap\text{ext}(\mathcal{I}_{j}))$ is the
set of edges with at least one endpoint in the open set
$\text{int}(\mathcal{I}_{j+1})\cap\text{ext}(\mathcal{I}_{j})$. We define
$X_{0}$ similarly, interpreting $\mathcal{I}_{i}$ to consist of the single
point $0$.

######  Proposition 4.17.

Assume either (4.1) or (4.2). Then, there exists $C,c,s>0$ so that, for all
$j\geq 0$ and $\lambda\geq 0$,

$\mathbb{P}(X_{j}\geq\lambda)\leq Cj^{5/3}e^{-c\lambda^{s}}.$

###### Proof.

Using the consistency of the geodesics $\gamma_{n}$ between open circuits
(Proposition 2.5(viii)),

$\displaystyle\quad\;\mathbb{P}(X_{j}\geq\lambda)$
$\displaystyle\leq\mathbb{P}\Big{(}\sum_{k=\mathfrak{i}(\mathcal{I}_{j})}^{\mathfrak{o}(\mathcal{I}_{j+1})}D_{k,\mathfrak{o}(\mathcal{I}_{j+1})}\geq\lambda\Big{)}$
$\displaystyle=\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\mathbb{P}\Big{(}\sum_{k=a}^{a+b}D_{k,a+b}\geq\lambda,\mathfrak{i}(\mathcal{I}_{j})=a,\mathfrak{o}(\mathcal{I}_{j+1})-\mathfrak{i}(\mathcal{I}_{j})=b\Big{)}$
$\displaystyle\leq\sum_{a=0}^{\infty}\sum_{b=0}^{\infty}\mathbb{P}\Big{(}\sum_{k=a}^{a+b}D_{k,a+b}\geq\lambda\Big{)}^{1/3}\mathbb{P}\Big{(}\mathfrak{i}(\mathcal{I}_{j})=a\Big{)}^{1/3}\mathbb{P}\Big{(}\mathfrak{o}(\mathcal{I}_{j+1})-\mathfrak{i}(\mathcal{I}_{j})=b\Big{)}^{1/3}.$

Then by a union bound on the sum inside the probability combined with
Proposition 4.10 and Lemma 4.14, the above is bounded from above by

$\displaystyle\quad\;Cj^{1/3}\Big{(}\sum_{a=0}^{\infty}\mathbb{P}\Big{(}\mathfrak{i}(\mathcal{I}_{j})=a\Big{)}^{1/3}\Big{)}\sum_{b=0}^{\infty}(b+1)^{1/3}e^{-c_{0}\big{(}\frac{\lambda}{b+1}\big{)}^{s}}e^{-c_{1}b}$
$\displaystyle\leq
Cj^{1/3}\Big{(}\sum_{a=0}^{\infty}\mathbb{P}\Big{(}\mathfrak{i}(\mathcal{I}_{j})=a\Big{)}^{1/3}\Big{)}\sum_{b=0}^{\infty}(b+1)^{1/3}e^{-c_{0}\big{(}\frac{\lambda}{b+1}\big{)}^{s}}e^{-c_{1}b}$
$\displaystyle\leq
Cj^{1/3}\Big{(}\sum_{a=0}^{\infty}\mathbb{P}\Big{(}\mathfrak{i}(\mathcal{I}_{j})\geq
a\Big{)}^{1/3}\Big{)}\sum_{b=0}^{\infty}e^{-c_{0}\big{(}\frac{\lambda}{b+1}\big{)}^{s}}e^{-c_{2}b},$
(4.27)

where the constants changed from line to line. To bound the sum in $a$, we use
Lemma 4.11. Namely,
$\mathfrak{i}(\mathcal{I}_{j})\leq\mathfrak{o}(\mathcal{I}_{j})$ is
stochastically dominated by the sum of $j$ i.i.d. geometric random variables,
which we will denote as $X_{i}$. Then, by a union bound,

$\mathbb{P}(\mathfrak{i}(\mathcal{I}_{j})\geq
a)^{1/3}\leq\mathbb{P}\Bigl{(}\sum_{i=1}^{j}X_{i}\geq a\Bigr{)}^{1/3}\leq
j^{1/3}\mathbb{P}(X_{i}\geq a/j)^{1/3}\leq Cj^{1/3}e^{-ca/j},$

for constants $c,C>0$. Summing $e^{-ca/j}$ in $a$ yields $(1-e^{-c/j})^{-1}$,
which is bounded by a constant times $j$, as can be seen from a Taylor
expansion. Thus, (4.27) is bounded from above by

$Cj^{5/3}\sum_{b=0}^{\infty}e^{-c_{0}\big{(}\frac{\lambda}{b+1}\big{)}^{s}}e^{-c_{2}b}$
(4.28)

Now we bound the final sum using a similar approach as in the proof of
Proposition 4.10:

$\displaystyle\sum_{b=0}^{\infty}e^{-c_{0}\big{(}\frac{\lambda}{b+1}\big{)}^{s}}e^{-c_{2}b}=\sum_{b=0}^{\lceil\sqrt{\lambda}\rceil-1}e^{-c_{0}\big{(}\frac{\lambda}{b+1}\big{)}^{s}}e^{-c_{2}b}+\sum_{b=\lceil\sqrt{\lambda}\rceil}^{\infty}e^{-c_{0}\big{(}\frac{\lambda}{b+1}\big{)}^{s}}e^{-c_{2}b}$
(4.29)
$\displaystyle\leq\sum_{b=0}^{\lceil\sqrt{\lambda}\rceil-1}e^{-c_{0}\big{(}\frac{\lambda}{\lceil\sqrt{\lambda}\rceil}\big{)}^{s}}e^{-c_{2}b}+\sum_{b=\lceil\sqrt{\lambda}\rceil}^{\infty}e^{-c_{2}b}\leq
e^{-c_{3}\lambda^{s/2}}.$

Substituting into (4.28) completes the proof. ∎

We next define a random variable that controls the number of nonzero edges on
a geodesic between the last open circuit $\mathcal{I}_{Y_{2^{n+1}}}$ and
$\partial B_{2^{n+1}}$.

$X^{n}=\\#\\{e\in\gamma_{n}\cap
E(B_{2^{n+1}}\cap\text{ext}(\mathcal{I}_{Y_{2^{n+1}}})):t_{e}>0\\}.$ (4.30)

In what follows we give a tail bound for the variable $X^{n}$.

######  Proposition 4.18.

Assume either (4.1) or (4.2). There exists $C,c,s>0$ so that for each $n\geq
1$ and $\lambda\geq 0$

$\mathbb{P}(X^{n}\geq\lambda)\leq Cn^{5/4}e^{-c\lambda^{s}}.$

###### Proof.

Using Proposition 4.10 and Lemma 4.16,

$\displaystyle\mathbb{P}(X^{n}\geq\lambda)$
$\displaystyle\leq\mathbb{P}\Big{(}\sum_{k=\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})}^{n}D_{k,n}\geq\lambda\Big{)}=\sum_{a=0}^{n}\mathbb{P}\Big{(}\sum_{k=n-a}^{n}D_{k,n}\geq\lambda,n-\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})=a\Big{)}$
$\displaystyle\leq\sum_{a=0}^{n}\mathbb{P}\Bigl{(}\sum_{k=n-a}^{n}D_{k,n}\geq\lambda\Bigr{)}^{1/2}\mathbb{P}(n-\mathfrak{i}(\mathcal{I}_{Y_{2^{n+1}}})=a)^{1/2}$
$\displaystyle\leq
C\sum_{a=0}^{\infty}(a+1)^{1/2}e^{-c\big{(}\frac{\lambda}{a+1}\big{)}^{s}}n^{5/4}e^{-ca}\leq
Cn^{5/4}e^{-c_{1}\lambda^{s/2}},$

where the last line follows from the computation in (4.29). ∎

We conclude this section by proving our last main result.

###### Proof of Theorem 1.4.

As $\pi_{3}(cR)\asymp\pi_{3}(R)$ for any positive constant $c$ (see Remark
3.2), it suffices to prove the statement when $R$ is a power of $2$. Seting
$R=2^{n+1}$, it suffices to prove the statement when for the geodesic
$\gamma_{n}$ constructed from Proposition 2.5. In this formulation, the
statement to be proven is that there exist constants $c,C>0$ so that for all
$\lambda\geq 1$ and $R\geq 1$,

$\mathbb{P}(|\gamma_{n}|\geq\lambda 2^{(n+1)\varepsilon}\pi_{3}(2^{n+1}))\leq
Ce^{-c(\log(\lambda)+n)^{s}}.$ (4.31)

Let $(\mathcal{I}_{i})_{i=1}^{Y_{2^{n+1}}}$ denote the collection of open
circuits from Proposition 2.5 with $\mathcal{A}=\\{0\\}$ and
$\mathcal{B}=\partial B_{2^{n+1}}$. We let $\zeta_{n}$ denote the dual path
from Proposition 2.5 that is closed except for those edges which cross one of
the circuits $\mathcal{I}_{j}$. From the construction,
$\mathcal{I}_{1},\ldots,\mathcal{I}_{Y_{2^{n+1}}}$ are the successive
innermost edge-disjoint open circuits in the box $B_{2^{n+1}}$, and these
satisfy
$0\in\mathsf{int}(\mathcal{I}_{i})\subset\mathsf{int}(\mathcal{I}_{i+1})$ for
all $i=1,...,Y_{2^{n+1}}$. Next, recall (4.26) and (4.30), and define

$\mathfrak{X}_{n}=\max\Big{\\{}X^{n},\max_{1\leq i<Y_{2^{n+1}}}X_{i}\Big{\\}}$

to serve as an upper bound on the number of nonzero edges a geodesic will take
between two consecutive open circuits $\mathcal{I}_{i}$ and
$\mathcal{I}_{i+1}$ or between the last open circuit
$\mathcal{I}_{Y_{2^{n+1}}}$ and $\partial B_{2^{n+1}}$. Finally set
$x_{n}^{\lambda}=(\log(\lambda)+n)^{1/3}$. We prove (4.31) by bounding
$\mathbb{P}\Big{(}|\gamma_{n}|\geq\lambda(2^{n+1})^{2+\varepsilon}\pi_{3}(2^{n+1}),\mathfrak{X}_{n}\leq
x_{n}^{\lambda}\Big{)}$ and $\mathbb{P}(\mathfrak{X}_{n}\geq x_{n}^{\lambda})$
separately.

For an edge $e\subseteq B_{2^{n+1}}$, let
$M=\min\\{\operatorname{dist}(e,0),\operatorname{dist}(e,\partial
B_{2^{n+1}})\\}$. We first claim that for every such edge,

$\mathbb{P}(\\{e\in\gamma_{n}\\}\cap\\{\mathfrak{X}_{n}\leq
x_{n}^{\lambda}\\})\leq C\pi_{3}^{(x_{n}^{\lambda})}(M)$ (4.32)

where $\pi_{3}^{(k)}$ is the probability of a 3-arm polychromatic event to
distance $M$ with at most $k$ “defects” along each of the arms. This means
that there exists three disjoint arms to distance $M$; two of the arms are
primal paths which are open except at at most $k$ edges each. The other arm is
a dual arm that is closed except at at most $k$ edges. The key estimate that
will be used is [21, Prop. 18] which gives