# The random field Ising chain domain-wall structure in the large interaction
limit

Orphée Collin, Giambattista Giacomin and Yueyun Hu  Université Paris Cité,
Laboratoire de Probabilités, Statistiques et Modélisation, UMR 8001, F-75205
Paris, France  Université Paris XIII, LAGA, UMR 7539, Institut Galilée,
F-93430 Villetaneuse, France

###### Abstract.

We study the configurations of the nearest neighbor Ising ferromagnetic chain
with IID centered and square integrable external random field in the limit in
which the pairwise interaction tends to infinity. The available free energy
estimates for this model show a strong form of disorder relevance – i.e., a
strong effect of disorder on the free energy behavior – and our aim is to make
explicit how the disorder affects the spin configurations. We give a
quantitative estimate that shows that the infinite volume spin configurations
are close to one explicit disorder dependent configuration when the
interaction is large. Our results confirm predictions on this model obtained
in [18] by applying the renormalization group method introduced by D. S.
Fisher [17].

_AMS subject classification (2010 MSC)_ : 60K37, 82B44, 60K35, 82B27

_Keywords_ : disordered systems, transfer matrix method, random matrix
products, infinite disorder RG fixed point, random walk excursion theory

This version:

##  1\. The model and the results

###  1.1. The Ising chain with disordered external field

For $N\in{\mathbb{N}}:=\\{1,2,\ldots\\}$ we introduce the partition function
of the Ising chain of length $N$, interaction $J$ and external magnetization
$h=(h_{n})$

$Z_{N,J,h}^{ab}\,=\,\sum_{\sigma\in\\{-1,+1\\}^{\\{1,2,\ldots,N\\}}}\exp\left(H^{ab}_{N,J,h}(\sigma)\right)\,,$
(1.1)

where

$H^{ab}_{N,J,h}(\sigma)\,:=\,J\sum_{n=1}^{N+1}\sigma_{n-1}\sigma_{n}+\sum_{n=1}^{N}h_{n}\sigma_{n}\,,$
(1.2)

with $\sigma_{0}=a$ and $\sigma_{N+1}=b$ ($a,b\in\\{-,+\\}\cong\\{-1,+1\\}$),
and $h=(h_{n})$ is a sequence of real numbers. Our results will be for the
case in which $(h_{n})$ is a realization of an IID sequence of centered random
variables with bounded non zero variance

${\mathbb{E}}[h_{1}]\,=0\ \text{ and
}\vartheta^{2}\,:=\,{\mathbb{E}}\left[h_{1}^{2}\right]\in(0,\infty)\,.$ (1.3)

In spite of the fact that for $(Z_{N,J,h})_{N\in{\mathbb{N}}}$ we only need
$(h_{n})_{n\in{\mathbb{N}}}$, we will often deal with the Ising model on
arbitrary subsets of ${\mathbb{Z}}$, so we consider
$(h_{n})_{n\in{\mathbb{Z}}}$ from the start.

The Gibbs measure ${\mathbf{P}}^{ab}_{N,J,h}$ associated to the partition
function (1.1) is the probability on $\\{-1,+1\\}^{\\{1,\ldots,N\\}}$
identified by the discrete density

${\mathbf{P}}^{ab}_{N,J,h}\left(\sigma\right)\,=\,\frac{\exp\left(H^{ab}_{N,J,h}(\sigma)\right)}{Z_{N,J,h}^{ab}}\,.$
(1.4)

Of course a spin configuration $\sigma\in\\{-1,+1\\}^{\\{0,1,\ldots\\}}$,
given $\sigma_{0}$, can be encoded in the increasing sequence $(n_{j})$
defined by $n_{1}:=\inf\\{n\in{\mathbb{N}}:\,\sigma_{n}\neq\sigma_{0}\\}$ and,
for $j>1$, $n_{j}:=\inf\\{n>n_{j-1}:\,\sigma_{n}\neq\sigma_{n-1}\\}$.
Therefore one can see the spin configuration $\sigma$ as a sequence of domains
in which the spins are constant. The boundaries (or walls) of the domains are
at $(n_{j}-1/2)_{j\in{\mathbb{N}}}$ and the sign of the spins switches
crossing the domain walls.

We aim at understanding the behavior of ${\mathbf{P}}^{ab}_{N,J,h}$ in the
thermodynamical limit $N\to\infty$ when the interaction $J$ is large.

For $J$ large one certainly expects that neighboring spins that have the same
sign are greatly favored, so the observed trajectories should contain _very
long_ domains.

This is easily established when $\vartheta=0$, i.e. $h_{n}=0$ for every $n$.
In fact in this case, see Remark 2.1, $(n_{j})$ is a renewal process with
geometric inter-arrival distribution: the expectation of the inter-arrival
variable, i.e. the expected length of the domains, is $\exp(2J)+1$.

What is expected in presence of centered disorder with finite variance agrees
with the non disordered case only because the domain sizes diverge with
$J\to\infty$, but it is otherwise radically different. The _Imry-Ma argument_
– by now a textbook argument in physics explained in our context for example
in [23, p. 373], but also in the introduction of [10] – strongly suggests that

  1. (1)

the typical length of the domains diverges with $J$, but only like $J^{2}$;

  2. (2)

the location of the walls between domains heavily depends on the disordered
variables realization.

In the physical literature the analysis of this problem has been pushed well
beyond the Imry-Ma predictions: the key word here is Renormalization Group
(RG) and notably the substantial refinement by D. S. Fisher [16, 17] of ideas
initiated in particular by C. Dasgupta and S.-K. Ma [12]. In the next
subsection, § 1.2, we are going to give the definitions that are necessary to
state our results: these definitions actually introduce some stochastic
processes that can be directly related to the RG predictions. And our aim is
proving a statement that shows that Fisher RG predictions [18] do hold for the
infinite volume ferromagnetic Ising chain at equilibrium with centered
external field, in the limit in which the interaction tends to infinity.

###  1.2. The Fisher domain-wall configuration

Let us set

$\Gamma\,:=\,2J\,,$ (1.5)

and let us introduce from now the two-sided random walk
$(S_{n})_{n\in{\mathbb{Z}}}$ defined by

$S_{n}\,:=\begin{cases}\sum_{j=1}^{n}h_{j}&\text{ if }n\in{\mathbb{N}}\,,\\\
0&\text{ if }n=0\,,\\\ -\sum_{j=n+1}^{0}h_{j}&\text{ if
}-n\in{\mathbb{N}}\,,\end{cases}$ (1.6)

even if the two-sided aspect will play an explicit role only from Section 3.
Let us introduce also for $m\leq n$

$S^{\uparrow}_{m,n}\,:=\,\max_{m\leq i\leq j\leq n}(S_{j}-S_{i}),\qquad
S^{\downarrow}_{m,n}:=\max_{m\leq i\leq j\leq n}(S_{i}-S_{j})\,,$ (1.7)

and for every $\Gamma>0$ we introduce the _first time of $\Gamma$-decrease_

$\mathtt{t}_{1}(\Gamma)\,:=\,\inf\left\\{n>0:S^{\downarrow}_{0,n}\geq\Gamma\right\\}\,.$
(1.8)

For almost all realizations of $(h_{n})$ we have that
$\limsup_{n}S_{n}=-\liminf_{n}S_{n}=\infty$: we assume that we work on such
realizations, so the infimum in (1.8) can be replaced by a minimum and
$\mathtt{t}_{1}(\Gamma)<\infty$. Next we introduce (with the notation
$\llbracket j,k\rrbracket:=[j,k]\cap{\mathbb{Z}}$ for the integers $j\leq k$)

$\mathtt{u}_{1}(\Gamma)\,:=\,\min\left\\{n\in\llbracket
0,\mathtt{t}_{1}(\Gamma)\rrbracket:S_{n}=\max_{i\in\llbracket
0,\mathtt{t}_{1}(\Gamma)\rrbracket}S_{i}\right\\}\,,$ (1.9)

so $\mathtt{u}_{1}(\Gamma)$ is the first time that $S$ reaches its maximum
before having a decrease of at least $\Gamma$: this is what we call _location
of the first $\Gamma$-maximum_. Note however that if the law of $h_{1}$ is
lattice supported, then $S$ may reach this maximum also at later times (before
$\mathtt{t}_{1}(\Gamma)$). In fact, this happens beyond the case in which
$h_{1}$ is lattice supported: it happens with positive probability if and only
if the law of $h_{1}$ is equal to $pP+(1-p)P^{\prime}$ with $p\in(0,1]$, $P$
and $P^{\prime}$ two probability measures and the support of $P$ spans
$x{\mathbb{Z}}$ for a value of $x>0$. So we introduce also

$\mathtt{u}^{+}_{1}(\Gamma)\,:=\,\max\left\\{n\in\llbracket
0,\mathtt{t}_{1}(\Gamma)\rrbracket:\,S_{n}=\max_{i\in\llbracket
0,\mathtt{t}_{1}(\Gamma)\rrbracket}S_{i}\right\\}\,.$ (1.10)

Now we proceed by looking for the first time of $\Gamma$-increase after
$\mathtt{t}_{1}(\Gamma)$

$\mathtt{t}_{2}(\Gamma)\,:=\,\min\left\\{n>\mathtt{t}_{1}(\Gamma):\,S^{\uparrow}_{\mathtt{t}_{1}(\Gamma),n}\geq\Gamma\right\\}\,,$
(1.11)

and we set

$\mathtt{u}_{2}(\Gamma)\,:=\,\min\left\\{n\in\llbracket\mathtt{t}_{1}(\Gamma),\mathtt{t}_{2}(\Gamma)\rrbracket:\,S_{n}=\min_{i\in\llbracket\mathtt{t}_{1}(\Gamma),\mathtt{t}_{2}(\Gamma)\rrbracket}S_{i}\right\\}\,,$
(1.12)

and

$\mathtt{u}^{+}_{2}(\Gamma)\,:=\,\max\left\\{n\in\llbracket\mathtt{t}_{1}(\Gamma),\mathtt{t}_{2}(\Gamma)\rrbracket:\,S_{n}=\min_{i\in\llbracket\mathtt{t}_{1}(\Gamma),\mathtt{t}_{2}(\Gamma)\rrbracket}S_{i}\right\\}\,.$
(1.13)

So $\mathtt{t}_{2}(\Gamma)$ is the first time at which there is an increase of
$S$ at least $\Gamma$ after the decrease time $\mathtt{t}_{1}(\Gamma)$. And
$\mathtt{u}_{2}(\Gamma)$, respectively $\mathtt{u}^{+}_{2}(\Gamma)$, is the
first (respectively last) absolute minimum of the walk between
$\mathtt{t}_{1}(\Gamma)$ and $\mathtt{t}_{2}(\Gamma)$.

Now (almost surely in the realization of the $h$ sequence) we can iterate this
procedure to build the increasing sequences
$(\mathtt{t}_{j}(\Gamma))_{j\in{\mathbb{N}}}$,
$(\mathtt{u}_{j}(\Gamma))_{j\in{\mathbb{N}}}$ and
$(\mathtt{u}^{+}_{j}(\Gamma))_{j\in{\mathbb{N}}}$ with

$\mathtt{t}_{j}(\Gamma)\,\leq\mathtt{u}_{j}(\Gamma)\,\leq\,\mathtt{u}^{+}_{j}(\Gamma)\,<\,\mathtt{t}_{j+1}(\Gamma)\,,$
(1.14)

with $\mathtt{u}_{j}(\Gamma)$ and $\mathtt{u}^{+}_{j}(\Gamma)$ that are the
locations of the (first and last) maxima (respectively minima) of $S$, more
precisely the first and last $S$ absolute maxima in
$[\mathtt{t}_{j}(\Gamma);\mathtt{t}_{j+1}]$, if $j$ is odd (respectively
even). Explicitly

$\mathtt{t}_{j+1}(\Gamma)\,:=\begin{cases}\min\big{\\{}n>\mathtt{t}_{j}(\Gamma):\,S^{\downarrow}_{\mathtt{t}_{j}(\Gamma),n}\geq\Gamma\big{\\}}&\textrm{
if $j$ is even}\,,\\\
\min\big{\\{}n>\mathtt{t}_{j}(\Gamma):\,S^{\uparrow}_{\mathtt{t}_{j}(\Gamma),n}\geq\Gamma\big{\\}}&\textrm{
if $j$ is odd}\,,\end{cases}$ (1.15)
$\mathtt{u}_{j+1}(\Gamma)\,:=\begin{cases}\min\left\\{n\in\llbracket\mathtt{t}_{j}(\Gamma),\mathtt{t}_{j+1}(\Gamma)\rrbracket:\,S_{n}=\max_{i\in\llbracket\mathtt{t}_{j}(\Gamma),\mathtt{t}_{j+1}(\Gamma)\rrbracket}S_{i}\right\\}&\textrm{
if $j$ is even}\,,\\\
\min\left\\{n\in\llbracket\mathtt{t}_{j}(\Gamma),\mathtt{t}_{j+1}(\Gamma)\rrbracket:\,S_{n}=\min_{i\in\llbracket\mathtt{t}_{j}(\Gamma),\mathtt{t}_{j+1}(\Gamma)\rrbracket}S_{i}\right\\}&\textrm{
if $j$ is odd}\,,\end{cases}$ (1.16)

and $\mathtt{u}_{j+1}^{+}(\Gamma)$ is defined like $\mathtt{u}_{j+1}(\Gamma)$
with the minimum for
$n\in\llbracket\mathtt{t}_{j}(\Gamma),\mathtt{t}_{j+1}(\Gamma)\rrbracket$
replaced by a maximum.

Like before, we say that $\mathtt{u}_{j}(\Gamma)$ and
$\mathtt{u}_{j}^{+}(\Gamma)$ are location of $\Gamma$-extrema: maxima
(respectively minima) if $j$ is odd (respectively even). In
$\llbracket\mathtt{u}_{j}(\Gamma),\mathtt{u}_{j}^{+}(\Gamma)\rrbracket$ there
can be other $\Gamma$-maxima if $j$ is odd (or $\Gamma$-minima if $j$ is
even).

######  Remark 1.1.

Of course the choice of starting with $\mathtt{t}_{1}(\Gamma)$ which is a time
of $\Gamma$-decrease is arbitrary: we could have chosen to start by looking
for a time of $\Gamma$-increase. This leads to a sequence of $\Gamma$-extrema
locations $(\mathtt{u}_{j})$ and $(\mathtt{u}^{+}_{j})$ that differs only for
the the first entry of the sequence and for the labeling of the sequence. More
precisely, if we set

$\mathtt{t}^{\downarrow}(\Gamma)\,:=\,\inf\left\\{n\geq
1:\,S^{\downarrow}_{0,n}\geq\Gamma\right\\}\ \text{ and }\
\mathtt{t}^{\uparrow}(\Gamma)\,:=\,\inf\left\\{n\geq
1:\,S^{\uparrow}_{0,n}\geq\Gamma\right\\}\,,$ (1.17)

and if we choose the opposite convention of looking first for a
$\Gamma$-increase (we call standard the other convention)

  * •

either $\mathtt{t}^{\uparrow}(\Gamma)<\mathtt{t}^{\downarrow}(\Gamma)$ and the
first detected $\Gamma$-extremum is the first of the absolute minima in
$\llbracket 0,\mathtt{t}^{\uparrow}(\Gamma)\rrbracket$, and the second one is
the $\Gamma$-maximum we found at $\mathtt{t}_{1}(\Gamma)$ with the standard
convention: after that the two procedures coincide except for the shift of $1$
in the labels;

  * •

or $\mathtt{t}^{\downarrow}(\Gamma)<\mathtt{t}^{\uparrow}(\Gamma)$ and
$\mathtt{t}_{1}(\Gamma)=\mathtt{t}^{\downarrow}(\Gamma)$ goes undetected. The
first detected extremum is the first absolute minimum in $\llbracket
0,\mathtt{t}^{\uparrow}(\Gamma)\rrbracket$, which coincides with the first
absolute minimum in
$\llbracket\mathtt{t}^{\downarrow}(\Gamma),\mathtt{t}^{\uparrow}(\Gamma)\rrbracket$
and it is the first minimum, at $\mathtt{t}_{2}(\Gamma)$, detected with the
standard convention: also in this case the two procedures coincide at all
later times, expect for the label shift of $1$.

We are now ready to define $\left(s_{n}^{(F,+)}\right)_{n\in{\mathbb{N}}}$:

$s_{n}^{(F,+)}\,:=\,\begin{cases}+1&\text{ if }\exists j\text{ even such that
}n\in[\mathtt{u}^{+}_{j}(\Gamma),\mathtt{u}_{j+1}(\Gamma))\text{ and
}S_{\mathtt{u}_{j+1}(\Gamma)}-S_{\mathtt{u}^{+}_{j}(\Gamma)}>\Gamma,\\\
-1&\text{ if }\exists j\text{ odd\,\, such that
}n\in[\mathtt{u}^{+}_{j}(\Gamma),\mathtt{u}_{j+1}(\Gamma))\text{ and
}S_{\mathtt{u}^{+}_{j}(\Gamma)}-S_{\mathtt{u}_{j+1}(\Gamma)}>\Gamma,\\\
\phantom{1}0&\text{ otherwise.}\end{cases}$ (1.18)

Figure 1.  The sequence of $\Gamma$-extrema for the simple symmetric random
walk with $\Gamma=5/2$: in fact, $\Gamma\in(2,3]$ leads to the very same
sequence of $\Gamma$ extrema.

###  1.3. The main result

We let $\log_{\circ 2}(\cdot):=\log(\log(\cdot))$.

######  Theorem 1.2.

Assume that (1.3) holds. Then there exists a positive (deterministic) function
$\Gamma\to D_{\Gamma}$ (see (3.19) for a concrete expression) such that for
almost every realization of $(h_{n})$ and every choice of the boundary spins

$\lim_{N\to\infty}\frac{1}{N}\left|\left\\{n=1,2\ldots,N:\,\sigma_{n}\neq
s^{(F,+)}_{n}\right\\}\right|\,=\,D_{\Gamma}\,,$ (1.19)

in ${\mathbf{P}}^{ab}_{N,J,h}$ probability. Moreover for $\Gamma\to\infty$

$D_{\Gamma}\,=\,O\left(\frac{\log_{\circ 2}\Gamma}{\Gamma}\right)\,.$ (1.20)

Note that $D_{\Gamma}$ is naturally interpreted as _discrepancy density_
between the spin configuration and the Fisher configuration. We will discuss
this result at length in § 1.4 notably about its relation with the analogous
result proven in [10], which deals with a continuum version of the model we
consider. The continuum model in [10] arises as a weak disorder scaling limit
of the disordered Ising model we consider. The continuum model is easily
guessed if one first remarks that the Gibbs measure we consider, i.e. (1.4),
is an exponential modification involving the disorder that is given by the IID
sequence $(h_{n})$ – or, equivalently, by the associated random walk $(S_{n})$
– of an underlying free spin process in which the spins are IID Rademacher
variables. The continuum model consists in replacing the free spin process by
a continuous time Markov chain on the state space $\\{-1,+1\\}$ with unit
(Poisson) jump rates, and the disorder $(S_{n})$ by a Brownian motion. In [10,
App. C] is shown that the partition function (1.1) converges when
$N\to\infty$, $\vartheta\to 0$ and $J\to\infty$ in a suitable joint way to the
Continuum partition function in [10]. It is somewhat surprising that this weak
disorder limit model preserves the main features of the underlying discrete
model: for the continuum model a result qualitatively (and, to a certain
extent, also quantitatively) close to Theorem 1.2 does hold. This will be
discussed further in § 1.4, but we anticipate here that the analogous
construction of the sequence of $\Gamma$-extrema of § 1.2 – we call it Neveu-
Pitman process – was first performed in [30] by J. Neveu and J. Pitman for a
Brownian trajectory, without the nuisance of having to deal with the
possibility that $\mathtt{u}_{j}<\mathtt{u}_{j}^{+}$. Moreover, in the
Brownian case, by scaling, the value of $\Gamma$ is inessential and one can
focus on $1$-extrema. We do not reproduce the construction here and we refer
of course to [30], but also to [8, 5, 10]. For the location of the $1$-extrema
of a standard Brownian motion trajectory $B$ we use the notation
$\upsilon_{1},\upsilon_{2},\ldots$. In fact, according to [30], the sequence
of $1$-extrema times $(\upsilon_{j})_{j\in{\mathbb{N}}}$ forms a renewal
sequence and
$((\upsilon_{j+1}-\upsilon_{j},|B_{\upsilon_{j+1}}-B_{\upsilon_{j}}|))_{n\in{\mathbb{N}}}$
is an IID sequence too.
$(\upsilon_{2}-\upsilon_{1},|B_{\upsilon_{2}}-B_{\upsilon_{1}}|)$ is a random
variable in $(0,\infty)\times(1,\infty)$ whose law is explicitly known. More
precisely the Laplace transform of this two dimensional law is known
explicitly: the two marginal laws are explicit, notably
$|B_{\upsilon_{2}}-B_{\upsilon_{1}}|-1$ is an exponential random variable of
parameter $1$ ([30], see also [18]).

Going back to the discrete case, the situation is different, but not
substantially different, in part thanks to Donsker Invariance Principle. We
collect the random walk analog of the Pitman-Neveu result in the next
statement.

######  Proposition 1.3.

We have that

$\left(\mathtt{u}_{n+1}(\Gamma)-\mathtt{u}_{n}(\Gamma),\left|S_{\mathtt{u}_{n+1}(\Gamma)}-S_{\mathtt{u}_{n}(\Gamma)}\right|\right)_{n\in{\mathbb{N}}}\,,$
(1.21)

is a sequence of independent random vectors taking values in
${\mathbb{N}}\times[\Gamma,\infty)$. Moreover the subsequence with even
(respectively odd) indices, i.e. $n\in 2{\mathbb{N}}$ (respectively $n\in
2{\mathbb{N}}-1$), is IID and, with the usual notation for convergence in law,
we have that for every $n\in{\mathbb{N}}$

$\left(\frac{\vartheta^{2}}{\Gamma^{2}}\left(\mathtt{u}_{n+1}(\Gamma)-\mathtt{u}_{n}(\Gamma)\right),\frac{\left|S_{\mathtt{u}_{n+1}(\Gamma)}-S_{\mathtt{u}_{n}(\Gamma)}\right|}{\Gamma}\right)\stackrel{{\scriptstyle\Gamma\to\infty}}{{\Longrightarrow}}\left(\upsilon_{2}-\upsilon_{1},|B_{\upsilon_{2}}-B_{\upsilon_{1}}|\right)\,.$
(1.22)

Proposition 1.3 is in part already present in [5] and we prove it in App. C.
It is of great conceptual importance because it provides an explicit
quantitative link with the RG predictions [18]: in particular Proposition 1.3
makes explicit the $\Gamma^{2}$ (spatial) scale for the Fisher configuration,
as predicted by the Imry-Ma argument.

###  1.4. Discussion of the result and overview of the rest of the paper

#### A quick overview of our main arguments

The proof of Theorem 1.2 relies on the transfer matrix representation of
$Z_{N,J,h}$: in transfer matrix terms we have

$Z^{++}_{N,J,h}\,=\,(M_{0}M_{1}M_{2}\ldots
M_{N})_{1,1}\exp\left(-h_{0}\right)\,,$ (1.23)

where $M_{k}=T(h_{k})Q$, with $T(h_{k})$ and $Q$ that are $2\times 2$ matrices
with non negative entries explicitly given in (2.9). We are therefore dealing
with a product of IID random matrices and the key tool in analyzing random
matrix products is the analysis of the action of these matrices on the
(projective space of the) direction of vectors, see e.g. [33]. Since the
dimension is two the direction of a vector
$v\in{\mathbb{R}}^{2}\setminus\\{0\\}$ is encoded by an angle
$\theta\in[0,\pi]$ and, since the matrices have positive entries, one can
restrict to $[0,\pi/2]$: $v=|v|(\cos(\theta),\sin(\theta))$ and $\theta$
identifies the direction. In more explicit terms, $(v_{n})$ defined
recursively, given $v_{0}$ (say, non random), by $v_{n}=M_{n}v_{n-1}$ is a
Markov chain with state space $[0,\infty)^{2}\setminus\\{(0,0)\\}$, and it
directly induces the _projective_ Markov chain $(\theta_{n})$, with state
space $[0,\pi/2]$. For us it is more practical to work with
$l_{1,n}:=\log\tan\theta_{n}$, a Markov chain on ${\mathbb{R}}$ (in reality,
we will see that the state space is smaller: $(-\Gamma,\Gamma)$). We point out
that $l_{1,n}$ depends on $h_{1},\ldots,h_{n}$ and the subscript $1$ is there
to recall that $n=1,2,\ldots$. In fact we will need to consider also the time
reversed Markov chain $(r_{N-n,N})_{n=0,1,\ldots}$, with initial condition
$r_{N,N}$, which has the _same_ transition mechanism as $(l_{1,n})$, except
that $h$ is replaced by $-h$: by this we mean that
$l_{1,n}=f_{h_{n}}(l_{1,n-1})$ and $r_{N-n,N}=f_{-h_{N-n}}(r_{N-n+1,N})$, for
a suitable function $f_{h}$ (given in (2.14)). So $r_{n+1,N}$ depends on
$h_{n+1},h_{n+2},\ldots,h_{N}$.

We are going to explain in Section 2.1 that for every $n\in\\{1,\ldots,N\\}$
we have the identity

${\mathbf{P}}^{ab}_{N,J,h}\left(\sigma_{n}=+1\right)\,=\,\frac{1}{1+\exp\left(-\left(l_{1,n-1}+2h_{n}+r_{n+1,N}\right)\right)}\,,$
(1.24)

and $l_{1,n-1}$, $h_{n}$ and $r_{n+1,N}$ are independent random variables
because of the their dependence on the sequence $(h_{n})$ of independent
variables. The boundary condition $a$ (resp. $b$) determines $l_{1,1}$ (resp.
$r_{N,N}$). The intuitive explanation is that $l_{1,\cdot}$, respectively
$r_{\cdot,N}$, carries the _disorder_ information from the left (respectively
right) boundary to the bulk of the system.

Here are three important facts for our analysis:

  1. (1)

We are going to consider $N\to\infty$ and sites $n$ away from the boundary, so
the processes $l$ and $r$ may be considered to be stationary (and independent
of the boundary conditions: so we omit the subscripts $1$ and $N$). Obtaining
properties on their common invariant probability is one of the important steps
in our arguments.

  2. (2)

The Markov kernels of $(l_{n})$ and of $(r_{-n})$ depend also on $J$. In fact
$(l_{n})$ is a random walk with increments $(2h_{n})$ and a _repulsion
mechanism_ that forces the walk never to leave $(-\Gamma,\Gamma)$. This
repulsion effectively acts only when the walk is at finite distance from
$\pm\Gamma$, even if, strictly speaking, it is felt in the whole of
$(-\Gamma,\Gamma)$. As $J\to\infty$, the two walls become farther and farther
so, on a large scale, this repulsion becomes more and more similar to the
(simpler) hard wall repulsion, i.e. the case in which the process behaves
exactly like a random walk, and if a jump would make it leave
$(-\Gamma,\Gamma)$, the walk is just stopped at the boundary (i.e., at
$\Gamma$ or at $-\Gamma$): of course the very same remarks apply to $(r_{n})$,
by time reversal. The hard wall processes are denoted by $(\widehat{l}_{n})$
and $\left(\widehat{r}_{n}\right)$. A second important step for us is
controlling the error in replacing $l$ and $r$ with $\widehat{l}$ and
$\widehat{r}$.

  3. (3)

The limit we consider is $N\to\infty$ before $J$. Since $n$ is away from the
boundary, for large $J$ we expect $|l_{n-1}|$ and $|r_{n+1}|$ to be typically
very large. Therefore $|l_{n-1}+2h_{n}+r_{n+1}|$ is expected to be large so
the probability in (1.24) is essentially $0$ or $1$. But whether it is $0$ or
$1$ depends crucially on the sign of $l_{n-1}+2h_{n}+r_{n+1}$. From these
observations we infer that, with high probability (as $J\to\infty$),
$\sigma_{n}$ is going to be equal to
$\operatorname{\mathrm{sign}}(l_{n-1}+2h_{n}+r_{n+1})$ and the spin
configuration $(\operatorname{\mathrm{sign}}(l_{n-1}+2h_{n}+r_{n+1}))_{n}$ is
our best bet for the behavior of the true spin configuration $(\sigma_{n})$.
On the basis of this discussion, one expects that $s^{(F,+)}_{n}$ is equal or
very close to be equal to
$\operatorname{\mathrm{sign}}(l_{n-1}+2h_{n}+r_{n+1})$. What we will show is
slightly different and very sharp:
$\operatorname{\mathrm{sign}}\left(\widehat{l}_{n-1}+2h_{n}+\widehat{r}_{n+1}\right)=s^{(F,+)}_{n}$
for $n\geq n_{0}$ with $n_{0}$ disorder dependent, but a.s. finite. Needless
to say, this is another crucial step to control the discrepancies between the
spin configuration and the Fisher configuration.

#### About the main result

Theorem 1.2 is analogous to the result proven in [10] for the continuum model
arising in the weak disorder rescaling limit. Our approach is highly robust:
in the discrete setup and assuming only (1.3), we match the result obtained in
the continuum and Gaussian context. While the argument of proof that we
present here is built from the strategy exploited in [10], we draw the
reader’s attention to the following facts.

  * •

In [10] the analog of the processes $(l_{n})$ and $(r_{-n})$ solve stochastic
differential equations that are (to a certain extent) solvable: in particular,
the invariant probability has an explicit expression in terms of Bessel
functions. We do not have an explicit expression of the invariant probability
of the Markov chains we deal with for the Ising model: it is possible to show
that the rescaled process $(l_{\lceil\Gamma^{2}t\rceil}/\Gamma)_{t\geq 0}$
converges (as $\Gamma\to\infty$) to a Brownian motion reflected at the
boundary of $[-1,1]$, and from this fact one can extract that the invariant
probability of the chain converges in the same limit to the uniform
probability over $[-1,1]$. But such a _macroscopic_ control on the invariant
probability, i.e. on intervals of length $\propto\\!\Gamma$, is of no help
and, in general, on small scales the invariant probability is not expected to
be close to the uniform measure (see Fig. 3 and its caption): and we need a
control on the invariant measure down to (almost _microscopic_) intervals of
length $\propto\\!\log_{\circ 2}\Gamma$.

  * •

Rather than relying on Brownian approximations of random walks (that appear to
yield rougher and much less general results), we perform the whole procedure
at the level of Markov chains: this demands a number of estimates that are
really discrete process estimates. Donsker Invariance Principle is used only
in Proposition 1.3 that states a fact that is very important in order to
appreciate the behavior of the Fisher configuration $s^{(F)}$ and the link
with Fisher RG. But, mathematically, Proposition 1.3 is a result independent
of Theorem 1.2.

  * •

About the law of $h_{1}$ we just ask for $h_{1}\in{\mathbb{L}}^{2}$. This
shows a remarkable universality of the result and it is even somewhat
surprising from the statistical mechanics viewpoint: note that the annealed
version does not exist at all under such a general assumption on the disorder.
And we are also able to deal with cases in which the law of $h$ has a lattice
component: the difficulty in this case is that there can be several extrema at
the same height.

  * •

Working with Markov chains causes a priori more troubles than dealing with
Brownian motion like in [10], but it gives also more flexibility at very small
scales: in particular, the fact that we can work with processes $l$ and $r$ on
a bounded state space $(-\Gamma,\Gamma)$ follows by a transfer matrix/Markov
chain trick. The boundedness property does simplify certain estimates.

We do not repeat here the detailed discussion in [10, Sec. 2.4] about the
relation of the pathwise results we present with the sharp free energy
estimates available in the literature. But we recall that for the free energy
density $\textsc{f}(J):=\lim_{N\to\infty}(1/N){\mathbb{E}}\log
Z_{N,J,h}^{ab}$, which does not depend on the choice of the boundary spins $a$
and $b$, we have the sharp estimate for $J\to\infty$ [21]

$\textsc{f}(J)\,=\,J+\frac{\kappa_{1}}{2J+\kappa_{2}}+O(\exp(-CJ))\,=\,J+\kappa_{1}\sum_{j=0}^{\infty}\frac{\kappa_{2}^{j}}{(2J)^{j+1}}+O(\exp(-CJ))\,,$
(1.25)

with $\kappa_{1}>0$, $\kappa_{2}\in{\mathbb{R}}$ and $C>0$, under the
hypothesis that there exists $c>0$ such that
${\mathbb{E}}[\exp(c|h_{1}|)]<\infty$ and that $h_{1}$ has a density that is
uniformly Hölder continuous. One of us has actually shown [9] that, assuming
(1.3) and that ${\mathbb{E}}[|h_{1}|^{p}]<\infty$ for $p>(3+\sqrt{5})/2\approx
2.618$ we have $\textsc{f}(J)\,=\,J+{\vartheta^{2}}/{(2J)}+o(1/J)$. This
yields in particular that when (1.25) holds, then $\kappa_{1}=\vartheta^{2}$.
The coefficient $\vartheta^{2}$ is strongly suggested, but not proven, by
Fisher RG and by Theorem 1.2 that say that the observed spin configurations
are close to $s^{(F,+)}$. In fact, one can show, by restricting the
computation of the partition function to $\sigma=s^{(F,+)}$, that in great
generality $\liminf_{J}2J(\textsc{f}(J)-J)\geq\vartheta^{2}$, but the
corresponding upper bound does not appear to be straightforward and can be
proven under the assumptions of [9]. In [18] a computation of the free energy
to order $1/J^{2}$ is given, but the result is not reliable because the
$1/J^{2}$ correction to the free energy depends on the trajectory that are
close to the optimal trajectory $s^{(F,+)}$ and not just on $s^{(F,+)}$ [21].
As a matter of fact, it is quite clear that the RG prediction cannot go beyond
the $1/J$ term in the free energy expansion and the finer corrections are
model dependent (and, in our case, dependent on the details of the law of
$h_{1}$ beyond the variance), even if the whole series in $1/J$ for the free
energy density (1.25) has the remarkable (and for now rather mysterious)
property of just depending on the constant $\kappa_{2}$.

The evident gap between the pathwise results we present here, assuming only
(1.3) on the law of $h_{1}$, and the free energy estimate (1.25), that demands
strong conditions on the law of $h_{1}$, is a natural source of open
questions, notably

  * •

the exponentially small remainder in (1.25) is directly linked to the
assumption of exponential integrability of $h_{1}$: what is the generalization
of (1.25) with weaker integrability assumptions?

  * •

What is the generalization of (1.25) under weaker regularity assumptions on
the law of $h_{1}$, for example in absence of density? [9] presents a result
in this direction.

We add to these open questions the following two open questions (discussed in
more detail in the introduction of [10]):

  1. (1)

The issue of whether the estimate $D_{\Gamma}=O(\log_{\circ
2}(\Gamma)/\Gamma)$ is optimal or not is open. It is worth stressing that also
in our case, and assuming $h_{1}$ to be Gaussian, the overlap estimate ([10,
Rem. 2.7]) yields $\liminf\Gamma\,D_{\Gamma}\geq\vartheta/2$ holds (see Remark
1.4).

  2. (2)

The case ${\mathbb{E}}[h_{1}]\neq 0$ is tackled via RG too in [18]. However
the results claimed in [18] are not expected to be exact (and proven not to be
exact [20]): Fisher RG is expected to be only _asymptotically_ exact (see also
[23, 34]) when ${\mathbb{E}}[h_{1}]\neq 0$ (meaning a suitable limit as
${\mathbb{E}}[h_{1}]\to 0$) and a free energy result in this direction is
proven in [20], but results beyond the free energy behavior are open issues.

######  Remark 1.4.

If $h_{1}\sim N(0,\vartheta^{2})$ the assumptions of [21, 9] are satisfied, so
(1.25) holds, in particular
$\textsc{f}_{\vartheta}(J):=\textsc{f}(J)=J+\vartheta^{2}/(2J)+O(1/J^{2})$.
The arguments in [21] yield that $\kappa_{2}$ is continuous in $\vartheta$ and
that the rest $O(\exp(-CJ)$ is locally uniform in $\vartheta$, so the bound on
the term $O(1/J^{2})$ is locally uniform in $\vartheta$. By a standard
Gaussian integration by parts trick in “replica computations” [4], we obtain
[10, (C.15)]
$\partial_{\vartheta}\textsc{f}_{\vartheta}(J)=2\lim_{N\to\infty}{\mathbb{E}}{\mathbf{E}}^{\otimes
2}_{N,J,h}\left[\frac{1}{N}\sum_{n=1}^{N}\mathbf{1}_{\sigma^{(1)}_{n}\neq\sigma^{(2)}_{n}}\right]$,
if $\textsc{f}_{\vartheta}(J)$ is differentiable at $\vartheta$ (which, by
convexity, holds except for at most a countable amount of values of
$\vartheta$: we tacitly assume that we are out of this countable set when we
write $\partial_{\vartheta}\textsc{f}_{\vartheta}(J)$). Now remarking that
with $s_{n}:=s^{(F,+)}_{n}$

${\mathbf{P}}_{N,J,h}^{\otimes
2}\left(\sigma_{n}^{(1)}\neq\sigma_{n}^{(2)}\right)\leq{\mathbf{P}}_{N,J,h}^{\otimes
2}\left(\sigma_{n}^{(1)}\neq s_{n}\text{ or }\sigma_{n}^{(1)}\neq
s_{n}\right)\leq 2{\mathbf{P}}_{N,J,h}\left(\sigma_{n}\neq s_{n}\right)\,,$
(1.26)

we obtain $\partial_{\vartheta}\textsc{f}_{\vartheta}(J)/4\leq
D_{\Gamma}(\vartheta)$, where we have made explicit the dependence on
$\vartheta$ in $D_{\Gamma}$. By convexity for
$\vartheta^{\prime}\geq\vartheta$ and every $\delta\in(0,\vartheta)$ we have
$\partial_{\vartheta^{\prime}}\textsc{f}_{\vartheta^{\prime}}(J)\geq(\textsc{f}_{\vartheta}(J)-\textsc{f}_{\vartheta-\delta}(J))/\delta=(\vartheta^{2}-(\vartheta-\delta)^{2})/(2J\delta)+O(1/J^{2})$.
Hence $\Gamma D_{\Gamma}(\vartheta)\,\geq\,(2\vartheta-\delta)/4+O(1/\Gamma)$
and we conclude that $\liminf_{\Gamma}\Gamma
D_{\Gamma}(\vartheta)\geq\vartheta/2$ for every $\vartheta>0$.

The review of the literature cannot be complete without mentioning that the
Fisher RG prediction have been established for Sinai random walks (and
diffusions) in random environment [8, 5]. The Sinai walk has a centering
condition that is analogous to requiring that $h_{1}$ is centered in our case.
Likewise, the results available for transient one dimensional random walks in
random environments [15], corresponding to ${\mathbb{E}}[h_{1}]\neq 0$, expose
the limits of Fisher RG outside the centered disorder case.

Moreover the case of Ising model with long range interaction and external
magnetic field is treated in [1, 6, 7] (and references therein). There the
focus is on the case is which the interaction is decaying so slowly that there
is still a (first order) phase transition in the non disordered model and on
the issue of whether the first order transition survives to the presence of a
centered disordered field. We refer to these works for details and we limit
ourselves to signaling the open problem of obtaining results analogous to ours
for Ising models with interactions beyond nearest neighbors.

#### Organization of the rest of the paper

In Section 2 we introduce the transfer matrix formalism, along with the
arising processes $l$ and $r$. In this section we also build the infinite
volume Gibbs measure and we relate it to the processes just introduced.
Moreover we introduce the ergodic set-up that we repeatedly exploit in the
later sections. We conclude Section 2 with Proposition 2.5 which is a result
in the spirit of Theorem 1.2: it is an important intermediate step in the
proof of Theorem 1.2 and it exploits the control of the invariant probability
of $l$ and $r$ (2.35) that is consequence of Lemma 4.2.

Section 3 is devoted to introducing the Fisher domain-wall configuration
$s^{(F)}_{\cdot}$ on ${\mathbb{Z}}$ that is the natural translationally
covariant companion to $s^{(F,+)}_{\cdot}$. In this section we also introduce
the hard wall processes $\widehat{l}$ and $\widehat{r}$ and we state three
results – Proposition 3.5, Proposition 3.6 and Lemma-3.7 – involving these
processes. With these results in our hand (and using once again the control on
the invariant probability (2.35)), we give the proof of Theorem 1.2 at the end
of Section 3 .

Section 4 is devoted to the proof of Lemma 3.7 (about the closeness of $l_{n}$
and $\widehat{l}_{n}$) and to the proof of Lemma 4.2 (about the invariant
probability).

In Section 5 we give the details about the ergodic statements stated in
Section 2.

App. A is devoted to the proof of various random walk estimates we need.
Proposition 3.5 and Proposition 3.6 are proven in App. B. The proof of the
convergence to the Neveu-Pitman process is in App. C.

##  2\. Transfer matrices, infinite volume limit and ergodic properties

It is practical to consider models on
$\llbracket\mathcalb{l},\mathcalc{r}\rrbracket=\\{\mathcalb{l},\mathcalb{l}+1,\ldots,\mathcalc{r}\\}$
with $\mathcalb{l}\leq\mathcalc{r}$, so we set

$Z_{\mathcalb{l},\mathcalc{r},J,h}^{ab}\,:=\,\sum_{\sigma\in\\{-1,+1\\}^{\llbracket\mathcalb{l},\mathcalc{r}\rrbracket}}\exp\left(H^{ab}_{\mathcalb{l},\mathcalc{r},J,h}(\sigma)\right)\,,$
(2.1)

and

$H^{ab}_{\mathcalb{l},\mathcalc{r},J,h}(\sigma)\,:=\,J\sum_{n=\mathcalb{l}}^{\mathcalc{r}+1}\sigma_{n-1}\sigma_{n}+\sum_{n=\mathcalb{l}}{r}h_{n}\sigma_{n},\
\ \ \text{ with }\sigma_{\mathcalb{l}-1}:=a\text{ and
}\sigma_{\mathcalc{r}+1}:=b\,.$ (2.2)

We recall that $a$ and $b$ are in $\\{-1,+1\\}\cong\\{-,+\\}$. We introduce
also the corresponding Gibbs measure
${\mathbf{P}}_{\mathcalb{l},\mathcalc{r},J,h}^{ab}$ on
$\\{-1,+1\\}^{\llbracket\mathcalb{l},\mathcalc{r}\rrbracket}$. Moreover for
every $\mathcalb{l}\in{\mathbb{Z}}$ we set also

$Z_{\mathcalb{l},\mathcalb{l}-1,J,h}^{ab}\,:=\,\exp\left(Jab\right)\,.$ (2.3)

With these notations we have for $\mathcalb{l}\leq n\leq\mathcalc{r}$

${\mathbf{P}}_{\mathcalb{l},\mathcalc{r},J,h}^{ab}\left(\sigma_{n}=+1\right)=\frac{{Z}_{\mathcalb{l},n-1,J,h}^{a+}e^{h_{n}}{Z}_{n+1,\mathcalc{r},J,h}^{+b}}{{Z}_{\mathcalb{l},n-1,J,h}^{a+}e^{h_{n}}{Z}_{n+1,\mathcalc{r},J,h}^{+b}+{Z}_{\mathcalb{l},n-1,J,h}^{a-}e^{-h_{n}}{Z}_{n+1,\mathcalc{r},J,h}^{-b}}\,.$
(2.4)

So if we introduce for $n\leq m+1$

$L^{(a)}_{n,m}\,:=\,\frac{{Z}_{n,m,J,h}^{a+}}{{Z}_{n,m,J,h}^{a-}}\ \ \ \text{
and }\ \ \
R^{(b)}_{n,m}\,:=\,\frac{{Z}_{n,m,J,h}^{+b}}{{Z}_{n,m,J,h}^{-b}}\,,$ (2.5)

as well as

$l^{(a)}_{n,m}\,:=\,\log L_{n,m}^{(a)}\ \ \ \text{ and }\ \ \
r^{(b)}_{n,m}\,:=\,\log R_{n,m}^{(b)}\,,$ (2.6)

we have for $\mathcalb{l}\leq n\leq\mathcalc{r}$

${\mathbf{P}}^{ab}_{\mathcalb{l},\mathcalc{r},J,h}\left(\sigma_{n}=+1\right)\,=\,\frac{L_{\mathcalb{l},n-1}^{(a)}e^{2h_{n}}R_{n+1,\mathcalc{r}}^{(b)}}{1+L_{\mathcalb{l},n-1}^{(a)}e^{2h_{n}}R_{n+1,\mathcalc{r}}^{(b)}}\,=\,\frac{1}{1+\exp\left(-\left(l_{\mathcalb{l},n-1}^{(a)}+2h_{n}+r_{n+1,\mathcalc{r}}^{(b)}\right)\right)}\,.$
(2.7)

Note that by (2.3) we have $l^{(a)}_{n,n-1}=2Ja$ and $r^{(b)}_{m+1,m}=2Jb$.

###  2.1. The transfer matrix formalism

Using the _quantum mechanics_ notation for vectors and adjoint vectors we set
$\langle+|:=(1,0)$, $|+\rangle:=(1,0)^{\mathtt{t}}$, $\langle-|:=(0,1)$ and
$|-\rangle:=(0,1)^{\mathtt{t}}$. We have

${Z}_{\mathcalb{l},\mathcalc{r},J,h}^{ab}\,=\,\langle
a|QT{l}QT_{\mathcalb{l}+1}Q\ldots QT_{\mathcalc{r}}Q|b\rangle\,,$ (2.8)

with $T_{n}:=T(h_{n})$ where for $h\in{\mathbb{R}}$

$T(h)\,:=\begin{pmatrix}e^{h}&0\\\ 0&e^{-h}\end{pmatrix}\ \ \ \text{ and }\ \
\ Q\,:=\begin{pmatrix}e^{J}&e^{-J}\\\ e^{-J}&e^{J}\end{pmatrix}\,.$ (2.9)

In other terms, if we index our $2\times 2$ matrices by
$\\{+1,-1\\}\times\\{+1,-1\\}$, then
${Z}_{\mathcalb{l},\mathcalc{r},J,h}^{ab}$ is the coefficient of
$QT{l}QT_{\mathcalb{l}+1}Q\ldots QT_{\mathcalc{r}}Q$ with index $(a,b)$. This
holds also when $\mathcalc{r}=\mathcalb{l}-1$, in which case the right-hand
side of (2.8) reduces to $\langle a|Q|b\rangle$.

######  Remark 2.1.

When $h_{n}=0$ for every $n$ the matrix product is reduced to a power of the
matrix $Q$ and

$Q\,=\,2\cosh(J)P\,:=\,2\cosh(J)\begin{pmatrix}p_{J}&1-p_{J}\\\
1-p_{J}&p_{J}\end{pmatrix}\ \text{ with }p_{J}\,=\,\frac{1}{1+\exp(-2J)}\,.$
(2.10)

It is therefore straightforward to see that in this case
${\mathbf{P}}^{ab}_{1,N,J,h}$ is simply the law of a Markov chain on the state
space $\\{+1,-1\\}$ with transition matrix $P$, initial condition $a$ at time
$0$, observed up to time $N$ and conditioned to be in the state $b$ at time
$N+1$. If we consider the limit $N\to\infty$, the differences between
switching state times is an IID sequence of geometric random variables of
parameter $1-p_{J}$.

In view of (2.5) and (2.8) we can write

$L_{n,m}^{(a)}\,=\,\frac{\langle a|QT_{n}Q\ldots QT_{m}Q|+\rangle}{\langle
a|QT_{n}Q\ldots QT_{m}Q|-\rangle}\,=\,\frac{\langle+|QT_{m}Q\ldots
QT_{n}Q|a\rangle}{\langle-|QT_{m}Q\ldots QT_{n}Q|a\rangle}\,,$ (2.11)

where we use that $Q$ and the $T_{n}$’s are self-adjoint. Note that if we set
for $j=0,1,2,\ldots$

$X_{j}\,:=\,QT_{n+j}QT_{n+j-1}Q\ldots QT_{n}Q|a\rangle\,,$ (2.12)

then $(X_{j})_{j=0,1,\ldots}$ is a Markov chain on $(0,\infty)^{2}$ with
initial condition $X_{0}=QT_{n}Q|a\rangle$ and
$Q|a\rangle=(\exp(Ja),\exp(-Ja))^{\mathtt{t}}$. Therefore
$(L_{n,n+j}^{(a)})_{j=0,1,\ldots}$ is just the (Markov) process with state
space $(0,\infty)$ that describes the evolution of the ratio of the two
coordinates of $(X_{j})_{j=0,1,\ldots}$. Note that with $h\in{\mathbb{R}}$ and
$T=T(h)$

$QT\begin{pmatrix}x\\\ 1\end{pmatrix}\,=\,\begin{pmatrix}e^{J+h}x+e^{-J-h}\\\
e^{-J+h}x+e^{J-h}\end{pmatrix}\,,$ (2.13)

which implies that
$x\mapsto\left(x+e^{-2J-2h}\right)/\left(e^{-2J}x+e^{-2h}\right)$ is the
random one step map from $(0,\infty)$ to $(0,\infty)$ encoding the evolution
of the ratio chain $(L_{n,n+j}^{(a)})_{j=0,1,\ldots}$. Since
$l_{n,n+j}^{(a)}=\log L_{n,n+j}^{(a)}$ it is straightforward that
$(l_{n,n+j}^{(a)})_{j=0,1,\ldots}$ is also a Markov chain. More precisely it
is the Markov chain with state space ${\mathbb{R}}$ generated by the one step
map (recall that $\Gamma=2J$)

$y\mapsto\log\left(\frac{e^{y}+e^{-\Gamma-2h}}{e^{-\Gamma+y}+e^{-2h}}\right)\,=\,y+2h+\log\left(\frac{1+e^{-\Gamma-(y+2h)}}{1+e^{-\Gamma+(y+2h)}}\right)\,=:\,f_{h}(y)\,=\,b_{\Gamma}\circ\theta_{h}(y)\,,$
(2.14)

where we have conveniently introduced (see Fig. 2)

$\theta_{h}(y)\,:=\,y+2h\ \ \text{ and
}b_{\Gamma}(y)\,:=\,y+\log\left(\frac{1+e^{-\Gamma-y}}{1+e^{-\Gamma+y}}\right)\,.$
(2.15)

Figure 2.  The function $b_{\Gamma}$ for $\Gamma=4$ (on the left) and
$\Gamma=10$ (on the right). The dashed lines plot the function
$\widehat{b}_{\Gamma}$ introduced in § 3.2.

Explicitly, for $j=0,1,\ldots$

$l^{(a)}_{n,n+j}\,=\,f_{h_{n+j}}\circ f_{h_{n+j-1}}\circ\ldots\circ
f_{h_{n}}\left(a\Gamma\right)\,,$ (2.16)

or (equivalently), for $n\leq m$

$l^{(a)}_{n,m}\,=\,f_{h_{m}}\circ f_{h_{m-1}}\circ\ldots\circ
f_{h_{n}}\left(a\Gamma\right)\,.$ (2.17)

It is at times practical to speak of Markov chains in terms of their (one
step) transition kernel: the transition kernel of
$(l_{n,m}^{(a)})_{m=n,n+1,\ldots}$ is
$\mathtt{p}_{\Gamma}:{\mathbb{R}}\times{\mathcal{B}}({\mathbb{R}})\to[0,1]$
given by $\mathtt{p}_{\Gamma}(l,B):={\mathbb{P}}(f_{h_{1}}(l)\in B)$, for
every $l\in{\mathbb{R}}$ and every $B\in{\mathcal{B}}({\mathbb{R}})$.

An important observation at this stage is that $b_{\Gamma}$ maps
${\mathbb{R}}$ to $(-\Gamma,\Gamma)$, so the state space of the Markov chain
$(l_{n,m}^{(a)})_{m=n,n+1,\ldots}$ may be taken to be $(-\Gamma,\Gamma)$ and
not the whole of ${\mathbb{R}}$. Consequently, the domain of the transition
kernel can be restricted to
$(-\Gamma,\Gamma)\times{\mathcal{B}}((-\Gamma,\Gamma))$.

We can now repeat these steps for the $R$ process: from (2.5) and (2.8) we
have for $n\leq m$

$R^{(b)}_{n,m}\,=\,\frac{{Z}_{n,m,J,h}^{+b}}{{Z}_{n,m,J,h}^{-b}}\,=\,\frac{\langle+|QT_{n}Q\ldots
QT_{m}Q|b\rangle}{\langle-|QT_{n}Q\ldots QT_{m}Q|b\rangle}\,,$ (2.18)

and we readily see that this time $(R^{(b)}_{m-j,m})_{j=0,1,\ldots}$ and
$(r^{(b)}_{m-j,m})_{j=0,1,\ldots}$ are Markov chain. More precisely, for
$j=0,1,\dots$

$r^{(b)}_{m-j,m}\,=\,f_{h_{m-j}}\circ f_{h_{m-j+1}}\circ\ldots\circ
f_{h_{m}}\left(b\Gamma\right)\,,$ (2.19)

or (equivalently), for $n\leq m$

$r^{(b)}_{n,m}\,=\,f_{h_{n}}\circ f_{h_{n+1}}\circ\ldots\circ
f_{h_{m}}\left(b\Gamma\right)\,.$ (2.20)

From (2.7), we derive the following formula, which holds for every
$s\in\\{+1,0,-1\\}$ and $\mathcalb{l},n,\mathcalc{r}\in{\mathbb{Z}}$ such that
$\mathcalb{l}\leq n\leq\mathcalc{r}$:

${\mathbf{P}}_{\mathcalb{l},\mathcalc{r},J,h}^{ab}\left(\sigma_{n}\neq
s\right)\,=\,\frac{1}{1+\exp\left(s\,m^{(ab)}_{\mathcalb{l},n,\mathcalc{r}}\right)}+\frac{1}{2}\mathbf{1}_{s=0}\,,$
(2.21)

with the notation
$m^{(ab)}_{\mathcalb{l},n,\mathcalc{r}}:=l^{(a)}_{\mathcalb{l},n-1}+2h_{n}+r^{(b)}_{n+1,\mathcalc{r}}$.
We mention that when $\mathcalb{l}=1$ and $\mathcalc{r}=N$,
${\mathbf{P}}_{\mathcalb{l},\mathcalc{r},J,h}^{ab}$ coincides with
${\mathbf{P}}_{N,J,h}^{ab}$ defined in (1.4). Note that $s$ may be chosen
depending on the sequence $h=(h_{n})_{n\in{\mathbb{Z}}}$.

###  2.2. The infinite volume system

We are interested in (2.21) with $\mathcalb{l}=1$ and $\mathcalc{r}=N$, but we
will also look at the limit as $\mathcalb{l}\to-\infty$ and
$\mathcalc{r}\to\infty$.

######  Lemma 2.2.

For every sequence $(h_{n})_{n\in{\mathbb{Z}}}\in{\mathbb{R}}^{\mathbb{Z}}$ we
have the following: for every $m$ the limit
$l_{m}:=\lim_{n\to-\infty}l^{(a)}_{n,m}$ exists and is independent of $a$.
Moreover $(l_{m})_{m\in{\mathbb{Z}}}$ is a sequence that satisfies
$l_{m}=f_{h_{m}}(l_{m-1})$ for every $m\in{\mathbb{Z}}$. Likewise, for every
$n$, the limit $r_{n}:=\lim_{m\to+\infty}r^{(b)}_{n,m}$ exists, is independent
of $b$ and $r_{n}=f_{h_{n}}(r_{n+1})$ for every $n\in{\mathbb{Z}}$.

If $h=(h_{n})_{n\in{\mathbb{Z}}}$ is an IID sequence of real random variables,
then $(l_{n})_{n\in{\mathbb{Z}}}$ and $(r_{-n})_{n\in{\mathbb{Z}}}$ are
stationary Markov chains with the same transition kernel $\mathtt{p}_{\Gamma}$
(given right after (2.17)), which admits a unique invariant probability.

######  Remark 2.3.

Of course the results proven for one of the two processes $l$ or $r$ can be
transferred to the other one. Let us be more explicit about this point (that
will be used several times) by introducing the random walk
$(S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}_{n})_{n\in{\mathbb{Z}}}$

$S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}_{n}\,:=\,S_{-n}\,,$
(2.22)

and $\mathtt{r}_{n}:=r_{-n+1}$. So
$\mathtt{r}_{n}=f_{h_{-n+1}}(\mathtt{r}_{n-1})$, but
$h_{-n+1}=-\left(S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}_{n}-S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}_{n-1}\right)$,
and in the end we have

$\mathtt{r}_{n}\,=\,f_{-\left(S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}_{n}-S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}_{n-1}\right)}(\mathtt{r}_{n-1})\,,$
(2.23)

that is to be compared with $l_{n}=f_{S_{n}-S_{n-1}}\left(l_{n-1}\right)$.
Therefore the (shifted) time reversal of the $r$ process coincides with the
$l$ process if we use the increments of
$-S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}$ rather than the
increments of $S$ as driving noise.

Another useful observation is the following: since the function $b_{\Gamma}$
is odd, the process $(-l_{n})_{n\in{\mathbb{Z}}}$ coincides with the process
$l$ if we use the increments of $-S$ as driving noise.

###### Proof.

$b_{\Gamma}$ is an odd increasing function and one readily sees that

$\max_{x}b_{\Gamma}^{\prime}(x)\,=\,b_{\Gamma}^{\prime}(0)\,=\,\frac{1-\varepsilon}{1+\varepsilon}\,\leq\exp(-\varepsilon)<1\,.$
(2.24)

Therefore $|f^{\prime}_{h}(x)|=f^{\prime}_{h}(x)\leq\exp(-\varepsilon)$
uniformly in $x$, so $|f_{h}(x)-f_{h}(y)|\leq\exp(-\varepsilon)|x-y|$ and

$\sup_{x,y\in[-\Gamma,\Gamma]}\left|f_{h_{m}}\circ f_{h_{m-1}}\circ\ldots\circ
f_{h_{n}}\left(x\right)-f_{h_{m}}\circ f_{h_{m-1}}\circ\ldots\circ
f_{h_{n}}\left(y\right)\right|\,\leq\,2\Gamma e^{-\varepsilon(m-n+1)}\,,$
(2.25)

and from such an estimate it is straightforward to see
$\lim_{n\to-\infty}l^{(a)}_{n,m}$ exists for every (deterministic!) choice of
the sequence $h\in{\mathbb{R}}^{\mathbb{Z}}$ and that the limit does not
depend on $a$. More than that: the limit does not depend on the initial
condition $l^{(a)}_{n,n}$ which may also depend on $n$ and on the sequence
$h$; note that in our case $l^{(a)}_{n,n}=f_{h_{n}}(a\Gamma)$. It is then
straightforward to check that $l_{m}=f_{h_{m}}(l_{m-1})$ for every
$m\in{\mathbb{Z}}$, as well as the other properties we claimed and this
completes the proof. We remark that the Markov chain we consider is a special
case in the class of _contractive Markov chains_ , see for example [14, Ch.
2], in particular Theorem 2.1.9, where the case of Markov chains such that
${\mathbb{E}}[\log|f^{\prime}_{h_{1}}(x)|]<0$ is considered: this _non
uniform_ contractive property suffices to see that the kernel
$\mathtt{p}_{\Gamma}$ has a unique invariant probability, to obtain that
$\lim_{n\to-\infty}l^{(a)}_{n,m}$ exists a.s. for every $m$ and that the
arising process $l_{\cdot}$, which does not depend on the value of $a$, is a
stationary Markov chain with kernel $\mathtt{p}_{\Gamma}$. ∎

Since it is straightforward to write the probability of every cylindrical
(i.e., local) event in $\\{-1,1\\}^{\mathbb{Z}}$ with the help of the
$(l_{n,m}^{(a)})$ and $(r_{n,m}^{(b)})$ processes (we have done so explicitly
in (2.7) for the event $\\{\sigma:\,\sigma_{j}=+1\\}$), it is easy to see
that, by Lemma 2.2, the two index sequence of probabilities
$({\mathbf{P}}^{ab}_{\mathcalb{l},\mathcalc{r},J,h})$ converges weakly – the
topology on $\\{-1,1\\}^{\mathbb{Z}}$ is the product one – as
$\mathcalb{l}\to-\infty$ and $\mathcalc{r}\to\infty$ and no matter how the
limit is taken. For definiteness we set
${\mathbf{P}}_{J,h}:=\lim_{\mathcalc{r}\to\infty}{\mathbf{P}}^{++}_{-\mathcalc{r},\mathcalc{r},J,h}$,
but the limit does not depend on the chosen sequence and it does not depend on
the boundary conditions either. From (2.7) and Lemma 2.2 we obtain

${\mathbf{P}}_{J,h}\left(\sigma_{n}=+1\right)\,=\,\frac{1}{1+\exp\left(-\left(l_{n-1}+2h_{n}+r_{n+1}\right)\right)}\,.$
(2.26)

or equivalently, for every $s\in\\{-1,0,+1\\}$,

${\mathbf{P}}_{J,h}\left(\sigma_{n}\neq
s\right)\,=\,\frac{1}{1+\exp\left(s\left(l_{n-1}+2h_{n}+r_{n+1}\right)\right)}+\frac{1}{2}\mathbf{1}_{s=0}\,.$
(2.27)

Thus, we denote for all $n\in{\mathbb{Z}}$, $m_{n}:=l_{n-1}+2h_{n}+r_{n+1}$,
in analogy with (2.21).

###  2.3. The ergodic system

As already hinted to, there is an ergodic system in play. Recall that $h$ is
the bi-infinite IID sequence $(h_{n})_{n\in{\mathbb{Z}}}$. We denote
$\Theta:{\mathbb{R}}^{\mathbb{Z}}\to{\mathbb{R}}^{\mathbb{Z}}$ the shift
application, defined by $\Theta h=(h_{n+1})_{n\in{\mathbb{Z}}}$ if
$h=(h_{n})_{n\in{\mathbb{Z}}}$. We have of course $\Theta^{\circ
m}h=(h_{n+m})_{n\in{\mathbb{Z}}}$ for every $m\in{\mathbb{Z}}$.
${\mathbb{R}}^{\mathbb{Z}}$ is equipped with the product topology and the
probability we consider on this space is the law ${\mathbb{P}}$ of the IID
sequence $h=(h_{n})_{n\in{\mathbb{Z}}}$ and the system is $\Theta$ invariant.
Furthermore the system satisfies the mixing property (because $(h_{n})$ is
IID), so it is ergodic. Hence, for every ${\mathbb{L}}^{1}$ function
$G:{\mathbb{R}}^{\mathbb{Z}}\to{\mathbb{R}}$ we have that
${\mathbb{P}}$-almost surely (see for example [14, Ch. 5])

$\frac{1}{N}\sum_{n=1}^{N}G(\Theta^{\circ
n}(h))\underset{N\to\infty}{\longrightarrow}{\mathbb{E}}[G(h)]\,.$ (2.28)

For the next result we need the notion of discrepancy density between two
configurations: for $\sigma$ and $\sigma^{\prime}$ in
$\\{-1,0,+1\\}^{\\{1,\ldots,N\\}}$ or in $\\{-1,0,+1\\}^{{\mathbb{N}}}$ we set

$D_{N}(\sigma,\sigma^{\prime})\,:=\,\frac{1}{N}\left|\left\\{n=1,2\ldots,N:\,\sigma_{n}\neq\sigma^{\prime}_{n}\right\\}\right|=\frac{1}{N}\sum_{n=1}^{N}\mathbf{1}_{\sigma_{n}\neq\sigma_{n}^{\prime}}\,.$
(2.29)

######  Proposition 2.4.

Consider a measurable function $K:{\mathbb{R}}^{\mathbb{Z}}\to\\{-1,0,+1\\}$
and set $s^{(K)}_{n}:=K(\Theta^{\circ n}h)$ for $n\in{\mathbb{Z}}$. Then
${\mathbb{P}}(\,\text{\rm d}h)$-a.s. (that is, almost surely in the
realization of the IID sequence $(h_{n})_{n\in{\mathbb{Z}}}$)

$D_{N}\left(\sigma,s^{(K)}\right)\underset{N\to\infty}{\longrightarrow}{\mathbb{E}}\left[\left(1+\exp\left(s^{(K)}_{0}m_{0}\right)\right)^{-1}+\frac{1}{2}\mathbf{1}_{s^{(K)}_{0}=0}\right]\,,$
(2.30)

in ${\mathbf{P}}^{ab}_{N,J,h}$-probability, as well as in
${\mathbf{P}}_{J,h}$-probability.

The proof requires no new ingredients. We however leave it to Section 5 for
presentation purposes. But let us mention that from the definitions we have
just given and by (2.27) and (2.28)

$\begin{split}{\mathbf{E}}_{J,h}\left[D_{N}\left(\sigma,s^{(K)}\right)\right]\,&=\,\frac{1}{N}\sum_{n=1}^{N}\frac{1}{1+\exp\left(s^{(K)}_{n}m_{n}\right)}+\frac{1}{2}\mathbf{1}_{s^{(K)}_{n}=0}\\\
&\underset{N\to\infty}{\overset{\mathrm{a.s.}}{\longrightarrow}}{\mathbb{E}}\left[\frac{1}{\left(1+\exp\left(s^{(K)}_{0}m_{0}\right)\right)}+\frac{1}{2}\mathbf{1}_{s^{(K)}_{0}=0}\right]\,.\end{split}$
(2.31)

The completion of the proof (in Section 5) is about controlling the boundary
effects and obtaining a concentration estimate, that is achieved by a variance
estimate.

###  2.4. A first discrepancy estimate

Recall that our aim is to show that the typical configurations under
${\mathbf{P}}_{N,J,h}^{ab}$ are close to one given configuration. A natural
candidate is the configuration $(s_{n})$ such that $s=s_{n}$ maximizes the
right-hand side of (2.21) with $\mathcalb{l}=1$ and $\mathcalc{r}=N$, that is
the configuration equal to the sign of $m^{(ab)}_{1,n,N}$. We make the
slightly different (and slightly sub-optimal) choice

$s^{(m)}_{n}\,=\begin{cases}+1&\text{ if }m_{n}\geq 0,\\\ -1&\text{
otherwise,}\end{cases}$ (2.32)

for $n\in{\mathbb{Z}}$. The configuration $s^{(m)}$ on ${\mathbb{Z}}$
satisfies the hypotheses of Proposition 2.4 and therefore, in
${\mathbf{P}}_{N,J,h}^{ab}$-probability, we have that

$D_{N}(\sigma,s^{(m)})\underset{N\to\infty}{\longrightarrow}{\mathbb{E}}\left[\frac{1}{1+\exp\left(s^{(m)}_{0}m_{0}\right)}\right]={\mathbb{E}}\left[\frac{1}{1+\exp\left(|m_{0}|\right)}\right]\,.$
(2.33)

######  Proposition 2.5.

We have that

${\mathbb{E}}\left[\frac{1}{1+\exp\left(|m_{0}|\right)}\right]\overset{\Gamma\to\infty}{=}O\left(\frac{\log_{\circ
2}\Gamma}{\Gamma}\right)\,.$ (2.34)

The proof of Proposition 2.5 depends on an estimate on the invariant
probability for the $\mathtt{p}_{\Gamma}$ transition kernel, that is the
common law of the random variables $l_{m}$ and $r_{n}$: this estimate, Lemma
4.2, is in section 4. Here and later on we will use the following result (that
can be extracted in a straightforward way from Lemma 4.2): there exist two
positive constants $C_{1}$ and $C_{2}$ such that for $\Gamma$ sufficiently
large we have

${\mathbb{P}}(l_{-1}\in[x,y])\,\leq\,C_{1}\frac{y-x}{\Gamma}\,,$ (2.35)

for every $x<y$ satisfying $y-x\geq C_{2}\log_{\circ 2}(\Gamma)$. We remark
that the same estimate holds for $r_{1}$, because it has the same law as
$l_{-1}$, but we are not using this in the proof.

###### Proof of Proposition 2.5 using (2.35).

With the constants involved in (2.35), we aim at showing the following _more
quantitative_ version of (2.34): for $\Gamma$ large

${\mathbb{E}}\left[\frac{1}{1+\exp\left(\left|m_{0}\right|\right)}\right]\,\leq\,3C_{1}C_{2}\frac{\log_{\circ
2}(\Gamma)}{\Gamma}\,.$ (2.36)

To show (2.36) we start by recalling that $m_{0}=l_{-1}+2h_{0}+r_{1}$ and by
observing that the three terms in this sum are independent: in fact, $l_{-1}$
is measurable with respect to $(h_{n})_{n=-1,-2,\dots}$ and $r_{1}$ is
measurable with respect to $(h_{n})_{n=1,2,\ldots}$. It is therefore enough to
show that for each value $\lambda\in{\mathbb{R}}$:

${\mathbb{E}}\left[\frac{1}{1+\exp\left(\left|l_{-1}+\lambda\right|\right)}\right]\,\leq\,3C_{1}C_{2}\frac{\log_{\circ
2}(\Gamma)}{\Gamma}\,.$ (2.37)

Setting $c_{\Gamma}:=C_{2}\log_{\circ 2}(\Gamma)$, let us observe that if we
introduce for every $\lambda\in{\mathbb{R}}$ and for $j=0,1,\ldots$

$A_{\lambda,j}\,:=\,\left\\{l\in{\mathbb{R}}:\,jc_{\Gamma}\leq|l+\lambda|\leq(j+1)c_{\Gamma}\right\\}\,,$
(2.38)

we have

${\mathbb{E}}\left[\frac{1}{1+\exp\left(\left|l_{-1}+\lambda\right|\right)}\right]\,\leq\,\sum_{j=0}^{\infty}\exp\left(-jc_{\Gamma}\right){\mathbb{P}}\left(l_{-1}\in
A_{\lambda,j}\right)\,.$ (2.39)

Since
$A_{\lambda,j}=[-(j+1)c_{\Gamma}-\lambda,-jc_{\Gamma}-\lambda]\cup[jc_{\Gamma}-\lambda,(j+1)c_{\Gamma}-\lambda]$,
we apply (2.35) and get that ${\mathbb{P}}\left(l_{-1}\in
A_{\lambda,j}\right)\leq 2C_{1}C_{2}\log_{\circ 2}(\Gamma)/\Gamma$ for every
$\lambda$. Therefore, we obtain that (2.36) holds for $\Gamma$ large and the
proof of Proposition 2.5 is complete. ∎

######  Remark 2.6.

Section 2 has been developed with the language of Markov chains, and the (one
step) Markov character is due to the fact that $(h_{n})$ is a sequence of
independent random variables. But the first part of Lemma 2.2 is stated for
general deterministic sequences $(h_{n})$ and the proof of Prop. 2.4 just
requires ergodicity of the sequence $(h_{n})$: as a consequence (2.33) holds
for ergodic sequences. In this generalized ergodic set-up one deals with a
more general discrete time stochastic process, or stochastic dynamical system,
always defined by iterated application of a random function. But we stress
that estimating the right-most term in (2.33), like in Proposition 2.5,
demands an understanding of the invariant probability of the stochastic
dynamical system, which we achieve under the hypothesis that $(h_{n})$ is IID.
And obtaining Theorem 1.2 demands a substantially deeper understanding of the
stochastic dynamical system.

##  3\. Fisher domain-wall configuration on ${\mathbb{Z}}$ and reflected
random walk

In the previous section, we have identified $s^{(m)}$ as a configuration
around which the typical configurations are _concentrated_ (in the sense of
the discrepancy density). This configuration has the disadvantage of being
rather implicit: it is a complicated function of the disorder $(h_{n})$,
involving Markov chains driven by the disorder and limit procedures. The aim
of the rest of the paper is to show that $s^{(m)}$ can be replaced with the
Fisher configuration $s^{(F,+)}$ (given in (1.18)), and this will yield
Theorem 1.2

A first step in this program is to define a translation invariant version
$s^{(F)}$ of the Fisher configuration $s^{(F,+)}$.

###  3.1. The Fisher domain-wall configuration on ${\mathbb{Z}}$

We introduce the Fisher configuration $s^{(F)}$ on ${\mathbb{Z}}$ by relying
on the two-sided random walk $S$ defined in (1.6). For every
$N\in{\mathbb{N}}$ we can repeat the construction of the one-sided Fisher
configuration $s^{(F,+)}$, replacing $(S_{n})_{n\in{\mathbb{Z}}}$ with
$(S_{n-N}-S_{-N})_{n\in{\mathbb{Z}}}$. This generates a new sequence of
$\Gamma$-extrema locations
$\mathtt{u}_{1,N}(\Gamma),\mathtt{u}_{2,N}(\Gamma),\ldots$, along with the
corresponding $+$ sequence
$\mathtt{u}^{+}_{1,N}(\Gamma),\mathtt{u}^{+}_{2,N}(\Gamma),\ldots$. In order
to take into account the translation we have made, the (ordered) set of
$\Gamma$-extrema we are interested in is

$\mathtt{u}_{1,N}(\Gamma)-N\,\leq\,\mathtt{u}^{+}_{1,N}(\Gamma)-N\,<\,\mathtt{u}_{2,N}(\Gamma)-N\,\leq\,\mathtt{u}^{+}_{2,N}(\Gamma)-N\,<\,\ldots$
(3.1)

but it is practical to relabel this sequence so that the location of the
$\Gamma$-extremum with label $0$ is the one at the largest negative location.
More formally we introduce
$J_{N}:=|\\{j\in{\mathbb{N}}:\,\mathtt{u}_{j,N}(\Gamma)-N<0\\}|$ and then we
set for $k\in\\{-J_{N}+1,-J_{N}+2,\ldots\\}$

${\mathbf{u}}_{k,N}\,:=\,\mathtt{u}_{J_{N}+k,N}(\Gamma)-N\ \ \text{ and }\ \
{\mathbf{u}}^{+}_{k,N}\,:=\,\mathtt{u}^{+}_{J_{N}+k,N}(\Gamma)-N\,.$ (3.2)

For the next statement let us introduce
$\Omega:=\\{(h_{n})\in{\mathbb{R}}^{\mathbb{Z}}:\,\limsup_{n}S_{n}=\limsup_{n}S_{-n}=-\liminf_{n}S_{n}=-\liminf_{n}S_{-n}=\infty\\}$
so (with abuse of notation) ${\mathbb{P}}(\Omega)=1$. Restricting to $\Omega$
ensures in particular that $({\mathbf{u}}_{k,N})_{k}$ and
$({\mathbf{u}}^{+}_{k,N})_{k}$ are infinite sequences, for every $N$.

######  Lemma 3.1.

Assume that $(h_{n})\in\Omega$. Then for every $k\in{\mathbb{Z}}$ we have the
existence of the limits

${\mathbf{u}}_{k}(\Gamma)\,:=\,\lim_{N\to\infty}{\mathbf{u}}_{k,N}\ \ \text{
and }\ \
{\mathbf{u}}^{+}_{k}(\Gamma)\,:=\,\lim_{N\to\infty}{\mathbf{u}}^{+}_{k,N}\,,$
(3.3)

and there exists $\mathtt{s}\in\\{-1,0,+1\\}$ such that
$\mathtt{u}_{k}(\Gamma)={\mathbf{u}}_{k+\mathtt{s}}(\Gamma)$, as well as
$\mathtt{u}^{+}_{k}(\Gamma)={\mathbf{u}}^{+}_{k+\mathtt{s}}(\Gamma)$, for
every $k\geq\max(1,1-\mathtt{s})$.

In short, Lemma 3.1 builds the bilateral sequence of $\Gamma$-extrema and the
$\Gamma$-extrema at non-negative locations coincide with the one-sided
sequence of $\Gamma$-extrema, except possibly for the first entry. And, in any
case, at $(\mathtt{u}_{k})_{k=2,3,\ldots}$ there are $\Gamma$-extrema both in
the one and two-sided case.

###### Proof.

First of all (3.3) directly follows because for every $L\in{\mathbb{N}}$ and
every $M>N\geq L$ we have

${\mathbf{u}}_{k,M}\,=\,{\mathbf{u}}_{k,N}\ \ \text{ and }\ \
{\mathbf{u}}^{+}_{k,M}\,=\,{\mathbf{u}}^{+}_{k,N}$ (3.4)

for $k>\min\\{j:\,{\mathbf{u}}_{j,N}\geq-L\\}=:j_{N,L}$. In fact, (3.4) holds
since there is a $\Gamma$-extremum at ${\mathbf{u}}_{j_{N,L},N}$ (say, a
maximum: the argument is strictly analogous if it is a minimum) so there
exists $n\in[\\![{\mathbf{u}}_{j_{N,L},N},{\mathbf{u}}_{j_{N,L}+1,N}]\\!]$
such that $S_{n}-S_{{\mathbf{u}}_{j_{N,L}+1,N}}\geq\Gamma$. And of course
there is
$n^{\prime}\in[\\![{\mathbf{u}}_{j_{N,L}+1,N},{\mathbf{u}}_{j_{N,L}+2,N}]\\!]$
such that $S_{n^{\prime}}-S_{{\mathbf{u}}_{j_{N,L}+1,N}}\geq\Gamma$, therefore
it follows from the construction in Section 1.2 that, regardless from where
the sequential procedure to discover the $\Gamma$-extrema starts (before
$-L$), the $\Gamma$-minimum at ${\mathbf{u}}_{j_{N,L}+1,N}$ is going to be
detected.

For the second part of the statement we start from the fact that at
$\mathtt{u}_{2}(\Gamma)$ there is a $\Gamma$-minimum for the one-sided case
(by definition), but also for the two-sided cases because of the argument we
just developed. Then recall the notation introduced in (1.17): either
$\mathtt{t}^{\downarrow}(\Gamma)<\mathtt{t}^{\uparrow}(\Gamma)$ or
$\mathtt{t}^{\downarrow}(\Gamma)>\mathtt{t}^{\uparrow}(\Gamma)$.

  * •

If $\mathtt{t}^{\downarrow}(\Gamma)<\mathtt{t}^{\uparrow}(\Gamma)$ then either
$\mathtt{u}_{1}=\mathbf{u}_{1}$ and in this case
$\mathtt{u}_{k}=\mathbf{u}_{k}$ and $\mathtt{u}^{+}_{k}=\mathbf{u}^{+}_{k}$
for every $k\in{\mathbb{N}}$, or at $\mathtt{u}_{1}$ is not the location of a
$\Gamma$-extremum in ${\mathbb{Z}}$ and $\mathtt{u}_{k+1}=\mathbf{u}_{k}$, as
well as $\mathtt{u}^{+}_{k+1}=\mathbf{u}^{+}_{k}$, for every
$k\in{\mathbb{N}}$.

  * •

If $\mathtt{t}^{\downarrow}(\Gamma)>\mathtt{t}^{\uparrow}(\Gamma)$ then
$\mathtt{u}_{1}$ is the location of a $\Gamma$-maximum also in the whole of
${\mathbb{Z}}$. If in $[\\![0,\mathtt{u}_{1}]\\!]$ there is no
$\Gamma$-minimum (in ${\mathbb{Z}}$) then $\mathtt{u}_{k}=\mathbf{u}_{k}$ and
$\mathtt{u}^{+}_{k}=\mathbf{u}^{+}_{k}$ for every $k\in{\mathbb{N}}$.
Otherwise $\mathtt{u}_{k}=\mathbf{u}_{k+1}$ and
$\mathtt{u}^{+}_{k}=\mathbf{u}^{+}_{k+1}$ for every $k\in{\mathbb{N}}$.

This completes the proof of Lemma 3.1. ∎

For $(h_{n})\in\Omega$ and $n\in{\mathbb{Z}}$ we set

$s_{n}^{(F)}\,:=\,\begin{cases}+1&\text{ if there exists }k\text{ such that
}n\in[\\![{\mathbf{u}}^{+}_{k}(\Gamma),{\mathbf{u}}_{k+1}(\Gamma)-1]\\!],\\\
&\ \ \ {\mathbf{u}}^{+}_{k}(\Gamma)\text{ is the location of a
}\Gamma\text{-minimum and
}S_{{\mathbf{u}}_{k+1}(\Gamma)}-S_{{\mathbf{u}}^{+}_{k}(\Gamma)}>\Gamma\\\
-1&\text{ if there exists }k\text{ such that
}n\in[\\![{\mathbf{u}}^{+}_{k}(\Gamma),{\mathbf{u}}_{k+1}(\Gamma)-1]\\!],\\\
&\ \ \ {\mathbf{u}}^{+}_{k}(\Gamma)\text{ is the location of a
}\Gamma\text{-maximum and
}S_{{\mathbf{u}}^{+}_{k}(\Gamma)}-S_{{\mathbf{u}}_{k+1}(\Gamma)}>\Gamma\\\ \
0&\text{ otherwise.}\end{cases}$ (3.5)

If $(h_{n})\not\in\Omega$ we set $s_{n}^{(F)}=0$ for every $n$. Note that a
direct consequence of Lemma 3.1 is that in $\Omega$ we have

$s_{n}^{(F)}\,=\,s_{n}^{(F,+)}\ \text{ for }\ n\geq\mathtt{u}_{2}\,,$ (3.6)

and we recall that $s_{n}^{(F,+)}$ is given in (1.18).

######  Remark 3.2.

We do not need nor want to enter the issue of whether the Fisher configuration
is the configuration which maximises the energy: this statement requires in
any case more precision at the boundary if one works in finite volume. Note
however that in the case of multiple $\Gamma$-extrema, i.e. when
$\mathtt{u}_{k}(\Gamma)<\mathtt{u}_{k}^{+}(\Gamma)$, or when a $\Gamma$-slope
has height exactly equal to $\Gamma$, there are several configurations with
the same “maximal” energy. On the sites where the spin of “the” Fisher
configuration is ambiguous, we decided to set spin 0, so that for every
configuration $\sigma\in\\{+1,-1\\}^{N}$, $D_{N}(\sigma,s)$ is not larger than
$D_{N}(\sigma,s^{(F)})$ for all choices of $s$ among the configurations with
“maximal” energy. Our analysis will show in particular that the regions where
we set $s_{n}^{(F)}=0$ have a density dominated by ${\log_{\circ
2}\Gamma}/\Gamma$.

###  3.2. The (hard wall) reflected random walk

In strict analogy with the processes $(l_{n})$ and $(r_{n})$ in Section 2.1 we
introduce also the Markov chain

$\widehat{l}_{n}\,=\,\widehat{b}_{\Gamma}\circ\theta_{h_{n}}\left(\widehat{l}_{n-1}\right)=:\,\widehat{f}_{h_{n}}\left(\widehat{l}_{n-1}\right),$
(3.7)

as well as the process

$\widehat{r}_{n}\,=\,\widehat{b}_{\Gamma}\circ\theta_{h_{n}}\left(\widehat{r}_{n+1}\right)=:\,\widehat{f}_{h_{n}}\left(\widehat{r}_{n+1}\right)\,,$
(3.8)

with $\widehat{b}_{\Gamma}(x)=\max(\min(x,\Gamma),-\Gamma)$, see Fig. 2. In
particular the time reversal discussion that one can find at the end of
Section 2.1 applies also to $(\widehat{l}_{n})$ and $(\widehat{r}_{n})$, so
results established on $(\widehat{l}_{n})$ can be transferred to
$(\widehat{r}_{n})$ via time reversal.

######  Remark 3.3.

To motivate the introduction of the reflected random walks, we argue that they
correspond to $zero$-temperature analogues of $(l_{n})$ and $(r_{n})$. To see
this consider the Gibbs measure
${\mathbf{P}}_{\mathcalb{l},\mathcalc{r},J,h,\beta}^{ab}$ defined as in (1.4)
but with the Hamiltonian $H$ multiplied by a parameter $\beta>0$. This means
in fact that $\Gamma$ is replaced by $\beta\Gamma$ and $h_{n}$ by $\beta
h_{n}$, so we can still use our analysis. Denote by $l^{\beta}$ the process
$l$ with these new parameters and _divided by $\beta$_, similarly for
$r^{\beta}$. Then, (2.7) becomes:

${\mathbf{P}}_{\mathcalb{l},\mathcalc{r},J,h,\beta}^{ab}\left(\sigma_{n}=+1\right)\,=\,\frac{1}{1+\exp\left(-\beta\left(l_{\mathcalb{l},n-1}^{\beta}+2h_{n}+r_{n+1,\mathcalc{r}}^{\beta}\right)\right)}\,.$
(3.9)

and $l^{\beta}$ and $r^{\beta}$ evolve in $(-\Gamma,\Gamma)$, with function
$f^{\beta}_{h}=b_{\Gamma}^{\beta}\circ\theta_{h}$, where $b_{\Gamma}^{\beta}$
is defined by:

$b_{\Gamma}^{\beta}(x)=x+\frac{1}{\beta}\log\left(\frac{1+e^{-\beta(\Gamma+x)}}{1+e^{-\beta(\Gamma-x)}}\right)$
(3.10)

for every $x\in{\mathbb{R}}$. We observe that as $\beta$ goes to infinity
$b_{\Gamma}^{\beta}$ converges pointwise to $\widehat{b}_{\Gamma}$, so if
$l^{\beta}$ and $\widehat{l}$ are started at the same point, $l^{\beta}$
converges pointwise towards $\widehat{l}$. We can even push our comment a bit
further: it suggests that in the zero-temperature model, the spins are aligned
with the sign of $\widehat{l}_{n-1}+2h_{n}+\widehat{r}_{n+1}$ (except for the
case in which $\widehat{l}_{n-1}+2h_{n}+\widehat{r}_{n+1}=0$, in which there
is clearly a ground state degeneracy).

We consider both $\widehat{l}$ and $\widehat{r}$ to be running at
_stationarity_ : the uniqueness of the invariant measure is given in the next
statement.

######  Lemma 3.4.

The Markov chain defined by the random iteration (3.7) has a unique invariant
probability and there exists a unique stationary Markov chain
$(\widehat{l}_{n})_{n\in{\mathbb{Z}}}$ that obeys (3.7) for every $n$.

By time inversion, Lemma 3.4 provides the definition of the stationary process
$(\widehat{r}_{n})_{n\in{\mathbb{Z}}}$.

###### Proof.

Since $\widehat{f}_{h}(\cdot)$ is $C^{0}$ for every $h\in{\mathbb{R}}$, one
readily sees that this Markov chain is Feller, i.e.
$x\mapsto{\mathbb{E}}\left[F(\widehat{f}_{h_{1}}(x)\right]$ is $C^{0}$ (and
bounded) whenever $x\mapsto F(x)$ is $C^{0}$ and bounded. Since the state
space $[-\Gamma,\Gamma]$ is compact one readily infers that there exists an
invariant probability (see for example [14, Sec. 12.3]). Let us remark that if
the Markov chain $\left(\widehat{l}_{n,-N,x}\right)_{n=-N,-N+1,\ldots}$ is
defined by the same recursion obeyed by the $\widehat{l}$ chain, but with
$\widehat{l}_{-N,-N,x}=x\in[-\Gamma,\Gamma]$, then
$\lim_{N\to\infty}\widehat{l}_{n,-N,x}$ exists a.s. and it is independent of
the value of $x$. To see this it suffices to observe that if we set

$\mathtt{t}^{\uparrow}(\Gamma,-N)\,:=\,\inf\left\\{n>-N:S^{\uparrow}_{n}\geq\Gamma\right\\}\
\text{ and }\
\mathtt{t}^{\downarrow}(\Gamma,-N)\,:=\,\inf\left\\{n>-N:S^{\downarrow}_{n}\geq\Gamma\right\\},$
(3.11)

then if
$n\geq\mathtt{t}^{\uparrow}(\Gamma,-N)\wedge\mathtt{t}^{\downarrow}(\Gamma,-N)$
we have that $\widehat{l}_{n,-N,x}$ becomes independent of the value of $x$.
In particular we can consider a process $\widehat{l}_{k,-M,y}$ with $M>N$, for
$k=-N,-N+1,\ldots$ and $\widehat{l}_{k,-M,y}=\widehat{l}_{n,-N,x}$ for every
$n\geq\mathtt{t}^{\uparrow}(\Gamma,-N)\wedge\mathtt{t}^{\downarrow}(\Gamma,-N)$.
This implies that $\lim_{N\to\infty}\widehat{l}_{n,-N,x}=:\widehat{l}_{n}$
exists a.s. for every $n$ and it is straightforward to check that
$\widehat{l}_{n+1}=\widehat{f}_{h_{n+1}}\left(\widehat{l}_{n}\right)$.
Moreover, the very same coupling argument yields uniqueness of the invariant
probability, trajectorial uniqueness of the process and that
$\left(\widehat{l}_{n}\right)_{n\in{\mathbb{Z}}}$ is stationary. ∎

We are now going to state three results on the reflected walks $\widehat{l}$
and $\widehat{r}$ and then show that Theorem 1.2 is a consequence of these
results.

The first one gives a rather explicit formula for $\widehat{l}_{0}$ and
$\widehat{r}_{1}$ (and, by translation, one can obtain a formula for
$\widehat{l}_{n}$ and $\widehat{r}_{n}$, $n\in{\mathbb{Z}}$). For this we
recall the notations given in (1.17) and add

$\mathtt{u}^{\downarrow}(\Gamma)\,:=\,\min\left\\{n\in\llbracket
0,\mathtt{t}^{\downarrow}(\Gamma)\rrbracket:S_{n}=\max_{i\in\llbracket
0,\mathtt{t}^{\downarrow}(\Gamma)\rrbracket}S_{i}\right\\}\,,$ (3.12)

and well as

$\mathtt{u}^{\uparrow}(\Gamma)\,:=\,\min\left\\{n\in\llbracket
0,\mathtt{t}^{\uparrow}(\Gamma)\rrbracket:S_{n}=\min_{i\in\llbracket
0,\mathtt{t}^{\uparrow}(\Gamma)\rrbracket}S_{i}\right\\}\,.$ (3.13)

Then we set

$\mathtt{s}^{\uparrow}(\Gamma)\,:=\,-\mathtt{t}^{\uparrow}_{S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}}(\Gamma),\
\mathtt{s}^{\downarrow}(\Gamma)\,:=\,-\mathtt{t}^{\downarrow}_{S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}}(\Gamma),\
\mathtt{v}^{\downarrow}(\Gamma)\,:=\,-\mathtt{u}^{\downarrow}_{S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}}(\Gamma)\text{
and
}\mathtt{v}^{\uparrow}(\Gamma)\,:=\,-\mathtt{u}^{\uparrow}_{S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}}(\Gamma)\,.$
(3.14)

######  Proposition 3.5.

We have that

$\widehat{l}_{0}\,=\begin{cases}+\Gamma-2S_{\mathtt{v}^{\downarrow}(\Gamma)}&\text{
if }\mathtt{s}^{\downarrow}(\Gamma)>\mathtt{s}^{\uparrow}(\Gamma)\,,\\\
-\Gamma-2S_{\mathtt{v}^{\uparrow}(\Gamma)}&\text{ if
}\mathtt{s}^{\uparrow}(\Gamma)>\mathtt{s}^{\downarrow}(\Gamma)\,,\end{cases}$
(3.15)

and

$\widehat{r}_{1}\,=\begin{cases}-\Gamma+2S_{\mathtt{u}^{\downarrow}(\Gamma)}&\text{
if }\mathtt{t}^{\downarrow}(\Gamma)<\mathtt{t}^{\uparrow}(\Gamma)\,,\\\
+\Gamma+2S_{\mathtt{u}^{\uparrow}(\Gamma)}&\text{ if
}\mathtt{t}^{\uparrow}(\Gamma)<\mathtt{t}^{\downarrow}(\Gamma)\,.\end{cases}$
(3.16)

Proposition 3.5 is crucial to the arguments in Section 4.1. The relevance of
the next result needs no comment:

######  Proposition 3.6.

We have for all $n\in{\mathbb{Z}}$ that

$s^{(F)}_{n}\,=\,\operatorname{\mathrm{sign}}\left(\widehat{l}_{n-1}+2h_{n}+\widehat{r}_{n+1}\right)\,.$
(3.17)

with the convention $\operatorname{\mathrm{sign}}(0)=0$.

The third result is about the proximity of the reflected walks with the
processes that arise from the transfer matrix analysis:

######  Lemma 3.7.

For every $\kappa>0$, there exists $C_{\kappa}>0$ such that with probability
$1-O(\Gamma^{-\kappa})$ we have

$\left|l_{0}-\widehat{l}_{0}\right|\,\leq\,\log_{\circ
2}\Gamma+C_{\kappa}\qquad\text{ and
}\qquad\left|r_{0}-\widehat{r}_{0}\right|\,\leq\,\log_{\circ
2}\Gamma+C_{\kappa}\,.$ (3.18)

By translating, Lemma 3.7 holds also if $0$ is replaced by any
$n\in{\mathbb{Z}}$ (the event with probability $1-O(\Gamma^{-\kappa})$ on
which the result holds depends on $n$).

The proofs of Proposition 3.5 and of Proposition 3.6 are in Appendix B. The
proof of Lemma 3.7 is in Section 4.

###  3.3. Proof of Theorem 1.2

Recall from § 1.1 the definition of
$\left(s^{(F,+)}_{n}\right)_{n\in\\{1,2,\ldots}$ and of
$\left(s^{(F)}_{n}\right)_{n\in{\mathbb{Z}}}$. Let us set

$D_{\Gamma}\,:=\,{\mathbb{E}}\left[{\mathbf{P}}_{J,h}\left(\sigma_{0}\neq
s^{(F)}_{0}\right)\right]={\mathbb{E}}\left[\frac{1}{1+\exp\left(s^{(F)}_{0}\left(l_{-1}+2h_{0}+r_{1}\right)\right)}\right]+\frac{1}{2}{\mathbb{P}}\left(s^{(F)}_{0}=0\right)\,.$
(3.19)

Then we remark that, since $s^{(F,+)}$ and $s^{(F)}$ differ only at finitely
many sites (see (3.6)), (1.19) of Theorem 1.2 is equivalent to showing that
for almost every realization of $(h_{n})$

$\lim_{N\to\infty}D_{N}\left(\sigma,s^{(F)}\right)\,=\,\lim_{N\to\infty}\frac{1}{N}\left|\left\\{n=1,2\ldots,N:\,\sigma_{n}\neq
s^{(F)}_{n}\right\\}\right|\,=\,D_{\Gamma}\,,$ (3.20)

in ${\mathbf{P}}_{N,J,h}^{ab}$ probability. But (3.20) is a direct consequence
of Proposition 2.4, so (1.19) is established with explicit expression for
$D_{\Gamma}$.

We are therefore left with showing (1.20), that is $D_{\Gamma}=O(\log_{\circ
2}\Gamma/\Gamma)$. For this we first remark by (2.28) that for almost every
realization of $(h_{n})_{n\in{\mathbb{Z}}}$

$D_{N}\left(s^{(m)},s^{(F)}\right)\underset{N\to\infty}{\longrightarrow}{\mathbb{P}}\left(s_{0}^{(m)}\neq
s_{0}^{(F)}\right)\,.$ (3.21)

So, in view of Proposition 2.5 and observing that
$D_{N}(\sigma,s^{(F)})\;\leqslant\;D_{N}(\sigma,s^{(m)})+D_{N}(s^{(m)},s^{(F)})$,
it suffices to show that

${\mathbb{P}}\left(s_{0}^{(m)}\neq
s_{0}^{(F)}\right)\,=\,O\left(\frac{\log_{\circ 2}\Gamma}{\Gamma}\right)\,.$
(3.22)

To establish (3.22) we use Lemma 3.7 that yields that with probability
$1-O(1/\Gamma)$ and $\Gamma$ sufficiently large

$\left|l_{-1}-\widehat{l}_{-1}\right|\,\leq\,2\log_{\circ 2}\Gamma\qquad\text{
and }\qquad\left|r_{1}-\widehat{r}_{1}\right|\,\leq\,2\log_{\circ 2}\Gamma\,.$
(3.23)

which implies in particular that

$\left|\left(l_{-1}+2h_{0}+r_{1}\right)-\left(\widehat{l}_{-1}+2h_{0}+\widehat{r}_{1}\right)\right|\,\leq\,4\log_{\circ
2}\Gamma\,.$ (3.24)

Recall now that $s_{0}^{(m)}=+1$ if $l_{-1}+2h_{0}+r_{1}\geq 0$ and $-1$
otherwise (see (2.32)) and that
$s_{0}^{(F)}=\operatorname{\mathrm{sign}}(\widehat{l}_{-1}+2h_{0}+\widehat{r}_{1})$,
with the convention $\operatorname{\mathrm{sign}}(0)=0$ (see Proposition 3.6),
so on the event on which (3.24) holds, $s^{(m)}_{0}\neq s^{(F)}_{0}$ implies
$|l_{-1}+2h_{0}+r_{1}|\leq 4\log_{\circ 2}\Gamma$, i.e., $l_{-1}$ belongs to
$[-2h_{0}-r_{1}-4\log_{\circ 2}\Gamma,-2h_{0}-r_{1}+4\log_{\circ 2}\Gamma]$.
Furthermore, (2.35) and the independence of $l_{-1}$, $h_{0}$ and $r_{1}$
imply that

${\mathbb{P}}\left(l_{-1}\in\left[-2h_{0}-r_{1}-4\log_{\circ
2}\Gamma,-2h_{0}-r_{1}+4\log_{\circ
2}\Gamma\right]\right)\,=\,O\left(\frac{\log_{\circ
2}\Gamma}{\Gamma}\right)\,,$ (3.25)

which yields (3.22) and therefore (1.20). The proof of Theorem 1.2 is
therefore complete. ∎

##  4\. Comparison estimates and the control of the invariant probability

###  4.1. On the proximity of random walks with different reflection
mechanisms

We consider the Markov chain $(l_{n})$: $(r_{-n+1})$ is the same process. We
recall that it satisfies

$l_{n+1}\,=\,l_{n}+2h_{n+1}+\log\left(\frac{1+e^{-\Gamma-
l_{n}-2h_{n+1}}}{1+e^{-\Gamma+l_{n}+2h_{n+1}}}\right)\,=\,b_{\Gamma}\circ\theta_{h_{n+1}}(l_{n}).$
(4.1)

The main result is about the stationary sequence $(l_{n})_{n\in{\mathbb{Z}}}$,
but this is clearly inessential for the first result that applies to
$(l_{n})_{n=n_{0},n_{0}+1,\dots}$ with arbitrary law of $l_{n_{0}}$.

Recall that $(S_{k})_{k\in{\mathbb{Z}}}$ is the random walk with $S_{0}=0$ and
$S_{m}-S_{n}=\sum_{j=n+1}^{m}h_{j}$ for every $n<m$.

######  Lemma 4.1.

We have for all time $n_{0}$ and all $n\geq n_{0}$,

$\displaystyle l_{n}\,$
$\displaystyle\leq\,\log\left(e^{l_{n_{0}}+2(S_{n}-S_{n_{0}})}+\varepsilon\sum_{k={n_{0}}+1}^{n}e^{-2(S_{k}-S_{n})}\right),$
(4.2) $\displaystyle l_{n}\,$
$\displaystyle\geq\,-\log\left(e^{-(l_{n_{0}}+2(S_{n}-S_{n_{0}}))}+\varepsilon\sum_{k={n_{0}}+1}^{n}e^{2(S_{k}-S_{n})}\right),$
(4.3)

with the convention $\sum_{\emptyset}:=0$ and $\prod_{\emptyset}:=1$.

###### Proof.

We observe that for every $j$,

$\exp\left(l_{j+1}\right)\,\leq\,\exp\left(l_{j}\right)\exp\left(2h_{j+1}\right)+\varepsilon\,.$
(4.4)

By iterating this bound starting from $j=n_{0}$ we obtain that for every
$n=n_{0},n_{0}+1,\ldots$

$\begin{split}\exp\left(l_{n}\right)\,&\leq\,\exp\left(l_{n_{0}}\right)\prod_{k={n_{0}}+1}^{n}\exp(2h_{k})+\varepsilon\sum_{k={n_{0}}+1}^{n}\prod_{i=k+1}^{n+1}\exp(2h_{i})\\\
&=\,\exp\left(l_{n_{0}}\right)e^{2(S_{n}-S_{n_{0}})}+\varepsilon\sum_{k={n_{0}}+1}^{n}e^{2(S_{n}-S_{k})}\,,\end{split}$
(4.5)

which, by taking the logarithm, becomes (4.2).

To obtain (4.3) it suffices to remark that $(-l_{n})_{n\in{\mathbb{Z}}}$
satisfies the same recurrence relation as $(l_{n})_{n\in{\mathbb{Z}}}$, with
the only change that $S$ is replaced by $-S$ (see Remark 2.3). Therefore (4.3)
follows from (4.2). ∎

We are now ready to prove Lemma 3.7.

###### Proof of Lemma 3.7.

Choose $\kappa>0$. We give a proof of the bound on $l_{0}-\widehat{l}_{0}$ in
(3.18): the result on $r_{0}-\widehat{r}_{0}$ follows by changing $S$ in
$-S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}$, and then
applying a shift by 1 (see Remark 2.3). For the bound on
$l_{0}-\widehat{l}_{0}$ we focus on the event that
$\mathtt{s}^{\downarrow}(\Gamma)>\mathtt{s}^{\uparrow}(\Gamma)$. The result in
the case $\mathtt{s}^{\uparrow}(\Gamma)>\mathtt{s}^{\downarrow}(\Gamma)$
follows by changing $S$ to $-S$.

Recall from Proposition 3.5 that on the event
$\\{\mathtt{s}^{\downarrow}(\Gamma)>\mathtt{s}^{\uparrow}(\Gamma)\\}$, we have
$\widehat{l}_{0}=\Gamma-2S_{\mathtt{v}^{\downarrow}(\Gamma)}$. Our program is
resumed in two (independent) bounds: on the event
$\\{\mathtt{s}^{\downarrow}(\Gamma)>\mathtt{s}^{\uparrow}(\Gamma)\\}$,
excluding an event of probability $O(\Gamma^{-\kappa})$ and by suitably
choosing a constant $\widetilde{C}_{\kappa}<\infty$

  * •

_Upper bound_ : exploiting that
$l_{\mathtt{v}^{\downarrow}(\Gamma)}\leq\Gamma$ and that from time
$\mathtt{v}^{\downarrow}(\Gamma)$ to time 0 the process $l_{\cdot}$ evolves
approximately according to $2S_{\cdot}$ we aim at showing that

$l_{0}\,\leq\,\Gamma-2S_{\mathtt{v}^{\downarrow}(\Gamma)}+\log\left(1+\widetilde{C}_{\kappa}\log\Gamma\right)\,;$
(4.6)

  * •

_Lower bound_ : exploiting that there is a rise in $S$ of size not smaller
than $\Gamma$ between times $\mathtt{s}^{\downarrow}(\Gamma)$ and
$\mathtt{v}^{\downarrow}(\Gamma)$ and that afterwards, up to time 0, the
process $l_{\cdot}$ evolves approximately according to $2S_{\cdot}$ we aim at
showing that

$l_{0}\,\geq\,\Gamma-2S_{\mathtt{v}^{\downarrow}(\Gamma)}-\log\left(1+\widetilde{C}_{\kappa}\log\Gamma\right)\,.$
(4.7)

We proceed now with the proofs of the two bounds, which rely on Lemma 4.1 and
a control of the error terms provided by Lemma A.3 in the appendix. The proofs
are independent.

#### Proof of the upper bound

By applying Lemma 4.1 with $n_{0}=\mathtt{v}^{\downarrow}(\Gamma)$ and $n=0$,
we obtain that

$\displaystyle l_{0}$
$\displaystyle\leq\,\log\left(e^{l_{\mathtt{v}^{\downarrow}(\Gamma)}+2(S_{0}-S_{\mathtt{v}^{\downarrow}(\Gamma)})}+\varepsilon\sum_{n=\mathtt{v}^{\downarrow}(\Gamma)+1}^{0}e^{-2\left(S_{n}-S_{0}\right)}\right)\,$
(4.8)
$\displaystyle=\,\Gamma-2S_{\mathtt{v}^{\downarrow}(\Gamma)}+\log\left(e^{l_{\mathtt{v}^{\downarrow}(\Gamma)}-\Gamma}+\sum_{n=\mathtt{v}^{\downarrow}(\Gamma)+1}^{0}e^{2\left(S_{\mathtt{v}^{\downarrow}(\Gamma)}-S_{n}-\Gamma\right)}\right)\,.$

Since $l_{v^{\downarrow}(\Gamma)}\in[-\Gamma,\Gamma]$, the first term inside
the logarithm is bounded by 1. It follows from Lemma A.3(ii) (applied to
$S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}$), that with
probability $1-O(\Gamma^{-\kappa})$, on the event
$\\{\mathtt{s}^{\downarrow}(\Gamma)>\mathtt{s}^{\uparrow}(\Gamma)\\}$, we have

$\sum_{n=\mathtt{v}^{\downarrow}(\Gamma)+1}^{0}e^{2\left(S_{\mathtt{v}^{\downarrow}(\Gamma)}-S_{n}-\Gamma\right)}\,\leq\,\widetilde{C}_{\kappa}\log\Gamma\,.$
(4.9)

The upper bound (4.6) follows.

#### Proof of the lower bound

For the lower bound, we can actually work without the assumption that
$s^{\downarrow}(\Gamma)>s^{\uparrow}(\Gamma)$. The bound will of course remain
true on the event $\\{s^{\downarrow}(\Gamma)>s^{\uparrow}(\Gamma)\\}$. By
applying Lemma 4.1 with $n_{0}=s^{\downarrow}(\Gamma)$ and $n=0$, we have that

$\displaystyle l_{0}$
$\displaystyle\,\geq\,-\log\left(e^{-\left(l_{\mathtt{s}^{\downarrow}(\Gamma)}+2(S_{0}-S_{\mathtt{s}^{\downarrow}(\Gamma)})\right)}+\varepsilon\sum_{n=\mathtt{s}^{\downarrow}(\Gamma)+1}^{0}e^{2\left(S_{n}-S_{0}\right)}\right)$
(4.10)
$\displaystyle\,=\,\Gamma-2S_{\mathtt{v}^{\downarrow}(\Gamma)}-\log\left(e^{\Gamma-
l_{s^{\downarrow}(\Gamma)}-2(S_{\mathtt{v}^{\downarrow}(\Gamma)}-S_{\mathtt{s}^{\downarrow}(\Gamma)})}+\sum_{n=\mathtt{s}^{\downarrow}(\Gamma)+1}^{0}e^{2\left(S_{n}-S_{\mathtt{v}^{\downarrow}(\Gamma)}\right)}\right)\,,$

Since $l_{\mathtt{s}^{\downarrow}(\Gamma)}\in[-\Gamma,\Gamma]$ and
$S_{\mathtt{v}^{\downarrow}(\Gamma)}-S_{\mathtt{s}^{\downarrow}(\Gamma)}\geq\Gamma$,
the first term inside the logarithm is bounded by 1. It follows from Lemma
A.3(i) (applied to
$S\rule{0.0pt}{6.0pt}^{\scaleto{\mathtt{rv}\mathstrut}{5pt}}$) that with
probability $1-O(\Gamma^{-\kappa})$ we have

$\sum_{n=\mathtt{s}^{\downarrow}(\Gamma)+1}^{0}e^{2\left(S_{n}-S_{\mathtt{v}^{\downarrow}(\Gamma)}\right)}\,\leq\,\widetilde{C}_{\kappa}\log\Gamma\,.$
(4.11)

This completes the proof of the lower bound (4.7) and therefore also the proof
of Lemma 3.7 is complete. ∎

###  4.2. An estimate on the invariant probability

To establish (2.35) we use Lemma 3.7 to see that for every interval
$I=[a,b]\subset[-\Gamma,\Gamma]$

${\mathbb{P}}\left(r_{1}\in[a,b]\right)\leq
O(\Gamma^{-1})+{\mathbb{P}}\left(\widehat{r}_{1}\in[a-2\log\log\Gamma,b+2\log\log\Gamma\right)\,,$
(4.12)

so we reduce the issue to estimating
${\mathbb{P}}\left(\widehat{r}_{1}\in[a,b]\right)$ and we recall that for
$\widehat{r}_{1}$ we have the expression (3.16). We recall also that the
processes $\widehat{l}$ and $\widehat{r}$ are stationary sequences with the
same marginal law. A simulation of such an invariant law is in Fig. 3.

######  Lemma 4.2.

There exists $c_{1}>0$ and $\Gamma_{0}>0$ such that for every
$I=[a,b]\subset[-\Gamma,\Gamma]$ and $\Gamma\geq\Gamma_{0}$

${\mathbb{P}}\left(\widehat{r}_{1}\in I\right)\leq
c_{1}\frac{1+b-a}{\Gamma}\,,$ (4.13)

and

${\mathbb{P}}\left(r_{1}\in I\right)\leq c_{1}\frac{5\log_{\circ
2}\Gamma+b-a}{\Gamma}\,,$ (4.14)

###### Proof.

Let us introduce the (strictly) ascending ladder epochs: $\varrho_{0}:=0$ and

$\varrho_{k}:=\inf\\{n>\varrho_{k-1}:S_{n}>S_{\varrho_{k-1}}\\},\qquad
k=1,2,\ldots$ (4.15)

By definition of $\mathtt{u}^{\downarrow}(\Gamma)$ in (3.12), we have that
$\mathtt{u}^{\downarrow}(\Gamma)=\varrho_{\mathcal{K}}$ with

${\mathcal{K}}\,:=\,\inf\\{k\geq 0:\max_{\varrho_{k}\leq
n<\varrho_{k+1}}(S_{\varrho_{k}}-S_{n})\geq\Gamma\\}\,.$ (4.16)

Consider the case
$\\{\mathtt{t}^{\downarrow}(\Gamma)<\mathtt{t}^{\uparrow}(\Gamma)\\}$. By
(3.16), $\widehat{r}_{1}=-\Gamma+2S_{\varrho_{\mathcal{K}}}$. It follows that

$\begin{split}{\mathbb{P}}\left(\widehat{r}_{1}\in[a,b],\,\mathtt{t}^{\downarrow}(\Gamma)<\mathtt{t}^{\uparrow}(\Gamma)\right)\,&\leq\,{\mathbb{P}}\left(-\Gamma+2S_{\varrho_{\mathcal{K}}}\in[a,b]\right)\,\\\
&\leq\,\sum_{k=0}^{\infty}{\mathbb{P}}\left(-\Gamma+2S_{\varrho_{k}}\in[a,b],\min_{\varrho_{k}\leq
j<\varrho_{k+1}}S_{j}\leq S_{\varrho_{k}}-\Gamma\right)\\\
&=\,\sum_{k=0}^{\infty}{\mathbb{P}}\left(-\Gamma+2S_{\varrho_{k}}\in[a,b]\right)\,{\mathbb{P}}\left(\min_{\varrho_{k}\leq
j<\varrho_{k+1}}S_{j}\leq S_{\varrho_{k}}-\Gamma\right)\\\
&=:\,U\left(\left[(a+\Gamma)/2,(b+\Gamma)2\right]\right)\,{\mathbb{P}}\left(\min_{0\leq
j<\varrho_{1}}S_{j}\leq-\Gamma\right),\end{split}$ (4.17)

with standard definition of the renewal function $U$ associated with the
increasing random walk, or renewal process, $S_{\varrho_{k}}$. By [31, Ch. IV,
p. 199] we know that ${\mathbb{E}}[S_{\varrho_{1}}]<\infty$ and since
$U([x,y])\leq U([0,y-x])$ (see for example [2, Th. 2.4, Ch. V]), the
elementary version of the Renewal Theorem, i.e. $U([0,x])\sim
x/{\mathbb{E}}[S_{\varrho_{1}}]$ for $x\to\infty$ (a consequence of the Law of
Large Numbers), implies that there exists $c_{2}>0$ such that for every $0\leq
x<y$

$U([x,y])\,\leq\,c_{2}(1+y-x)\,.$ (4.18)

Since by (A.2) we have ${\mathbb{P}}(\min_{0\leq
j<\varrho_{1}}S_{j}\leq-\Gamma)\leq{c_{4}}/{\Gamma}$, we obtain (4.13). And,
by (4.12), from (4.13) we obtain (4.14). The proof of Lemma 4.2 is therefore
complete. ∎

Figure 3.  A simulation of the $(l_{n})_{n=0,\ldots,N}$, with $l_{0}=0$ and
$N=5\cdot 10^{7}$, $\Gamma=20$. In the first case the law of $h_{1}$ is
$(\delta_{-2}+\delta_{+2})/2$ and in the second case the law is
$(\delta_{-2}+\gamma)/2$ where $\gamma$ is the law of a Gaussian of mean $2$
and variance $1/\sqrt{2}$. These two histograms are expected to give an idea
of how the invariant probability looks like: in fact, the simulation is very
stable with respect to increasing $N$ and changing the randomness. Note that
this empirical observation is compatible in the second case with the invariant
probability being very close to (a multiple) of the Lebesgue measure away from
the boundaries (this result is proven in [21] under stronger regularity
properties of the law of $h_{1}$). However, in the first case a periodic
phenomenon seems to be present and proximity to the Lebesgue measure appears
to be plausible only if averages over large boxes are taken.

##  5\. Proof of Proposition 2.4

We are going to prove Proposition 2.4. So we are given a mesurable function
$K:{\mathbb{R}}^{\mathbb{Z}}\to\\{-1,0,+1\\}$, and we set
$s_{n}^{(K)}=K(\Theta^{\circ n}h)$. Our aim is to show the convergence of

$D_{N}\left(\sigma,s^{(K)}\right)=\frac{1}{N}\
\left|\left\\{n=1,2\ldots,N:\,\sigma_{n}\neq
s_{n}^{(K)}\right\\}\right|=\frac{1}{N}\sum_{n=1}^{N}\mathbf{1}_{\sigma_{n}\neq
s_{n}^{(K)}}\,,$ (5.1)

which we will write for short $D_{N}$, towards

${\mathbb{E}}\left[\frac{1}{1+\exp\left(s^{(K)}_{0}m_{0}\right)}\right]+\frac{1}{2}{\mathbb{P}}\left(s_{0}^{(K)}=0\right)\,,$
(5.2)

in ${\mathbf{P}}^{ab}_{N,J,h}$-probability, as well as in
${\mathbf{P}}_{J,h}$-probability.

Our strategy is to get rid of the boundary effects to turn to the infinite
volume model, and then use ergodicity. Paramount to our proof is the uniform
contractive estimate (2.25) in the proof of Lemma 2.2. We reproduce it here:
for every $k\geq 0$ and every $h_{1},\dots,h_{k}\in\mathbb{R}$

$\sup_{s,t\in[-\Gamma,\Gamma]}\left|f_{h_{k}}\circ f_{h_{k-1}}\circ\dots\circ
f_{h_{1}}(t)-f_{h_{k}}\circ f_{h_{k-1}}\circ\dots\circ
f_{h_{1}}(s)\right|\,\leq\,2\Gamma\exp(-k\varepsilon)\,.$ (5.3)

Of course this observation directly applies to control the difference of two
$l$ (or $r$) processes started at different initial values and/or different
initial times.

We first have a look at the expectation of $D_{N}$ under
${\mathbf{P}}_{N,J,h}^{ab}$. We are going to get rid of the boundaries by
turning to the infinite volume model; precisely we are going to show that:

${\mathbf{E}}^{ab}_{N,J,h}\left[D_{N}\right]\,=\,{\mathbf{E}}_{J,h}\left[D_{N}\right]+O\left(\frac{1}{N}\right),$
(5.4)

where $O\left(1/N\right)$ denotes a quantity whose absolute value is bounded
by $1/N$ multiplied by a constant depending on $\Gamma$. Observe that the
quantity on the left-hand-side of (5.4) is:

$\displaystyle{\mathbf{E}}^{ab}_{N,J,h}\left[D_{N}\right]$
$\displaystyle\,=\,\frac{1}{N}\sum_{n=1}^{N}{\mathbf{P}}^{ab}_{N,J,h}\left(\sigma_{n}\neq
s^{(K)}_{n}\right)$ (5.5)
$\displaystyle\,=\,\frac{1}{N}\sum_{n=1}^{N}\frac{1}{1+\exp\left(s^{(K)}_{n}\,m^{(ab)}_{1,n,N}\right)}+\frac{1}{2}\mathbf{1}_{s_{n}^{(K)}=0}$

and similarly, ${\mathbf{E}}_{J,h}\left[D_{N}(\sigma)\right]$ has the same
expression with $m^{(ab)}_{1,n,N}$ replaced by $m_{n}$. We recall that
$m^{(ab)}_{1,n,N}=l^{(a)}_{1,n-1}+2h_{n}+r^{(b)}_{n+1,N}$ and that
$m_{n}=l_{n-1}+2h_{n}+r_{n+1}$. Using that $x\mapsto 1/{(1+\exp(x))}$ is
1-Lipschitz and then (5.3) we deduce, for $n=1,\dots,N$,

$\displaystyle\left|\frac{1}{1+\exp\left(s^{(F)}_{n}m^{(ab)}_{1,n,N}\right)}-\frac{1}{1+\exp\left(s^{(F)}_{n}m_{n}\right)}\right|$
$\displaystyle\leq|m^{(ab)}_{1,n,N}-m_{n}|$ (5.6)
$\displaystyle\leq|l^{(a)}_{1,n-1}-l_{n-1}|+|r^{(b)}_{n+1,N}-r_{n+1}|$
$\displaystyle\leq
2\Gamma\left(\exp(-(n-1)\varepsilon)+\exp(-(N-n)\varepsilon)\right),$

and therefore (5.4) follows by summing over $n=1,\dots,N$.

Now, let us turn to controlling the second moment of $D_{N}$ under
${\mathbf{P}}^{ab}_{N,J,h}$. We have

$\displaystyle{\mathbf{E}}^{ab}_{N,J,h}\left[D_{N}^{2}\right]=\frac{1}{N}{\mathbf{E}}^{ab}_{N,J,h}[D_{N}]+\frac{2}{N^{2}}\sum_{1\leq
n<m\leq N}{\mathbf{P}}^{ab}_{N,J,h}\left(\sigma_{n}\neq
s^{(K)}_{n},\sigma_{m}\neq s^{(K)}_{m}\right).$ (5.7)

We note that for $1\leq n<m\leq N$

${\mathbf{P}}_{N,J,h}^{ab}\left(\sigma_{n}\neq s^{(K)}_{n},\sigma_{m}\neq
s^{(K)}_{m}\right)\,=\,{\mathbf{P}}_{N,J,h}^{ab}\left(\sigma_{m}\neq
s^{(K)}_{m}\right)\,{\mathbf{P}}_{m-1,J,h}^{a(-s^{(K)}_{m})}\left(\sigma_{n}\neq
s^{(K)}_{n}\right).$ (5.8)

Proceeding as in (5.6), we use (5.3) to bound the difference between
${\mathbf{P}}_{N,J,h}^{ab}\left(\sigma_{m}\neq s^{(K)}_{m}\right)$ and
${\mathbf{P}}_{J,h}\left(\sigma_{m}\neq s^{(K)}_{m}\right)$. Analogously we
bound the difference between
${\mathbf{P}}_{m-1,J,h}^{a(-s^{(K)}_{m})}\left(\sigma_{n}\neq
s^{(K)}_{n}\right)$ and ${\mathbf{P}}_{J,h}\left(\sigma_{n}\neq
s^{(K)}_{n}\right)$. Observing that
$|xy-x^{\prime}y^{\prime}|\leq|x-x^{\prime}|+|y-y^{\prime}|$ when
$x,y,x^{\prime},y^{\prime}\in[0,1]$, we derive from (5.7) and (5.8) that

${\mathbf{E}}^{ab}_{N,J,h}\left[D_{N}^{2}\right]=\left({\mathbf{E}}_{J,h}\left[D_{N}\right]\right)^{2}+O\left(\frac{1}{N}\right)\,.$
(5.9)

Putting together (5.4) and (5.9) and using that $D_{N}$ is bounded, the
variance of $D_{N}$ under ${\mathbf{P}}^{ab}_{N,J,h}$ is $O\left(1/N\right)$.
Furthermore, ${\mathbf{E}}^{ab}_{N,J,h}[D_{N}]$ converges to the announced
quantity since ${\mathbf{E}}_{J,h}[D_{N}]$ does, as already mentioned in
(2.31). This completes the proof of Proposition 2.4 for what concerns the
statements in ${\mathbf{P}}_{N,J,h}^{ab}$-probability. The proof in
${\mathbf{P}}_{J,h}$-probability can be obtained either by modifications of
the argument we just developed or by remarking that in reality we have
established the convergence for all boundary conditions in $\\{-1,+1\\}^{2}$,
even random ones, and from this the case of ${\mathbf{P}}_{J,h}$ follows. ∎

##  Appendix A Random walk estimates

Recall that, by (1.3) and (1.6), $(S_{n})$ is a random walk with centered
finite variance increments and $S_{0}=0$. In this section we consider also the
case in which $S_{0}=x\in{\mathbb{R}}$ and the law of this process is denoted
by ${\mathbb{P}}_{x}$. When $x=0$ we just drop de dependence on $x$:
${\mathbb{P}}={\mathbb{P}}_{0}$. We will use the following estimate several
times, referred to as the Gambler’s ruin problem. Let

$T^{+}_{0}\,:=\,\inf\left\\{n\geq 1:S_{n}>0\right\\}\,.$ (A.1)

By [25, Theorem 5.1.7] there exist positive constants $x_{0}$, $c_{3}$ and
$c_{4}$ such that for any $x\geq x_{0},$

$\,{\mathbb{P}}\left(\min_{0\leq k\leq
T^{+}_{0}}S_{k}\leq-x\right)\,\in\,\left[\frac{c_{3}}{x},\frac{c_{4}}{x}\right].$
(A.2)

The objective of this subsection is to establish Lemma A.3, which serves as
the main technical tool in proving Lemma 3.7.

We begin with a general fact that follows from the study in [22] on a
stochastic recurrence equation, in the case where the solution exhibits a thin
tail.

######  Lemma A.1.

Let $(\eta_{j},\Delta_{j})_{j\geq 1}$ be an IID sequence of random vectors
distributed like $(\eta,\Delta)$. Suppose that $\eta$ and $\Delta$ are
nonnegative random variables such that ${\mathbb{P}}(\Delta>0)>0$ and that
${\mathbb{E}}e^{c\,\eta}<\infty$ for some $c>0$. Then there exists $\rho>0$
such that

$\sup_{k\geq
0}{\mathbb{E}}\exp\left(\rho\,\sum_{j=1}^{k+1}\eta_{j}e^{-\sum_{\mathcalb{l}=j}^{k}\Delta{l}}\right)<\infty\,,$
(A.3)

with the convention $\sum_{\mathcalb{l}=k+1}^{k}\Delta{l}=0$.

###### Proof.

Let $Y_{j}:=\Delta_{1}+...+\Delta_{j}$ for any $j\geq 1$ and $Y_{0}:=0$. Since
$(\eta_{k+1-j},\Delta_{k+1-j})_{j=1,\ldots,k+1}$ has the same law as
$(\eta_{j},\Delta_{j})_{j=0,1,\ldots,k}$, we see that, for each fixed $k\geq
0$, $\sum_{j=1}^{k+1}\eta_{j}e^{-\sum_{\mathcalb{l}=j}^{k}\Delta{l}}$ has the
same law as $\sum_{\mathcalb{l}=0}^{k}\eta{l}e^{-Y{l}}$, which is bounded
above by
$\eta_{0}+\sum_{\mathcalb{l}=1}^{\infty}\eta{l}e^{-Y_{\mathcalb{l}-1}}$. By
[22], Theorem 2.1, there exists some $\rho>0$ such that
${\mathbb{E}}\exp(\rho\sum_{\mathcalb{l}=1}^{\infty}\eta{l}e^{-Y_{\mathcalb{l}-1}})<\infty$,
yielding the Lemma. ∎

The following estimate ensures the existence of small exponential moments, a
requirement for the assumption in Lemma A.1.

######  Lemma A.2.

Assume (1.3). There exist positive constants $\rho$ and $c_{5}$ such that for
every $x\leq 0$

${\mathbb{E}}_{x}\left[e^{\rho\sum_{n=0}^{\infty}e^{2S_{n}}\mathbf{1}_{\\{\max_{0\leq
i\leq n}S_{i}\leq 0\\}}}\right]\,\leq\,c_{5}(1+|x|)\,.$ (A.4)

###### Proof.

By applying [24, Corollary 4.2] for $\lambda<2$ we obtain that there exists
$c_{6}>0$ such that for every $x\leq 0$

${\mathbb{E}}_{x}\left[e^{2S_{n}}(1+|S_{n}|)\mathbf{1}_{\\{\max_{0\leq i\leq
n}S_{i}\leq 0\\}}\right]\,\leq\,c_{6}\,(1+|x|)\,n^{-3/2}\,,$ (A.5)

which yields that there exists $c_{7}>0$ such that

${\mathbb{E}}_{x}\left[\sum_{n=0}^{\infty}e^{2S_{n}}(1+|S_{n}|)\mathbf{1}_{\\{\max_{0\leq
i\leq n}S_{i}\leq 0\\}}\right]\,\leq\,c_{7}\,(1+|x|),\qquad\text{for every
}x\leq 0\,.$ (A.6)

For any $m\geq 1$, applying the Markov property successively at
${\mathcalb{l}_{m-1}},...,\mathcalb{l}_{1}$ in the following sum, we deduce
from (A.6) that for any $x\leq 0$,

$\sum_{0\leq\mathcalb{l}_{1}<...<\mathcalb{l}_{m}}{\mathbb{E}}_{x}\left[\prod_{i=1}^{m}e^{2S_{\mathcalb{l}_{i}}}\mathbf{1}_{\\{\max_{0\leq
j\leq\mathcalb{l}_{m}}S_{j}\leq 0\\}}\right]\,\leq\,c_{7}^{m}\,(1+|x|)\,.$
(A.7)

It follows that for every integer $p\geq 1$ and $x\leq 0$

$\begin{split}{\mathbb{E}}_{x}\Bigg{[}\Bigg{(}\sum_{n=0}^{\infty}e^{2S_{n}}&\mathbf{1}_{\\{\max_{0\leq
i\leq n}S_{i}\leq 0\\}}\Bigg{)}^{p}\Bigg{]}\,\leq\,p!\sum_{0\leq
n_{1}\leq...\leq n_{p-1}\leq
n_{p}}{\mathbb{E}}_{x}\left[\prod_{i=1}^{p}e^{2S_{n_{i}}}\mathbf{1}_{\\{\max_{0\leq
j\leq n_{p}}S_{j}\leq 0\\}}\right]\\\
&\phantom{move}=\,p!\sum_{m=1}^{p}\sum_{\begin{subarray}{c}r_{1}+...+r_{m}=p\\\
r_{1},...,r_{m}\geq
1\end{subarray}}\sum_{0\leq\mathcalb{l}_{1}<...<\mathcalb{l}_{m}}{\mathbb{E}}_{x}\left[\prod_{i=1}^{m}e^{2r_{i}S_{\mathcalb{l}_{i}}}\mathbf{1}_{\\{\max_{0\leq
j\leq\mathcalb{l}_{m}}S_{j}\leq 0\\}}\right]\\\
&\phantom{move}\leq\,p!\sum_{m=1}^{p}\sum_{\begin{subarray}{c}r_{1}+...+r_{m}=p\\\
r_{1},...,r_{m}\geq
1\end{subarray}}\sum_{0\leq\mathcalb{l}_{1}<...<\mathcalb{l}_{m}}{\mathbb{E}}_{x}\left[\prod_{i=1}^{m}e^{2S_{\mathcalb{l}_{i}}}\mathbf{1}_{\\{\max_{0\leq
j\leq\mathcalb{l}_{m}}S_{j}\leq 0\\}}\right]\\\
&\phantom{move}\leq\,p!\sum_{m=1}^{p}\sum_{\begin{subarray}{c}r_{1}+...+r_{m}=p\\\
r_{1},...,r_{m}\geq 1\end{subarray}}c_{7}^{m}(1+|x|),\end{split}$ (A.8)

where the last inequality follows from (A.7). We further remark that

$\sum_{m=1}^{p}\sum_{\begin{subarray}{c}r_{1}+...+r_{m}=p\\\
r_{1},...,r_{m}\geq
1\end{subarray}}c_{7}^{m}\,\leq\,\,\sum_{m=1}^{p}C_{p}^{m}c_{7}^{m}\,\leq\,(1+c_{7})^{p}\,.$
(A.9)

It follows that

$\frac{1}{p!}{\mathbb{E}}_{x}\left[\left(\sum_{n=0}^{\infty}e^{2S_{n}}\mathbf{1}_{\\{\max_{0\leq
i\leq n}S_{i}\leq 0\\}}\right)^{p}\right]\,\leq\,(1+c_{7})^{p}(1+|x|)\,,$
(A.10)

which yields that for every $\rho\in(0,1/(1+c_{7}))$

${\mathbb{E}}_{x}\left[e^{\rho\sum_{n=0}^{\infty}e^{2S_{n}}\mathbf{1}_{\\{\max_{0\leq
i\leq n}S_{i}\leq 0\\}}}\right]\,\leq\,\frac{1+|x|}{1-\rho(1+c_{7})},$ (A.11)

and the proof of Lemma A.2 is complete. ∎

Below are our main technical estimates: they are used in the proof of Lemma
3.7. Recall (1.17) for the definitions of ${\tt t}^{\downarrow}(\Gamma)$ and
${\tt t}^{\uparrow}(\Gamma)$ and (3.12) for $\mathtt{u}^{\downarrow}(\Gamma)$,
as well as (1.8), (1.9) for ${\tt t}_{1}(\Gamma)$ and ${\tt u}_{1}(\Gamma)$.
In particular, ${\tt t}^{\downarrow}(\Gamma)={\tt t}_{1}(\Gamma)$ and
$\mathtt{u}^{\downarrow}(\Gamma)={\tt u}_{1}(\Gamma)$.

######  Proposition A.3.

Assume (1.3). Then for every $\kappa>0$, there exists
$\widetilde{C}_{\kappa}>0$ such that we have

(i)

${\mathbb{P}}\left(\sum_{n=0}^{{\tt
t}^{\downarrow}(\Gamma)-1}e^{2\left(S_{n}-S_{\mathtt{u}^{\downarrow}(\Gamma)}\right)}\geq\widetilde{C}_{\kappa}\log\Gamma\right)\,=\,O\left(\Gamma^{-\kappa}\right)\,.$
(A.12)

(ii)

${\mathbb{P}}\left(\sum_{n=0}^{\mathtt{u}^{\downarrow}(\Gamma)-1}e^{2\left(S_{\mathtt{u}^{\downarrow}(\Gamma)}-S_{n}-\Gamma\right)}\geq\widetilde{C}_{\kappa}\log\Gamma\,,\,{\tt
t}^{\downarrow}(\Gamma)<{\tt
t}^{\uparrow}(\Gamma)\right)\,=\,O\left(\Gamma^{-\kappa}\right)\,.$ (A.13)

###### Proof.

We start with item $(i)$. Recall (4.15), (4.16) and
$\mathtt{u}^{\downarrow}(\Gamma)=\varrho_{\mathcal{K}}$. By definition, we
have the following bound

$\sum_{n=0}^{{\tt
t}^{\downarrow}(\Gamma)-1}e^{2\left(S_{n}-S_{\mathtt{u}^{\downarrow}(\Gamma)}\right)}\,\leq\,\sum_{j=1}^{{\mathcal{K}}+1}\sum_{n=\varrho_{j-1}}^{\varrho_{j}-1}e^{2\left(S_{n}-S_{\varrho_{\mathcal{K}}}\right)}.$
(A.14)

For every $j=1,\dots,\mathcal{K}+1$ we have

$\sum_{n=\varrho_{j-1}}^{\varrho_{j}-1}e^{2(S_{n}-S_{\varrho_{\mathcal{K}}})}=e^{2(S_{\varrho_{j-1}}-S_{\varrho_{\mathcal{K}}})}\,\sum_{n=\varrho_{j-1}}^{\varrho_{j}-1}e^{2(S_{n}-S_{\varrho_{j-1}})}=\eta_{j}e^{-\sum_{\mathcalb{l}=j}^{\mathcal{K}}\Delta{l}},$
(A.15)

with the convention
$\sum_{\mathcalb{l}={\mathcal{K}}+1}^{\mathcal{K}}\Delta{l}=0$ and where we
denoted for $j=1,2,\ldots$

$\eta_{j}\,:=\,\sum_{n=\varrho_{j-1}}^{\varrho_{j}-1}e^{2(S_{n}-S_{\varrho_{j-1}})}\
\text{ and }\ \Delta_{j}\,:=\,2(S_{\varrho_{j}}-S_{\varrho_{j-1}})>0\,.$
(A.16)

Remark that the law of ${\mathcal{K}}$ is geometric:

${\mathbb{P}}({\mathcal{K}}=k)\,=\,p_{\Gamma}(1-p_{\Gamma})^{k},\qquad
k=0,1,\ldots,$ (A.17)

with $p_{\Gamma}\,:=\,{\mathbb{P}}\left(\min_{0\leq
k\leq\varrho_{1}}S_{k}\leq-\Gamma\right)$. By (A.2),
$p_{\Gamma}\,\in\,\left[\frac{c_{3}}{\Gamma},\frac{c_{4}}{\Gamma}\right]$ for
all large $\Gamma$. It follows that

${\mathbb{P}}\left({\mathcal{K}}\geq\Gamma^{2}\right)\,\leq\,\Gamma^{-\kappa}\,.$
(A.18)

By (A.14) and (A.18),

${\mathbb{P}}\Big{(}\sum_{n=0}^{{\tt
t}^{\downarrow}(\Gamma)-1}e^{2\left(S_{n}-S_{\mathtt{u}^{\downarrow}(\Gamma)}\right)}\geq\widetilde{C}_{\kappa}\log\Gamma\Big{)}\leq\Gamma^{-\kappa}+\sum_{k=0}^{\Gamma^{2}-1}{\mathbb{P}}\Big{(}\sum_{j=1}^{k+1}\eta_{j}e^{-\sum_{\mathcalb{l}=j}^{k}\Delta{l}}\geq
C_{\kappa}^{\prime}\log\Gamma\Big{)}.$ (A.19)

Note that $(\eta_{j},\Delta_{j})_{j\geq 1}$ are IID and that $\eta_{j}$ is
distributed as $\sum_{n=0}^{\varrho_{1}-1}e^{2S_{n}}$ which has finite small
exponential moments by Lemma A.2. Therefore we can apply Lemma A.1 and obtain
that for some positive constant $c_{8}$,

${\mathbb{P}}\Big{(}\sum_{n=0}^{{\tt
t}^{\downarrow}(\Gamma)-1}e^{2\left(S_{n}-S_{\mathtt{u}^{\downarrow}(\Gamma)}\right)}\geq\widetilde{C}_{\kappa}\log\Gamma\Big{)}\leq\Gamma^{-\kappa}+c_{8}\,\Gamma^{2}\,e^{-\rho\,\widetilde{C}_{\kappa}\log\Gamma}\,,$
(A.20)

yielding (A.12), as soon as $\widetilde{C}_{\kappa}>(\kappa+2)/\rho$.

For what concerns $(ii)$ first we claim that there exists some $b=b(\kappa)>0$
such that for every large $\Gamma$

${\mathbb{P}}\left(\mathtt{u}^{\downarrow}(\Gamma)\geq\Gamma^{b}\right)\,\leq\,2\,\Gamma^{-\kappa}\,.$
(A.21)

In fact, we use $\mathtt{u}^{\downarrow}(\Gamma)=\varrho_{\mathcal{K}}$ with
${\mathcal{K}}$ given in (4.16) and estimated in (A.18): we obtain

${\mathbb{P}}\left(\mathtt{u}^{\downarrow}(\Gamma)\geq\Gamma^{b}\right)\,\leq\,\Gamma^{-\kappa}+{\mathbb{P}}\left(\varrho_{\Gamma^{2}}\geq\Gamma^{b}\right)\,.$
(A.22)

By [24, Theorem 4.6], there exists $c_{9}>0$ such that

${\mathbb{P}}\left(\varrho_{1}\geq n\right)\stackrel{{\scriptstyle
n\to\infty}}{{\sim}}\frac{c_{9}}{\sqrt{n}}\,.$ (A.23)

Since $\varrho_{k}$ is distributed as the sum of $k$ IID copies of
$\varrho_{1}$, we have that there exists $c_{10}>0$ such that

${\mathbb{P}}\left(\varrho_{k}\geq
n\right)\,\leq\,k\,{\mathbb{P}}\left(\varrho_{1}\geq\frac{n}{k}\right)\,\leq\,c_{10}\frac{k^{3/2}}{\sqrt{n}}\,.$
(A.24)

This implies that ${\mathbb{P}}(\varrho_{\Gamma^{2}}\geq\Gamma^{b})\leq
c_{2}\Gamma^{3-b/2}\leq\Gamma^{-\kappa}$ if we choose $b>2\kappa+6$ ($\Gamma$
being large). Therefore (A.21) is established.

Now we give the proof of (A.13). On the event $\\{{\tt
t}^{\downarrow}(\Gamma)<{\tt t}^{\uparrow}(\Gamma)\\}$, for every
$n\leq\mathtt{u}^{\downarrow}(\Gamma)$,
$S_{\mathtt{u}^{\downarrow}(\Gamma)}-S_{n}<\Gamma$ and

${\mathbb{P}}\left(\sum_{n=0}^{\mathtt{u}^{\downarrow}(\Gamma)-1}e^{2\left(S_{\mathtt{u}^{\downarrow}(\Gamma)}-S_{n}-\Gamma\right)}\geq\widetilde{C}_{\kappa}\log\Gamma,\,{\tt
t}^{\downarrow}(\Gamma)<{\tt
t}^{\uparrow}(\Gamma)\right)\leq{\mathbb{P}}\left(\mathtt{u}^{\downarrow}(\Gamma)\geq\Gamma^{b}\right)+\sum_{k=0}^{\Gamma^{b}-1}J_{k},$
(A.25)

where

$J_{k}\,:=\,{\mathbb{P}}\left(\sum_{n=0}^{k-1}e^{2\left(S_{k}-S_{n}-\Gamma\right)}\geq\widetilde{C}_{\kappa}\log\Gamma,\,\max_{0\leq
n<k}(S_{k}-S_{n})<\Gamma\right)\,.$ (A.26)

Since $(S_{k}-S_{k-j})_{j=0,\ldots,k-1}$ is distributed like
$(S_{j})_{j=1,\ldots,k}$, we get that

$\displaystyle J_{k}$ $\displaystyle=$
$\displaystyle{\mathbb{P}}\left(\sum_{j=1}^{k}e^{2(S_{j}-\Gamma)}\geq\widetilde{C}_{\kappa}\log\Gamma,\,\max_{1\leq
j\leq k}S_{j}<\Gamma\right)$ (A.27) $\displaystyle=$
$\displaystyle{\mathbb{P}}_{-\Gamma}\left(\sum_{j=1}^{k}e^{2S_{j}}\times\mathbf{1}_{\\{\max_{1\leq
i\leq k}S_{i}<0\\}}\geq\widetilde{C}_{\kappa}\log\Gamma\right)$
$\displaystyle\leq$
$\displaystyle{\mathbb{P}}_{-\Gamma}\left(\sum_{j=0}^{\infty}e^{2S_{j}}\mathbf{1}_{\\{\max_{1\leq
i\leq
j}S_{i}<0\\}}\geq\widetilde{C}_{\kappa}\log\Gamma\right)\,\leq\,\Gamma^{-\kappa-b},$

for large $\Gamma$: the last inequality holds for $\widetilde{C}_{\kappa}$
large enough, depending on $\kappa$ and is due to Lemma A.2. By (A.25) and
(A.21), we get that

${\mathbb{P}}\left(\sum_{n=0}^{\mathtt{u}^{\downarrow}(\Gamma)}e^{-2(S_{\mathtt{u}^{\downarrow}(\Gamma)}-S_{n}-\Gamma)}\geq\widetilde{C}_{\kappa}\log\Gamma,\,{\tt
t}^{\downarrow}(\Gamma)<{\tt
t}^{\uparrow}(\Gamma)\right)\,\leq\,3\,\Gamma^{-\kappa}\,,$ (A.28)

and the proof of Proposition A.3 is complete. ∎

The tools introduced in this section are useful to establish also the
following result: set ${\tt t}_{0}(\Gamma):=0$ and recall the definition of
the sequences of random times $({\tt t}_{n}(\Gamma))_{n\in{\mathbb{N}}}$ and
$({\tt u}_{n}(\Gamma))_{n\in{\mathbb{N}}}$ from Section 1.2.

######  Lemma A.4.

The two sequences of random vectors in ${\mathbb{R}}^{2}$

$\left({\tt u}_{n}(\Gamma)-{\tt t}_{n-1}(\Gamma),S_{{\tt
u}_{n}(\Gamma)}-S_{{\tt t}_{n-1}(\Gamma)}\right)_{n\in{\mathbb{N}}}\text{ and
}\left({\tt t}_{n}(\Gamma)-{\tt u}_{n}(\Gamma),S_{{\tt t}_{n}(\Gamma)}-S_{{\tt
u}_{n}(\Gamma)}\right)_{n\in{\mathbb{N}}}$ (A.29)

are independent and, both of them, are sequences of independent random
vectors. Moreover the two (sub)sequences

$\left({\tt u}_{n}(\Gamma)-{\tt t}_{n-1}(\Gamma),S_{{\tt
u}_{n}(\Gamma)}-S_{{\tt t}_{n-1}(\Gamma)}\right)_{n\in 2{\mathbb{N}}}\text{
and }\left({\tt t}_{n}(\Gamma)-{\tt u}_{n}(\Gamma),S_{{\tt
t}_{n}(\Gamma)}-S_{{\tt u}_{n}(\Gamma)}\right)_{n\in 2{\mathbb{N}}}$ (A.30)

are IID and the same holds if we consider the (sub)sequences with $n\in
2{\mathbb{N}}-1$.

###### Proof.

Using the fact that ${\tt u}_{1}(\Gamma)=\varrho_{\mathcal{K}}$ with
${\mathcal{K}}$ defined in (4.16), we remark that, conditioned on $\sigma({\tt
u}_{1}(\Gamma),(S_{j})_{j=1,\ldots,{\tt u}_{1}(\Gamma)})$, the law of
$(S_{{\tt u}_{1}(\Gamma)+i}-S_{{\tt u}_{1}(\Gamma)})_{i=0,\ldots,{\tt
t}_{1}(\Gamma)}$ coincides with the law of
$(S_{i})_{i=0,\ldots,T_{S}(-\Gamma)}$ conditioned on
$\\{T_{S}(-\Gamma)<\varrho_{1}\\}$, where $T_{S}(-\Gamma):=\min\\{n\geq
0:S_{n}\leq-\Gamma\\}$ and $\varrho_{1}$ is defined in (4.15). It follows that
$({\tt t}_{1}(\Gamma)-{\tt u}_{1}(\Gamma),S_{{\tt t}_{1}(\Gamma)}-S_{{\tt
u}_{1}(\Gamma)})$ is independent of $({\tt u}_{1}(\Gamma),S_{{\tt
u}_{1}(\Gamma)})$.

Since ${\tt t}_{2}(\Gamma)-{\tt t}_{1}(\Gamma)$ is the first time of
$\Gamma$-increase of $(S_{{\tt t}_{1}(\Gamma)+i}-S_{{\tt
t}_{1}(\Gamma)})_{i=0,1,\ldots}$, and ${\tt u}_{2}(\Gamma)-{\tt
t}_{1}(\Gamma)$ is the corresponding time at which the minimum $\min_{0\leq
i\leq{\tt t}_{2}(\Gamma)-{\tt t}_{1}(\Gamma)}(S_{{\tt
t}_{1}(\Gamma)+i}-S_{{\tt t}_{1}(\Gamma)})$ is achieved, we deduce from the
strong Markov property that $({\tt u}_{2}(\Gamma)-{\tt t}_{1}(\Gamma),S_{{\tt
u}_{2}(\Gamma)}-S_{{\tt t}_{1}(\Gamma)})$ is independent of
$\sigma(S_{n},n=0,\ldots,{\tt t}_{1}(\Gamma))$, in particular independent of
$({\tt t}_{1}(\Gamma)-{\tt u}_{1}(\Gamma),S_{{\tt t}_{1}(\Gamma)}-S_{{\tt
u}_{1}(\Gamma)})$ and $({\tt u}_{1}(\Gamma),S_{{\tt u}_{1}(\Gamma)})$.

The proof is completed by iterating the arguments above. ∎

##  Appendix B Proofs for the reflected random walk

###### Proof of Proposition 3.5.

For what concerns (3.15) we consider the case
$\mathtt{s}^{\downarrow}(\Gamma)>\mathtt{s}^{\uparrow}(\Gamma)$ (see Fig. 4)
and observe that there is an increase of at least $\Gamma$ for
$(S_{n})_{n=\mathtt{s}^{\downarrow}(\Gamma),\ldots,\mathtt{v}^{\downarrow}(\Gamma)}$.
Recalling that $\widehat{l}_{n}\in[-\Gamma,\Gamma]$ and that $\widehat{l}$ is
driven by $2S$, we see that the process
$(\widehat{l}_{n})_{n=\mathtt{s}^{\downarrow}(\Gamma),\mathtt{s}^{\downarrow}(\Gamma)+1,\ldots}$
hits $\Gamma$ for a value of
$n\in\\{\mathtt{s}^{\downarrow}(\Gamma)+1,\ldots,\mathtt{v}^{\downarrow}(\Gamma)\\}$
and, using the definition of $\mathtt{v}^{\downarrow}(\Gamma)$, we see also
that $\widehat{l}_{\mathtt{v}^{\downarrow}(\Gamma)}=\Gamma$ too: the process
hits $\Gamma$ not later than $\mathtt{v}^{\downarrow}(\Gamma)$ because the
driving process increases of at least $2\Gamma$, but $\widehat{l}$ is set to
$\Gamma$ if the driving process _tries_ to make it larger than $\Gamma$. From
time $n$ the process tries again to follow the driving process, except that it
cannot become larger than $\Gamma$.

Figure 4.  In this figure $\Gamma=5.5$. The $\widehat{l}$ process, on the
right, is driven by twice the random walk trajectory, on the left. We look at
the trajectory from $\mathtt{s}^{\downarrow}(\Gamma)$ and we use
$\widehat{l}_{\mathtt{s}^{\downarrow}(\Gamma)}\in[-\Gamma,\Gamma]$. The fact
that the $S$ trajectory increases of at least $\Gamma$ from
$\mathtt{s}^{\downarrow}(\Gamma)$ to $\mathtt{v}^{\downarrow}(\Gamma)$
guarantees that the $\widehat{l}$ trajectory hits $\Gamma$ not later than
$\mathtt{v}^{\downarrow}(\Gamma)$. Moreover,
$\widehat{l}_{\mathtt{v}^{\downarrow}(\Gamma)}=\Gamma$ too. After
$\mathtt{v}^{\downarrow}(\Gamma)$ and up to time 0 the $\widehat{l}$
trajectory copies the increments of $2S$.

Now we observe that if $\mathtt{v}^{\downarrow}(\Gamma)=0$ we have
$\widehat{l}_{0}=\Gamma$ and therefore (3.15) holds in this case. If instead
$\mathtt{v}^{\downarrow}(\Gamma)<0$, see Fig. 4, the increments of
$(\widehat{l}_{n})_{n=\mathtt{v}^{\downarrow}(\Gamma),\ldots,0}$ coincide with
the increments of $(2S_{n})_{n=\mathtt{v}^{\downarrow}(\Gamma),\ldots,0}$, and
also in this case (3.15) holds.

Figure 5.  In this figure $\Gamma=5.5$. Equation (3.16) holds because the
evolution of $(\widehat{r}_{-n})$, i.e. we are looking from right to left,
repeats the increments of $(-2S_{-n-1})$ except that the $\widehat{r}$ process
is confined to $[-\Gamma,\Gamma]$. Always arguing with the reversed time
arrow, we see that, regardless of the value of
$\widehat{r}_{\mathtt{t}^{\downarrow}(\Gamma)+1}\in[-\Gamma,\Gamma]$, the
$\widehat{r}$ process is going to hit $\Gamma$ not later than
$\mathtt{u}^{\downarrow}(\Gamma)$ and that, in any case,
$\widehat{r}_{\mathtt{u}^{\downarrow}(\Gamma)+1}=-\Gamma$. After that moment
and up to time 1 the evolution of $\widehat{r}$ reproduces the increments of
$(-2S_{-n-1})$.

The argument for
$\mathtt{s}^{\downarrow}(\Gamma)<\mathtt{s}^{\uparrow}(\Gamma)$ is absolutely
analogous or by observing that we can map the case
$\mathtt{s}^{\downarrow}(\Gamma)<\mathtt{s}^{\uparrow}(\Gamma)$ to
$\mathtt{s}^{\downarrow}(\Gamma)>\mathtt{s}^{\uparrow}(\Gamma)$ by replacing
$S$ with $-S$ and $\widehat{l}$ with $-\widehat{l}$.

For what concerns (3.16) one can repeat the very same argument or use the time
reversal symmetry of the problem: in any case one has to take into account the
shift of one site pointed out in Remark 2.3, see Fig. 5 and its caption. ∎

###### Proof of Lemma 3.6.

We use the notion of $\Gamma$-extrema over ${\mathbb{Z}}$ introduced in
Section 3.1, but we present an interesting caracterisation of these which may
be of help for the reader: a site $\mathtt{u}\in{\mathbb{Z}}$ is a
$\Gamma$-maximum of $(S_{n})_{n\in{\mathbb{Z}}}$ over ${\mathbb{Z}}$ if and
only if there exist $\mathtt{s},\mathtt{t}\in{\mathbb{Z}}$ such that
$\mathtt{s}\leq\mathtt{u}\leq\mathtt{t}$, $S_{\mathtt{s}}\leq
S_{\mathtt{u}}-\Gamma$ and $S_{\mathtt{t}}\leq S_{\mathtt{u}}-\Gamma$.
$\Gamma$-minima over ${\mathbb{Z}}$ are caracterised similarly. With this
caracterisation in mind, we introduce a slightly richer notation for the
_multiple_ $\Gamma$-extrema. In fact, here we denote by
$\mathtt{U}_{j},j\in{\mathbb{Z}}$ all maximal sets of adjacent
$\Gamma$-extrema (adjacent means “not separated by opposite
$\Gamma$-extrema”). The sets $\mathtt{U}_{j},j\in{\mathbb{Z}}$ are ordered in
the natural way, inherited from the usual order on ${\mathbb{Z}}$. One can fix
the indexation by letting for example $\mathtt{U}_{1}$ be the class containing
the smallest positive $\Gamma$-extremum. Of course $S$ is constant over
$\mathtt{U}_{j}$ and we denote by $S_{\mathtt{U}_{j}}$ this common value.
Furthermore, we set $u_{j}^{-}=\min\mathtt{U}_{j}$ and
$u_{j}^{+}=\max\mathtt{U}_{j}$. Recall that $s^{(F)}$ is defined by:

  * •

$s^{(F)}_{n}=1$ if there exists $j\in{\mathbb{Z}}$ such that
$\mathtt{u}_{j}^{+}<n\leq\mathtt{u}_{j+1}^{-}$ and
$(\mathtt{u}_{j}^{+},\mathtt{u}_{j+1}^{-})$ is a $\Gamma$-ascending stretch of
height strictly larger than $\Gamma$;

  * •

$s^{(F)}_{n}=-1$ if there exists $j\in{\mathbb{Z}}$ such that
$\mathtt{u}_{j}^{+}<n\leq\mathtt{u}_{j+1}^{-}$ and
$(\mathtt{u}_{j}^{+},\mathtt{u}_{j+1}^{-})$ is a $\Gamma$-descending stretch
of height strictly larger than $\Gamma$;

  * •

$s^{(F)}_{n}=0$ otherwise.

We compute $\widehat{l}$ and $\widehat{r}$ on a $\Gamma$-ascending stretch,
see Fig. 6. Take $j\in{\mathbb{Z}}$ and assume that
$(\mathtt{u}_{j}^{+},\mathtt{u}_{j+1}^{-})$ is a $\Gamma$-ascending stretch,
i.e. $S_{\mathtt{u}_{j+1}^{-}}-S_{\mathtt{u}_{j}^{+}}\geq\Gamma$ and
$S_{n}-S_{m}>-\Gamma$ whenever $\mathtt{u}_{j}^{+}\leq
m<n\leq\mathtt{u}_{j+1}^{-}$. We are going to express $\widehat{l}$ on
$\\{\mathtt{u}_{j}^{-},\dots,\mathtt{u}_{j+1}^{-}\\}$, then $\widehat{r}$ on
$\\{\mathtt{u}_{j}^{-}+1,\dots,\mathtt{u}_{j+1}^{-}+1\\}$, and finally
$\widehat{m}$ on $\\{\mathtt{u}_{j}^{-}+1,\dots,\mathtt{u}_{j+1}^{-}\\}$.

Figure 6.  In this figure we present a $\Gamma$-ascending stretch of $S$, for
$\Gamma\in[3,8)$: the set of $\Gamma$-minima
$\mathtt{U}_{j}=\\{\mathtt{u}_{j}^{-},\mathtt{u}^{\prime}_{j},\mathtt{u}_{j}^{+}\\}$
and the set of $\Gamma$-maxima $\mathtt{U}_{j}=\\{\mathtt{u}_{j+1}^{-}\\}$ are
independent of the choice of $\Gamma$ in $[3,8)$. We use $\mathtt{t}_{j,1}$
for $\mathtt{t}_{j}(\Gamma)$ and $\mathtt{s}_{j,1}$ for
$\mathtt{s}_{j}(\Gamma)$ when $\Gamma\in[3,6)$. When $\Gamma\in[6.5,8)$
instead we use $\mathtt{t}_{j,2}$ for $\mathtt{t}_{j}(\Gamma)$ and
$\mathtt{s}_{j,2}$ for $\mathtt{s}_{j}(\Gamma)$. Note that
$\mathtt{t}_{j,1}<\mathtt{s}_{j,1}$, while
$\mathtt{t}_{j,2}>\mathtt{s}_{j,2}$. So if we consider the site $n$ for
$\Gamma\in[3,6)$ we have to apply the middle option in the right-hand side of
(B.4): we obtain that $\widehat{m}_{n}=2\Gamma-6>0$. If instead we consider
the site $n$ for $\Gamma\in[6.5,8)$, we have to apply the lowest option in the
right-hand side of (B.4): we obtain that $\widehat{m}_{n}=24>0$. Note that
when $\Gamma\in[6,6.5)$, then $\mathtt{t}_{j}(\Gamma)$ is $\mathtt{t}_{j,2}$,
but $\mathtt{s}_{j}(\Gamma)$ still coincides with $\mathtt{s}_{j,1}$. Note
also that is $\Gamma<3$ then $\tt{U}_{j}$ is no longer a set of
$\Gamma$-minima. And if $\Gamma\geq 8$ the portion of random walk in the image
is insufficient to determine the $\Gamma$-extrema.

We set :

$\begin{split}\mathtt{t}_{j}&=\inf\\{\mathtt{u}_{j}^{-}\leq
n\leq\mathtt{u}_{j+1}^{-}:S_{n}\geq S_{\mathtt{U}_{j}}+\Gamma\\}\\\
\mathtt{s}_{j}&=\sup\\{\mathtt{u}_{j}^{-}\leq
n\leq\mathtt{u}_{j+1}^{-}:S_{n}\leq
S_{\mathtt{U}_{j+1}}-\Gamma\\}\,.\end{split}$ (B.1)

Note that $\mathtt{t}_{j}$ and $\mathtt{s}_{j}$ both belong to
$\\{\mathtt{u}_{j}^{+},\dots,\mathtt{u}_{j+1}^{-}\\}$. We point out that they
are a priori not ordered: we may have either
$\mathtt{t}_{j}\leq\mathtt{s}_{j}$ or $\mathtt{s}_{j}<\mathtt{t}_{j}$.

By construction we have that $\widehat{l}_{\mathtt{u}_{j}^{-}}=-\Gamma$ and,
therefore, that

$\widehat{l}_{n}=\begin{cases}-\Gamma+2\left(S_{n}-S_{\mathtt{U}_{j}}\right)&\text{
for }\mathtt{u}_{j}^{-}\leq n<\mathtt{t}_{j}\\\
\Gamma-2\max_{\mathtt{u}_{j}^{-}\leq i\leq n}\left(S_{i}-S_{n}\right)&\text{
for }\mathtt{t}_{j}\leq n\leq\mathtt{u}_{j+1}^{-}\end{cases}$ (B.2)

By exploiting the time reversal properties (see Remark 2.3), we get the
similar result for $\widehat{r}$:

$\widehat{r}_{n}=\begin{cases}-\Gamma+2\left(S_{\mathtt{U}_{j+1}}-S_{n-1}\right)&\text{
for }\mathtt{s}_{j}<n\leq\mathtt{u}_{j+1}^{-}+1\\\ \Gamma-2\max_{n-1\leq
i\leq\mathtt{u}_{j+1}^{-}}\left(S_{n-1}-S_{i}\right)&\text{ for
}\mathtt{u}_{j}^{-}\leq n\leq\mathtt{s}_{j}\end{cases}$ (B.3)

Using these expressions, we can now compute $\widehat{m}_{n}$. We distinguish
between two cases. If $\mathtt{t}_{j}\leq\mathtt{s}_{j}$, then

$\widehat{m}_{n}=\widehat{l}_{n-1}+2h_{n}+\widehat{r}_{n+1}=\begin{cases}2\min_{n\leq
i\leq\mathtt{u}_{j+1}^{-}}\left(S_{i}-S_{\mathtt{U}_{j}}\right)&\text{ for
}\mathtt{u}_{j}^{-}<n<\mathtt{t}_{j}\,,\\\
2\Gamma-2\max_{\mathtt{u}_{j}^{-}\leq i\leq n\leq
j\leq\mathtt{u}_{j+1}^{-}}\left(S_{i}-S_{j}\right)&\text{ for
}\mathtt{t}_{j}\leq n\leq\mathtt{s}_{j}\,,\\\ 2\max_{\mathtt{u}_{j}^{-}\leq
i\leq n}\left(S_{\mathtt{U}_{j+1}}-S_{i}\right)&\text{ for
}\mathtt{s}_{j}<n\leq\mathtt{u}_{j+1}^{-}\,.\end{cases}$ (B.4)

If $\mathtt{s}_{j}<\mathtt{t}_{j}$ instead

$\widehat{m}_{n}=\widehat{l}_{n-1}+2h_{n}+\widehat{r}_{n+1}=\begin{cases}2\min_{n\leq
i\leq\mathtt{u}_{j+1}^{-}}\left(S_{i}-S_{\mathtt{U}_{j}}\right)&\text{ for
}\mathtt{u}_{j}^{-}<n\leq\mathtt{s}_{j}\,,\\\
-2\Gamma+2\left(S_{\mathtt{U}_{j+1}}-S_{\mathtt{U}_{j}}\right)&\text{ for
}\mathtt{s}_{j}<n<\mathtt{t}_{j}\,,\\\ 2\max_{\mathtt{u}_{j}^{-}\leq i\leq
n}\left(S_{\mathtt{U}_{j+1}}-S_{i}\right)&\text{ for
}\mathtt{t}_{j}<n\leq\mathtt{u}_{j+1}^{-}\,.\end{cases}$ (B.5)

Using these explicit expressions for $\widehat{m}_{n}$, we can complete the
proof. In fact one readily checks that

  * •

if $\mathtt{u}_{j}^{-}<n\leq\mathtt{u}_{j}^{+}$, then $\widehat{m}_{n}=0$;

  * •

if $S_{\mathtt{U}_{j+1}}-S_{\mathtt{U}_{j}}>\Gamma$, then we have
$\widehat{m}_{n}>0$ when $\mathtt{u}_{j}^{+}<n\leq\mathtt{u}_{j+1}^{-}$, using
the fact that $\max_{\mathtt{u}_{j}^{-}\leq i\leq
j\leq\mathtt{u}_{j+1}^{-}}\left(S_{i}-S_{j}\right)<\Gamma$;

  * •

if $S_{\mathtt{U}_{j+1}}-S_{\mathtt{U}_{j}}=\Gamma$, then we have
$\mathtt{s}_{j}=\mathtt{u}_{j}^{+}$ and $\mathtt{t}_{j}=\mathtt{u}_{j+1}^{-}$
and we derive $\widehat{m}_{n}=0$ for all $n$ such that
$\mathtt{u}_{j}^{+}<n\leq\mathtt{u}_{j+1}^{-}$.

To cover $\Gamma$-descending stretches, it suffices to replace $S$ by $-S$. ∎

##  Appendix C Scaling to the Neveu-Pitman process

This section relies on the Donsker Invariance Principle for one dimensional
centered random walks with finite variance increments. This is often
established in $C^{0}([0,1];{\mathbb{R}})$, but for us it is more practical to
work in $C^{0}([0,\infty);{\mathbb{R}})$ [32, Sec. 1.3]: Donsker Invariance
Principle in $C^{0}([0,\infty);{\mathbb{R}})$ is given as an exercise in [32,
Ex. 2.4.2] or it follows by applying the general result [32, Th. 11.2.3]. Here
is the statement we are going to use: set for $t\geq 0$ such that
$t\Gamma^{2}/\vartheta^{2}$ is integer

$B^{(\Gamma)}_{t}:=\,\frac{1}{\Gamma}S_{t\,\Gamma^{2}/\vartheta^{2}}\,,$ (C.1)

and extend this definition to every $t\geq 0$ by affine interpolation, so
$B^{(\Gamma)}$ is a stochastic process with trajectories in
$C^{0}([0,\infty);{\mathbb{R}})$, like the standard Brownian motion $B$.
Donsker Invariance Principle states, with standard notation [3], that
$B^{(\Gamma)}\Rightarrow B$ as $\Gamma\to\infty$.

###### Proof of Proposition 1.3.

We start by observing that Lemma A.4 directly implies that the sequence in
(1.21) is a sequence of independent random vectors and that the odd and even
subsequences are IID. We are therefore left with proving the convergence part
of the statement, i.e. (1.22). For this we introduce the continuum analog of
the notation of Section 1.2. We therefore consider the set $C^{0}_{\infty}$ of
functions $b_{\cdot}\in C^{0}([0,\infty);{\mathbb{R}})$ such that
$\limsup_{t\to\infty}b_{t}=\infty$ and $\liminf_{t\to\infty}b_{t}=-\infty$:
note that $C^{0}_{\infty}$ is open in $C^{0}([0,\infty);{\mathbb{R}})$. We
introduce the first time of first (unit) decrease for $b\in C^{0}_{\infty}$ as

$t_{1}(b)\,:=\,\min\left\\{t\geq 0:\,\text{there exists }s\in[0,t]\text{ such
that }b_{s}-b_{t}=1\right\\},$ (C.2)

and in turn the time of the first absolute maximum in $[0,t_{1}(b)]$:

$u_{1}(b)\,:=\,\min\left\\{t\in[0,t_{1}(b)]:\,b_{t}=\max_{s\in[0,t_{1}(b)]}b_{s}\right\\}.$
(C.3)

Then we keep going by looking for the first time of unit increase after
$t_{1}(b)$:

$t_{2}(b)\,:=\,\min\left\\{t\geq t_{1}(b):\,\text{there exists
}s\in[t_{1}(b),t]\text{ such that }b_{t}-b_{s}=1\right\\},$ (C.4)

and the first absolute minimum in $[t_{1}(b),t_{2}(b)]$

$u_{2}(b)\,:=\,\min\left\\{t\in[t_{1}(b),t_{2}(b)]:\,b_{t}=\min_{s\in[t_{1}(b),t_{2}(b)]}b_{s}\right\\}.$
(C.5)

And then one iterates this construction generating the infinite sequence of
$(t_{k}(b))_{k\in{\mathbb{N}}}$ of alternating unit decrease/increase and,
above all, the infinite sequence $(u_{k}(b))_{k\in{\mathbb{N}}}$ of
$1$-extrema for the function $b$.

A first key observation is that, with $B^{(\Gamma)}$ the continuous trajectory
built from the random walk trajectory $S$ in (C.1), we have

$u_{k}\left(B^{(\Gamma)}\right)\,=\,\frac{\vartheta^{2}}{\Gamma^{2}}\mathtt{u}_{k}(\Gamma)\
\ \text{ and }\ \
B^{(\Gamma)}_{u_{k}\left(B^{(\Gamma)}\right)}\,=\,\frac{1}{\Gamma}S_{\mathtt{u}_{k}(\Gamma)}\,.$
(C.6)

Now we remark that, for every $k$ the function
$u_{k}:C^{0}_{\infty}\to[0,\infty)$ is continuous in a neighborhood of $b$ if
for every $j=1,\ldots,k-1$ we have $|b_{u_{j+1}(b)}-b_{u_{j}(b)}|>1$ and at
$u_{k}(b)$ there is a unique absolute maximum (if $k$ is odd) or minimum (if
$k$ is even) for $b$ on the interval $[u_{k-1}(b),u_{k+1}(b)]$. This is an
event that happens with probability one if $b$ is a trajectory of a standard
Brownian motion. From this we directly infer an analog explicit condition on
the trajectory $b$ in order to have the continuity of

$b\mapsto\left(u_{j+1}(b)-u_{j}(b),\left|b_{u_{j+1}(b)}-b_{u_{j}(b)}\right|\right)_{j=1,2,\ldots,n},$
(C.7)

in the neighborhood of a given function $b$ and check that these properties
are almost surely verified if $b$ is a Brownian trajectory. Therefore by
Donsker Invariance Principle and by the Mapping Theorem [3, Th. 2.7, Ch. 1] we
obtain the convergence in law as $\Gamma\to\infty$ of the $2n$ dimensional
random vector

$\left(\frac{\vartheta^{2}}{\Gamma^{2}}\left(\mathtt{u}_{j+1}(\Gamma)-\mathtt{u}_{j}(\Gamma)\right),\frac{\left|S_{\mathtt{u}_{j+1}(\Gamma)}-S_{\mathtt{u}_{j}(\Gamma)}\right|}{\Gamma}\right)_{j=1,\ldots,n}\,,$
(C.8)

to

$\left(u_{j+1}(B)-u_{j}(B),|B_{u_{j+1}(B)}-B_{u_{j}(B)}|\right)_{j=1,\ldots,n}\,.$
(C.9)

This of course implies (1.22), so the proof of Proposition 1.3 is complete. It
is worth remarking that, given the independence and IID properties discussed
at the beginning of the proof, it suffices to establish the convergence for
the first two marginals of the sequence in (C.8): in particular it suffices to
establish the convergence of (C.8) to (C.9) for $n=2$. ∎

## Acknowledgments

Y.H. acknowledges the support of ANR, project LOCAL.

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