Corner percolation with preferential directions
Régine Marchand, Irène Marcovici and Pierrick Siest

Université de Lorraine, IECL

54000 Nancy

E-mail adress<EMAIL_ADDRESS>irene.marcovici@univ-
lorraine.fr<EMAIL_ADDRESS>

Abstract. Corner percolation is a dependent bond percolation model on
${\mathbb{Z}}^{2}$ introduced by Bálint Tóth, in which each vertex has exactly
two incident edges, perpendicular to each other. Gábor Pete has proven in 2008
that under the maximal entropy probability measure, all connected components
are finite cycles almost surely. We consider here a regime where West and
North directions are preferred with probability $p$ and $q$ respectively, with
$(p,q)\neq(\frac{1}{2},\frac{1}{2})$. We prove that there exists almost surely
an infinite number of infinite connected components, which are in fact
infinite paths. Furthermore, they all have the same asymptotic slope
$\frac{2q-1}{1-2p}$.

##  1 Introduction

Various constrained percolation models on $\mathbb{Z}^{2}$ have been studied,
including models with restrictions on the degree of each vertex, see for
example [4] or [2]. In the present work, we study an even more constrained
percolation model, called _corner percolation_ , where each vertex has exactly
two incident open edges, perpendicular to each other. It was introduced by
Bálint Tóth, and studied by Gábor Pete in the maximal entropy regime [6].

Corner percolation configurations are also known under the name of
_hitomezashi_ design, by analogy with a Japanese style of embroidery called
_sashiko_ , that creates patterns satisfying the same constraints, see Figure
2. Since at each vertex, there is exactly one horizontal and one vertical open
edge, the components are either finite cycles or bi-infinite paths, made in
both case of a perfect alternation of horizontal and vertical edges. The
mathematical properties of hitomezashi loops have been recently investigated
and still raise many questions. In particular, it was proven that their length
is always congruent to $4$ modulo $8$, but the proof is quite intricate, and
the enumeration of hitomezashi loops according to their length is still an
open question [3].

Although they exhibit a rich combinatorial structure, corner percolation
configurations can be created by a very simple procedure. Indeed, in order to
satisfy the constraint, there must be on each horizontal line a perfect
alternation of open and closed edges, and the same holds for vertical lines.
Consequently, a configuration can be parametrized by two binary sequences
specifying, for each horizontal and vertical line, which edges are open. A
natural question is then to ask whether there exists an infinite connected
component, depending on the probability distributions of the two binary
sequences. If the choice is made uniformly at random and independently for
each horizontal and vertical line, it can be interpreted as the maximal
entropy probability measure on all configurations satisfying the constraints.
Gábor Pete treated this case and proved that under the maximal entropy
probability measure, all connected components are finite cycles almost surely
[6].

Let us give an alternative interpretation of the maximal entropy probability
measure. We consider that each horizontal line of the grid $\mathbb{Z}^{2}$ is
a one-way road that can be either oriented to the East or to the West with
probability $\frac{1}{2}$, and the same for vertical lines with North and
South. This orientation is fixed once and for all. We start from the origin of
the grid and follow the horizontal road according to its orientation. At each
corner, we turn left or right according to the direction of the road
encountered, see Figure 2. This process describes a deterministic walk in a
random environment, and it can be seen that the path taken follows the
component of the origin in a corner percolation configuration distributed
under the maximal entropy probability measure. Consequently, Pete’s result
ensures that the trajectory necessarily comes back where it started.

Figure 1: A configuration of corner percolation.

$y$$x$$(0,0)$ Figure 2: An instance of a finite trajectory for the
deterministic walk starting from the origin, in an environment giving the
directions of the roads.

This point of view leads us to consider a case where some directions can be
preferred: we extend this model by considering that each vertical road is
oriented to the North with probability $q$, and that each horizontal road is
oriented to the West with probability $p$. We prove the following:

######  Theorem 1.1.

If $(p,q)\neq(\frac{1}{2},\frac{1}{2})$, then with positive probability the
path starting from the origin never goes back to its starting point. Moreover,
there almost surely exists an infinite number of infinite paths, and they all
have a same asymptotic slope, which is equal to $\frac{2q-1}{1-2p}$ (and if
$p=1/2$, the asymptotic slope is infinite).

This result will be proved in two steps, Theorem 3.1 and Theorem 4.1. In the
next section, we define the model more formally and we prove its ergodicity.

##  2 Definition of the model and ergodicity

###  2.1 Definition and notations

Let us denote by $(\mathbb{Z}^{2},\mathbb{E}^{2})$ the two-dimensional grid,
that is, the graph whose set of vertices is $\mathbb{Z}^{2}$ and whose set of
edges is
$\mathbb{E}^{2}=\\{\\{x,x+e_{1}\\}:x\in\mathbb{Z}^{2}\\}\cup\\{\\{x,x+e_{2}\\}:x\in\mathbb{Z}^{2}\\}$,
where $e_{1}=(1,0)$ and $e_{2}=(0,1)$.

Let $\mathcal{G}$ be the set of subgraphs $G=(\mathbb{Z}^{2},\mathcal{E})$ of
$(\mathbb{Z}^{2},\mathbb{E}^{2})$ such that each vertex has exactly one
horizontal and one vertical adjacent edge, that is: for any
$x\in\mathbb{Z}^{2}$, exactly one of the edges among $\\{x,x+e_{1}\\}$ and
$\\{x,x-e_{1}\\}$ belongs to $\mathcal{E}$, and exactly one of the edges among
$\\{x,x+e_{2}\\}$ and $\\{x,x-e_{2}\\}$ belongs to $\mathcal{E}$. Note that
this implies that connected components are all infinite simple paths or finite
circuits.

Let us introduce the following sets of _even_ and _odd_ elements of
$\mathbb{Z}^{2}$:

$\displaystyle\mathbb{Z}^{2}_{e}=\\{(x_{1},x_{2})\in\mathbb{Z}^{2}:\,x_{1}+x_{2}\in
2\mathbb{Z}\\},$
$\displaystyle\mathbb{Z}^{2}_{o}=\\{(x_{1},x_{2})\in\mathbb{Z}^{2}:\,x_{1}+x_{2}\in
1+2\mathbb{Z}\\}.$

Let $\Omega=\\{-1,1\\}^{\mathbb{Z}}\times\\{-1,1\\}^{\mathbb{Z}}$. Our
notations are inspired by those of Pete. We define a map
$X:\Omega\to\mathcal{G}$ as follows. For $\omega=(\xi,\eta)\in\Omega$, the
graph $X(\omega)=(\mathbb{Z}^{2},\mathcal{E})$ is defined this way (see Figure
3): $\mathcal{E}$ is the set of open edges, and for all $i\in\mathbb{Z}$,

  * •

if $\xi(i)=1$, then for all $k\in\mathbb{Z}$, the vertical edge
$\\{(i,k),(i,k)+e_{2}\\}$ is open if and only if $(i,k)\in\mathbb{Z}^{2}_{o}$.
If $\xi(i)=0$, then for all $k\in\mathbb{Z}$, the vertical edge
$\\{(i,k),(i,k)+e_{2}\\}$ is open if and only if $(i,k)\in\mathbb{Z}^{2}_{e}$;

  * •

if $\eta(i)=1$, then for all $k\in\mathbb{Z}$, the horizontal edge
$\\{(k,i),(k,i)+e_{1}\\}$ is open if and only if $(k,i)\in\mathbb{Z}^{2}_{o}$.
If $\eta(i)=0$, then for all $k\in\mathbb{Z}$, the horizontal edge
$\\{(k,i),(k,i)+e_{1}\\}$ is open if and only if $(k,i)\in\mathbb{Z}^{2}_{e}$.

$y$$x$0101 Figure 3: Construction of some open edges knowing that $\xi(1)=1$,
$\xi(-3)=0$, $\eta(1)=1$ and $\eta(2)=0$. Points of $\mathbb{Z}_{e}^{2}$ are
colored in gray, points of $\mathbb{Z}_{o}^{2}$ in red.

By construction, $X$ takes values in $\mathcal{G}$. Furthermore, the map
$X:\Omega\to\mathcal{G}$ is bijective. For $p,q\in]0,1[$, we introduce the
probability

$\mu^{p,q}=(q\delta_{1}+(1-q)\delta_{-1})^{\otimes\mathbb{Z}}\otimes(p\delta_{1}+(1-p)\delta_{-1})^{\otimes\mathbb{Z}}$

on $\Omega$ (with the product $\sigma$-algebra). We denote by
$\mathbb{P}^{p,q}$ the image of $\mu^{p,q}$ by $X$, and we call
$\mathbb{P}^{p,q}$ the _corner percolation model_ with parameters $(p,q)$.
When there is no ambiguity on the parameters, we write $\mu$ (respectively
$\mathbb{P}$) instead of $\mu^{p,q}$ (respectively $\mathbb{P}^{p,q}$). In the
special case $q=p=\frac{1}{2}$, this distribution can be interpreted as the
uniform distribution on $\mathcal{G}$. In this work, we are interested in the
properties of random graphs of $\mathcal{G}$ distributed according to
$\mathbb{P}^{p,q}$, and more specifically in the existence and properties of
infinite paths.

###  2.2 Invariance by even translations and ergodicity

For $v\in\mathbb{Z}^{2}$, the translation $\tau_{v}$ is the map:
$\tau_{v}:\mathcal{G}\to\mathcal{G}$ defined, for
$G=(\mathbb{Z}^{2},\mathcal{E})\in\mathcal{G}$, by:

$\tau_{v}(G)=(\mathbb{Z}^{2},v+\mathcal{E}),$

where $v+\mathcal{E}=\\{\\{x+v,y+v\\}:\\{x,y\\}\in\mathcal{E}\\}.$ Observe
that for any $v\in\mathbb{Z}^{2}$, the translation $\tau_{v}$ is bijective, of
inverse the translation $\tau_{-v}$. If $v\in\mathbb{Z}^{2}_{e}$, we call
$\tau_{v}$ an _even translation_.

######  Remark 2.1.

The law of the corner percolation model has some useful symmetry properties:

  1. 1.

The model of parameters $(p,q)$ has the same law as the model of parameters
$(1-p,1-q)$ translated by the vector $(1,0)$ or by the vector $(0,1)$.
Therefore, the distribution $\mathbb{P}$ is invariant under even translations.

  2. 2.

The model of parameters $(p,q)$ has the same law as the model obtained by
symmetry with respect to the $x$-axis (respectively $y$-axis) of the model of
parameters $(p,1-q)$ (respectively $(1-p,q)$).

  3. 3.

The model of parameters $(q,p)$ has the same law as the model obtained by
symmetry with respect to the line of equation $y=x$ of the model of parameters
$(p,q)$.

These transformations also let the number of infinite paths invariant.

######  Proposition 2.2.

For all $(k,\ell)\in\mathbb{Z}^{2}_{e}$, such that $k\not=0$ and $\ell\not=0$,
the dynamical system $(\mathcal{G},\mathfrak{F},\mathbb{P},\tau_{(k,\ell)})$
is _ergodic_ , that is: for any event $A$ such that
$\tau_{(k,\ell)}^{-1}(A)=A$, we have $\mathbb{P}(A)\in\\{0,1\\}$.

###### Proof.

We will show that $(\mathcal{G},\mathbb{P},\mathfrak{F},\tau_{(k,\ell)})$ is
_mixing_ , that is: for any events $A$ and $B$,

$\lim\limits_{n\to+\infty}\mathbb{P}(A\cap\tau_{(k,\ell)}^{-n}(B))=\mathbb{P}(A)\mathbb{P}(B).$

Indeed, Corollary 2.2.6 of [5] shows that it implies ergodicity.

Let us first assume that $A$ and $B$ depend only on finitely many lines and
columns. Let $I_{A}$ and $J_{A}$ (resp. $I_{B}$ and $J_{B}$) be finite subsets
of $\mathbb{Z}$ such that $A$ (resp. $B$) depends only on $\\{\eta(i):{i\in
I_{A}}\\}$ and $\\{\xi(j):{j\in J_{A}}\\}$ (resp. on $\\{\eta(i):{i\in
I_{B}}\\}$ and $\\{\xi(j):{j\in J_{B}}\\}$). Since $k\not=0$ and $\ell\not=0$,
for $n$ large enough, we have $I_{A}\cap(-kn+I_{B})=\emptyset$ and
$J_{A}\cap(-\ell n+J_{B})=\emptyset$, so that the events $A$ and
$\tau_{(k,\ell)}^{-n}(B)$ depend on disjoint sets of lines and columns. The
probability $\mu$ is a product distribution, so we have
$\mathbb{P}(A\cap\tau_{(k,\ell)}^{-n}(B))=\mathbb{P}(A)\times\mathbb{P}(\tau_{(k,\ell)}^{-n}(B))$.
Since $(k,\ell)\in\mathbb{Z}^{2}_{e}$, the translation $\tau_{(k,\ell)}$ is
even and preserves $\mathbb{P}$. So, we obtain

$\mathbb{P}(A\cap\tau_{(k,\ell)}^{-n}(B))=\mathbb{P}(A)\mathbb{P}(B).$

Events depending only on finitely many lines and columns generate the product
$\sigma$-algebra, so that by Lemma 2.2.7 of [5], we obtain that
$(\mathcal{G},\mathfrak{F},\mathbb{P},\tau_{(k,\ell)})$ is mixing, and
therefore ergodic. ∎

Now, we state two consequences of Birkhoff’s pointwise ergodic theorem, that
will be used multiple times thereafter. We first define

$\begin{array}[]{ccccc}\phi&:&\mathbb{Z}^{2}&\to&\mathbb{Z}^{2}_{e}\\\
&&x=(x_{1},x_{2})&\mapsto&(x_{1}-x_{2},x_{1}+x_{2})\\\ \end{array},$

which is a bijection. We also set (see Figure 4), for all
$N\in\mathbb{N}^{*}$, $B_{N}=\llbracket-N,N\rrbracket^{2}$ and

$C_{N}=\phi(B_{N})=\\{x\in\mathbb{Z}^{2}_{e}~{}:~{}-2N\leq x_{1}\leq 2N\text{
and }-2N+|x_{1}|\leq x_{2}\leq 2N-|x_{1}|\\}.$

$y$$x$$0$$2$$2$ Figure 4: Representation of $C_{N}$ (red circles) and $B_{N}$
(grey points), for $N=2$.

######  Theorem 2.3 (Birkhoff’s pointwise ergodic theorem).

  1. 1.

For all $(k,\ell)\in\mathbb{Z}^{2}_{e}$ such that $k\neq 0$ and $\ell\neq 0$,
if $f\in L^{1}(\mathcal{G})$, then

$\displaystyle\lim\limits_{n\to+\infty}\frac{1}{n}\sum\limits_{i=0}^{n-1}f\circ\tau_{(k,l)}^{{i}}=\mathbb{E}(f)\mbox{
a.s.}$

  2. 2.

If $f\in L^{1}(\mathcal{G})$, then
$\displaystyle\lim\limits_{N\to+\infty}\frac{1}{\\#C_{N}}\sum\limits_{x\in
C_{N}}f\circ\tau_{x}=\mathbb{E}(f)$ a.s.

###### Proof.

Since $\mathbb{P}$ is invariant under the even translation $\tau_{(k,l)}$, the
first point is exactly Birkhoff’s theorem (see for instance Theorem 2.1.5 of
[5]). By Proposition 2.2, the dynamical system
$(\mathcal{G},\mathbb{P},\mathfrak{F},\tau_{(k,\ell)})$ is ergodic, thus the
limit is constant.

In the second point, we apply a multi-dimensional ergodic theorem. Consider
the two commuting even translations $\tau_{1}=\tau_{(1,1)}$ and
$\tau_{2}=\tau_{(-1,1)}$. We can apply Theorem 2.1.5 of [5] to the set
$\mathcal{T}=\\{\tau_{1}^{x_{1}}\circ\tau_{2}^{x_{2}}=\tau_{\phi(x_{1},x_{2})},~{}(x_{1},x_{2})\in\mathbb{Z}^{2}\\}$.
The ergodicity of $(\mathcal{G},\mathfrak{F},\mathbb{P},\tau_{1})$ ensures
that the $\sigma$-algebra of the events invariant by $\mathcal{T}$ is trivial,
so the limit is equal to $\mathbb{E}(f)$. ∎

##  3 Height function and infinite paths

Now, we define the _height_ function
$H:\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2})\to\mathbb{Z}$, which was the key
ingredient introduced by Gábor Pete in [6] to study the case
$(\frac{1}{2},\frac{1}{2})$. We color the faces of $\mathbb{Z}^{2}$ in a
chessboard manner: a face $(n+\frac{1}{2},m+\frac{1}{2})$ is black if $n+m\in
2\mathbb{Z}$, otherwise it is white. Note that a path has only black faces
along one side and only white faces along the other. We set
$H\left(\frac{1}{2},\frac{1}{2}\right)=0$. For a face
$(n+\frac{1}{2},m+\frac{1}{2})$, consider a path in
$\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2})$, from $(\frac{1}{2},\frac{1}{2})$
to $(n+\frac{1}{2},m+\frac{1}{2})$. We follow this path starting from
$(\frac{1}{2},\frac{1}{2})$, and each time it crosses an open edge from a
black face to a white face (respectively from a white face from a black face),
we add (respectively substract) $1$ to the height. Note that it does not
depend on the choice of the path, and all black faces (respectively all white
faces) have the same height along a path. The _level of a path_ is the height
of a black face along this path.

Despite the strong dependencies of the model, the height function can be
expressed as a function of two independent simple random walks on
$\mathbb{Z}$. Remember that we denote by $(\xi(n))_{n\in\mathbb{Z}}$ (resp.
$(\eta(m))_{m\in\mathbb{Z}}$) the values of the vertical (resp. horizontal)
lines, and that its law is
$(q\delta_{1}+(1-q)\delta_{-1})^{\otimes\mathbb{Z}}$ (resp.
$(p\delta_{1}+(1-p)\delta_{-1})^{\otimes\mathbb{Z}}$). We define two
independent random walks $(X_{n})_{n\in\mathbb{Z}}$ and
$(Y_{m})_{m\in\mathbb{Z}}$ by setting:

$\displaystyle X_{0}=0,\quad\text{ for
}n>0,\,X_{n}=\sum\limits_{i=1}^{n}\xi(i),$ $\displaystyle\quad\text{ and for
}n<0,\,X_{n}=-\sum\limits_{i=n+1}^{0}\xi(i);$ $\displaystyle
Y_{0}=0,\quad\text{ for }m>0,\,Y_{m}=\sum\limits_{i=1}^{m}\eta(i),$
$\displaystyle\quad\text{ and for
}m<0,\,Y_{m}=-\sum\limits_{i=m+1}^{0}\eta(i).$

It is not difficult to check that:

$H\left(n+\frac{1}{2},m+\frac{1}{2}\right)=\left\lceil\frac{X_{n}+Y_{m}}{2}\right\rceil.$

We can now prove our main result:

######  Theorem 3.1.

There almost surely exists an infinite number of infinite paths in corner
percolation with parameters $(p,q)$, as soon as
$(p,q)\neq(\frac{1}{2},\frac{1}{2})$.

###### Proof.

Step 1. We want to construct an event having positive probability, ensuring
that the component of the origin is infinite. First of all, note that we have,
for all $n\in\mathbb{Z}$,

$\mathbb{P}(X_{n+1}=X_{n}+1)=q\text{\quad
and\quad}\mathbb{P}(Y_{n+1}=Y_{n}+1)=p.$

We start by assuming that $q>\frac{1}{2}$: by the symmetry properties $2$ and
$3$ of Remark 2.1, we will have the result for all parameters
$(p,q)\neq(\frac{1}{2},\frac{1}{2})$. By the law of large numbers, we have
$\lim\limits_{n\to+\infty}X_{n}=+\infty$ and
$\lim\limits_{n\to+\infty}X_{-n}=-\infty$. Then let $K>\frac{16}{2q-1}$ be
such that

$\forall n\geq
K,\quad\frac{X_{n}}{n}-\frac{2q-1}{2}>\frac{2q-1}{4}\quad\text{and}\quad\frac{X_{-n}}{n}-\frac{1-2q}{2}<\frac{1-2q}{4},$

with positive probability. We have, for all $n\geq K$ and $m\in\mathbb{N}$
such that $m\leq\frac{2q-1}{2}n$, on one hand

$\displaystyle\left\lceil\frac{X_{n}+Y_{m}}{2}\right\rceil$
$\displaystyle\geq\frac{X_{n}-\frac{2q-1}{2}n}{2}\geq\frac{n}{2}\left(\frac{X_{n}}{n}-\frac{2q-1}{2}\right)>1,$

and on the other hand

$\displaystyle\left\lceil\frac{X_{-n}+Y_{-m}}{2}\right\rceil$
$\displaystyle\leq\frac{X_{-n}+\frac{2q-1}{2}n}{2}+1\leq\frac{n}{2}\left(\frac{X_{-n}}{n}-\frac{1-2q}{2}\right)+1<-1.$

Therefore, the event

$A=\left\\{\forall n\geq K,~{}\forall 0\leq
m\leq\frac{2q-1}{2}n,\quad\left\lceil\frac{X_{n}+Y_{m}}{2}\right\rceil>1\text{
and }\left\lceil\frac{X_{-n}+Y_{-m}}{2}\right\rceil<-1\right\\}$

has positive probability. For
$(\xi,\eta)\in\left(\\{-1,1\\}^{\mathbb{Z}}\right)^{2}$, we denote by
$(\xi_{K},\eta_{K})\in\left(\\{-1,1\\}^{\llbracket-K,K\rrbracket}\right)^{2}$
the restrictions $\left((\xi_{n})_{|n|\leq K},(\eta_{n})_{|n|\leq K}\right)$.
Let us define the map

$\begin{array}[]{rcccl}\Pi&:&\left(\\{-1,1\\}^{\mathbb{Z}}\right)^{2}&\to&\left(\\{-1,1\\}^{\mathbb{Z}\setminus\llbracket-K,K\rrbracket}\right)^{2}\\\
&&(\xi,\eta)&\mapsto&\left((\xi_{n})_{|n|>K},(\eta_{n})_{|n|>K}\right)\end{array}.$

Now we set

$A_{St}=\left\\{(\xi,\eta)\in(\\{-1,1\\}^{\mathbb{Z}})^{2}~{}:~{}(\xi_{K},\eta_{K})=(\mathbf{1}_{K},\mathbf{1}_{K}),~{}\Pi(\xi,\eta)\in\Pi(A)\right\\}.$

As all vertical and horizontal lines are independent, and
$A\subset\Pi^{-1}\circ\Pi(A)$, we have

$\displaystyle\mathbb{P}\left(A_{St}\right)$
$\displaystyle\geq\mathbb{P}\left((\xi_{K},\eta_{K})=(\mathbf{1}_{K},\mathbf{1}_{K})\right)\mathbb{P}\left(\Pi^{-1}\circ\Pi(A)\right)\geq
p^{(2K+1)^{2}}\mathbb{P}(A)>0.$

Now let us prove that $A_{St}\subset A$. Let $(\xi,\eta)\in A_{St}$: there
exists $(\tilde{\xi},\tilde{\eta})\in A$ such that
$\Pi(\tilde{\xi},\tilde{\eta})=\Pi(\xi,\eta)$. We denote by
$(\tilde{X}_{n})_{n\in\mathbb{Z}}$ and $(\tilde{Y}_{m})_{m\in\mathbb{Z}}$ the
same walks as before for $(\tilde{\xi},\tilde{\eta})$. Then for all $n\geq K$
and $0\leq m\leq\frac{2q-1}{2}n$, we have

$\displaystyle H_{(\xi,\eta)}\left(n+\frac{1}{2},m+\frac{1}{2}\right)$
$\displaystyle=\left\lceil\frac{X_{n}+Y_{m}}{2}\right\rceil$
$\displaystyle=\left\lceil\frac{K+\sum\limits_{k=K+1}^{n}\xi(k)+\min(m,K)+\sum\limits_{k=\min(m,K)+1}^{m}\eta(k)}{2}\right\rceil$
$\displaystyle=\left\lceil\frac{K+\sum\limits_{k=K+1}^{n}\tilde{\xi}(k)+\min(m,K)+\sum\limits_{k=\min(m,K)+1}^{m}\tilde{\eta}(k)}{2}\right\rceil$
$\displaystyle\geq\left\lceil\frac{\sum\limits_{k=1}^{n}\tilde{\xi}(k)+\sum\limits_{k=1}^{m}\tilde{\eta}(k)}{2}\right\rceil$
$\displaystyle\geq\left\lceil\frac{\tilde{X}_{n}+\tilde{Y}_{m}}{2}\right\rceil=H_{(\tilde{\xi},\tilde{\eta})}\left(n+\frac{1}{2},m+\frac{1}{2}\right).$

In the same manner we have, for all $n,m\leq-K$,

$H_{(\xi,\eta)}\left(n+\frac{1}{2},m+\frac{1}{2}\right)\leq
H_{(\tilde{\xi},\tilde{\eta})}\left(n+\frac{1}{2},m+\frac{1}{2}\right),$

therefore $A_{St}\subset A$. We recall that
$B_{K}=\llbracket-K,K\rrbracket^{2}$. On $A_{St}$, the components crossing
$B_{K}$ are stairs in $B_{K}$, and all the heights of the faces in $B_{K}$ are
known. The path containing the origin has level $-1$, so if it is finite, then
the points $(-K,K)$ and $(K,-K)$ must be connected in $\mathbb{Z}^{2}\setminus
B_{K}$. But it is not possible, because such a path would go through a zone
where all the heights are positive or all the heights are negative, see Figure
5.

$0$$0$$0$$0$$0$$0$$0$$1$$1$$1$$1$$1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-1$$-2$$-2$$-2$$-2$$-2$$-3$$-3$$-3$$2$$2$$2$$-4$$3$$y=\frac{-(2q-1)}{2}x$$y=\frac{2q-1}{2}x$$y=\frac{2q-1}{2}x$$y=\frac{-(2q-1)}{2}x$$C_{1}$$C_{-1}$
Figure 5: Example of the construction of the event $A_{St}$ for $q=0.7$ and
$K=4$. The path containing the origin has level $-1$. If a point $(n,m)$ is in
$C_{-1}$ or $C_{1}$, then we have $H(n+\frac{1}{2},m+\frac{1}{2})>1$ or
$H(n+\frac{1}{2},m+\frac{1}{2})<-1$. The only face with height $-1$ on the
$x$-axis of $\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2})$ is
$(-\frac{3}{2},\frac{1}{2})$. Therefore, the two blue points cannot be in the
same finite cycle.

Therefore, on the event $A_{St}$, which has positive probability, the
connected component of the origin is infinite.

Step 2. Now we use Theorem 2.3 to construct an infinity of infinite paths. Let
$(\alpha_{1},\alpha_{2})\in\mathbb{N}\times\mathbb{N}^{*}$ be such that
$0<\frac{\alpha_{1}}{\alpha_{2}}<\frac{2q-1}{2}$. By Theorem 2.3, we have

$\lim\limits_{n\to+\infty}\frac{1}{n}\sum\limits_{i=0}^{n-1}\mathds{1}_{A_{St}}\circ\tau_{(\alpha_{2},\alpha_{1})}^{{i}}=\mathbb{P}(A_{St})>0~{}a.s.,$

which means that almost surely, there exists a sequence of integers
$(N_{k})_{k\in\mathbb{N}}$ such that for all $k\in\mathbb{N}$,
$\tau^{-{N_{k}}}(A_{St})$ is realised and $N_{k+1}-N_{k}>K$. The components of
the points $((N_{k},N_{k}))_{k\in\mathbb{N}}$ are infinite, and the last
condition ensures that they are disjoint two by two, since they all have
different levels. Therefore, we just proved that there almost surely exists an
infinite number of infinite connected components. ∎

In the next section, we prove the last part of Theorem 1.1, stating that all
infinite paths have a same asymptotic slope. The proof will require one more
time the height function.

##  4 Direction of the infinite paths

Figure 6: Simulations on the left (resp. right) have parameters $p=q=0.9$
(resp. $q=0.6$ and $p=0.5$). Faces are colored acccording to their heights,
with colors that repeat periodically.

Remember that the connected components of the graph are either finite circuits
or bi-infinite simple paths. We encode the connected component of the origin
by a bi-infinite, possibly periodic, path
$\Gamma=(\Gamma_{n})_{n\in\mathbb{Z}}$ of vertices. We choose its orientation
by taking $\Gamma_{1}$ such that when we walk along the path in the positive
direction, the face with the highest height is on the right side of the path.
Note that it amounts to choosing $\Gamma_{1}$ such that
$\\{\Gamma_{0},\Gamma_{1}\\}$ is a horizontal edge. The origin of the path is
$\Gamma_{0}=(0,0)$. If the connected component of the origin is infinite, then
$\Gamma$ is simple, while if the connected component of the origin is a
circuit, then $\Gamma$ is periodic.

Let us denote by $D_{R}$ the disk of radius $R\in\mathbb{R}^{*}_{+}$, and by
$\mathbb{Z}[i]$ the set $\\{a+ib~{}:~{}a,b\in\mathbb{Z}\\}$. We set
$z_{p,q}=2p-1+i(2q-1)$. We denote by $\theta_{p,q}$ the argument of $z_{p,q}$
in $\left[0,2\pi\right[$. Then we define, for $\varepsilon>0$, the following
discrete cone

$C_{p,q}^{\varepsilon}=\\{z\in\mathbb{Z}[i]~{}:~{}\langle\frac{z}{|z|},e^{i\theta_{p,q}}\rangle\leq\varepsilon\\}.$

Note that $\tan\left(\theta_{p,q}+\frac{\pi}{2}\right)=\frac{2q-1}{1-2p}$, if
$p\neq\frac{1}{2}$. Therefore, $C_{p,q}^{\varepsilon}$ is centered around the
line of equation $y=\frac{2q-1}{1-2p}x$ if $p\neq\frac{1}{2}$, and around the
$y$-axis if $p=\frac{1}{2}$.

######  Theorem 4.1.

Let $(p,q)\neq(\frac{1}{2},\frac{1}{2})$. Then we have, for all
$\varepsilon>0$,

$\mathbb{P}(\exists R\in\mathbb{R}^{*}_{+}~{}:~{}\Gamma\subset D_{R}\cup
C_{p,q}^{\varepsilon})=1.$

This implies that if $\Gamma$ is simple, then $\Gamma$ has an asymptotic
slope, which is $\frac{2q-1}{1-2p}$. By invariance by even translation, the
same result holds for all bi-infinite paths.

$y$$x$$C_{p,q}^{\varepsilon}$$C_{p,q}^{\varepsilon}$$D_{R}$ Figure 7:
Representation of the set described in Theorem 4.1. The path $\Gamma$ is
included in the colored area for some value of $R\in\mathbb{R}_{+}^{*}$.

###### Proof.

Note that $\Gamma$ has level $0$ or $-1$. This implies that the faces adjacent
to $\Gamma$ have height $0,1$ or $-1$. Using the asymptotic behaviour of the
random walks, we will see that almost surely, these faces are all in a
restricted part of the grid.

The key step to prove the theorem is to show that, for all $\varepsilon>0$ and
$0<\varepsilon_{0}<\frac{|z_{p,q}|\varepsilon}{2}$, almost surely there exists
$R_{0}\in\mathbb{R}^{*}_{+}$ such that, for all $\theta\in[0,2\pi[$ and $r\geq
R_{0}$ satisfying $(r\cos(\theta),r\sin(\theta))\in\mathbb{Z}^{2}$,

$\displaystyle\left|\frac{X_{r\cos(\theta)}+Y_{r\sin(\theta)}}{r}-\langle
z_{p,q},e^{i\theta}\rangle_{\mathbb{R}^{2}}\right|\leq\varepsilon_{0}.$ (1)

Let us start by showing that $(\ref{eq1})$ implies the result. We set
$\varepsilon>0$ and $R=\max\left(R_{0},\frac{4}{|z_{p,q}|\varepsilon}\right)$.
Let $(n,m)\in\mathbb{Z}^{2}\setminus D_{R}$ be such that $(n,m)\notin
C_{p,q}^{\varepsilon}$. There exist $r\in\mathbb{R}^{*}_{+}$ and
$\theta\in[0,2\pi[$ such that $(n,m)=(r\cos(\theta),r\sin(\theta))$, with
$r>R$ and $\left|\langle
e^{i\theta_{p,q}},e^{i\theta}\rangle_{\mathbb{R}^{2}}\right|>\varepsilon$.
Then we have $\left|\langle
z_{p,q},e^{i\theta}\rangle_{\mathbb{R}^{2}}\right|>|z_{p,q}|\varepsilon$, and

$\displaystyle\left|H\left(n+\frac{1}{2},m+\frac{1}{2}\right)\right|$
$\displaystyle\geq\frac{\left|X_{r\cos(\theta)}+Y_{r\sin(\theta)}\right|}{2}$
$\displaystyle\geq\frac{r}{2}\left|\;\left|\frac{\left|X_{r\cos(\theta)}+Y_{r\sin(\theta)}\right|}{r}-\langle
z_{p,q},e^{i\theta}\rangle_{\mathbb{R}^{2}}\right|-\left|\langle
z_{p,q},e^{i\theta}\rangle_{\mathbb{R}^{2}}\right|\;\right|$
$\displaystyle\geq\frac{r}{2}\left|\;\left|\langle
z_{p,q},e^{i\theta}\rangle_{\mathbb{R}^{2}}\right|-\varepsilon_{0}\right|$
$\displaystyle\geq\frac{r}{2}\left(|z_{p,q}|\varepsilon-\varepsilon_{0}\right)$
$\displaystyle\geq\frac{r|z_{p,q}|\varepsilon}{4}>1,$

the fourth and fifth inequalities coming from the fact that
$\frac{|z_{p,q}|\varepsilon}{2}>\varepsilon_{0}$. Since the level of $\Gamma$
is $0$ or $-1$, then we deduce that $(n,m)\notin\Gamma$. Therefore,
$\Gamma\subset D_{R}\cup C_{p,q}^{\varepsilon}$.

Now, let us prove (1). By the law of large numbers, there exists
$N\in\mathbb{N}^{*}$ such that for all $n\geq N$,

$\displaystyle\max\left(\left|\frac{X_{n}}{n}-(2p-1)\right|,\left|\frac{X_{-n}}{n}-(1-2p)\right|,\left|\frac{Y_{n}}{n}-(2q-1)\right|,\left|\frac{Y_{-n}}{n}-(1-2q)\right|\right)\leq\frac{\varepsilon_{0}}{2}.$
(2)

We set $R_{0}=\frac{4(N+1)}{\varepsilon_{0}}$. For $\theta\in[0,2\pi[$ and
$r\geq R_{0}$ such that $(r\cos(\theta),r\sin(\theta))\in\mathbb{Z}^{2}$, we
have

$\displaystyle\left|\frac{X_{r\cos(\theta)}+Y_{r\sin(\theta)}}{r}-\left[(2p-1)\cos(\theta)+(2q-1)\sin(\theta)\right]\right|$
$\displaystyle\leq\left|\frac{X_{r\cos(\theta)}}{r}-(2p-1)\cos(\theta)\right|+\left|\frac{Y_{r\sin(\theta)}}{r}-(2q-1)\sin(\theta)\right|.$
(3)

We now bound the first term from above. We have two cases:

Case 1: $|\cos(\theta)|\leq\frac{\varepsilon_{0}}{4}$.
In this case, we bound from above $X_{r\cos(\theta)}$ with the triangular
inequality:

$\displaystyle\left|\frac{X_{r\cos(\theta)}}{r}\right|$
$\displaystyle\leq|\cos(\theta)|.$

Since $|(2p-1)\cos(\theta)|\leq|\cos(\theta)|$, we have

$\displaystyle\left|\frac{X_{r\cos(\theta)}}{r}-(2p-1)\cos(\theta)\right|$
$\displaystyle\leq 2\cos(\theta)\leq\frac{\varepsilon_{0}}{2}.$

Case 2: $|\cos(\theta)|>\frac{\varepsilon_{0}}{4}$.
This time we have $r|\cos(\theta)|\geq N$, so by (2), we have

$\left|\frac{X_{r\cos(\theta)}}{r\cos(\theta)}-(2p-1)\right|\leq\frac{\varepsilon_{0}}{2}.$

Then we have

$\displaystyle\left|\frac{X_{r\cos(\theta)}}{r}-(2p-1)\cos(\theta)\right|$
$\displaystyle\leq\left|\frac{X_{r\cos(\theta)}}{r\cos(\theta)}\cos(\theta)-(2p-1)\cos(\theta)\right|$
$\displaystyle\leq|\cos(\theta)|\left|\frac{X_{r\cos(\theta)}}{r\cos(\theta)}-(2p-1)\right|\leq\frac{\varepsilon_{0}}{2}.$

Working in the same manner for the second term in (3), we obtain (1). ∎

######  Remark 4.2.

Remember that $\Gamma$ is a bi-infinite path. We define the _forward path_
$\Gamma^{+}$ (resp. the _backward path_ $\Gamma^{-}$) by

$\Gamma^{+}=(\Gamma_{n})_{n\in\mathbb{N}}\qquad\text{(resp.
}\Gamma^{-}=(\Gamma_{-n})_{n\in\mathbb{N}}).$

Note that Theorem 4.1 implies that, conditionally on the event
$\\{\Gamma\text{ is simple}\\}$, we have

$\lim\limits_{n\to+\infty}\frac{\Gamma^{+}_{n}}{|\Gamma^{+}_{n}|}\in\\{e^{i\tilde{\theta}_{p,q}},e^{-i\tilde{\theta}_{p,q}}\\}\text{\quad
and\quad}\lim\limits_{n\to+\infty}\frac{\Gamma^{-}_{n}}{|\Gamma^{-}_{n}|}\in\\{e^{i\tilde{\theta}_{p,q}},e^{-i\tilde{\theta}_{p,q}}\\},$

where $\tilde{\theta}_{p,q}=\theta_{p,q}+\frac{\pi}{2}$ (modulo $2\pi$).

In the proof of Theorem 3.1, we proved that with positive probability, we have

$\lim\limits_{n\to+\infty}\frac{\Gamma^{+}_{n}}{|\Gamma^{+}_{n}|}\neq\lim\limits_{n\to+\infty}\frac{\Gamma^{-}_{n}}{|\Gamma^{-}_{n}|}.$

Indeed, on the event we constructed (see Figure 5), vertices of $\Gamma^{+}$
(resp. $\Gamma^{-}$) are restricted to a certain area after a sufficient
number of steps.

Using Theorem $2$ of Burton and Keane [1], we can prove that in fact, forward
and backward paths always have the same asymptotic direction. In other words,
we have almost surely
$\lim\limits_{n\to+\infty}\frac{\Gamma^{+}_{n}}{|\Gamma^{+}_{n}|}\neq\lim\limits_{n\to+\infty}\frac{\Gamma^{-}_{n}}{|\Gamma^{-}_{n}|}$.
Indeed, Theorem $2$ of [1] asserts that an infinite connected component $C$
has exactly one neighbor on each connected component of
$\mathbb{Z}^{2}\setminus C$, ignoring finite cycles. If we assume by
contradiction that forward and backward paths may go to the same direction,
then we deduce that an event of the type of the one of Figure 8 has positive
probability, which leads to a contradiction.

Figure 8: Example of a configuration containing a bi-infinite path whose
forward and backward paths go to the same direction (the blue one). This path
has two neighbors on the same side (the red one and the green one).

##  5 Questions and discussions

###  5.1 Level of infinite paths

On Figure 6, it seems that two distinct infinite connected components have
different levels, and that each integer is the level of one infinite connected
component. This leads to the following conjecture:

######  Conjecture 5.1.

The function which maps each infinite connected component to its level is
bijective. Moreover, two infinite connected components are neighbors if and
only if the difference of their level is $-1$ or $1$.

###  5.2 Ergodicity of the processes of the signs

Remember that horizontal edges and vertical edges alternate along $\Gamma$,
and the edges $E_{2n}=\\{\Gamma_{2n},\Gamma_{2n+1}\\}$ (resp.
$E_{2n+1}=\\{\Gamma_{2n+1},\Gamma_{2n+2}\\}$), $n\in\mathbb{Z}$, are
horizontal (resp. vertical). We define the sign of a horizontal (resp.
vertical) edge by setting

$\Gamma_{2n+1}-\Gamma_{2n}=-\mathrm{sgn}(E_{2n})e_{1}\qquad\text{ (resp.
}\Gamma_{2n+2}-\Gamma_{2n+1}=\mathrm{sgn}(E_{2n+1})e_{2}).$

With the help of Theorem 2.3, we can prove the following:

######  Proposition 5.2.

Let $p,q\in]0,1[$. The process $(\mathrm{sgn}(E_{2n}))_{n\in\mathbb{Z}}$
(resp. $(\mathrm{sgn}(E_{2n+1}))_{n\in\mathbb{Z}}$) is _stationary_ , that is:

$\mathbb{P}(\mathrm{sgn}(E_{t_{1}})=x_{1},...,\mathrm{sgn}(E_{t_{n}})=x_{n})=\mathbb{P}(\mathrm{sgn}(E_{t_{1}+k})=x_{1},...,\mathrm{sgn}(E_{t_{n}+k})=x_{n})$

for all $n\in\mathbb{N}^{*}$, $x_{1},x_{2},...,x_{n}\in\\{-1,1\\}^{n}$, $k\in
2\mathbb{Z}$ and $t_{1}<t_{2}<...<t_{n}$ in $2\mathbb{Z}$ (resp. in
$2\mathbb{Z}+1$). In particular, we have

$\forall
n\in\mathbb{Z},\quad\mathbb{E}(\mathrm{sgn}(E_{2n}))=2p-1\quad\text{and}\quad\mathbb{E}(\mathrm{sgn}(E_{2n+1}))=2q-1.$

If $p\neq\frac{1}{2}$ and $q\neq\frac{1}{2}$, bi-infinite simple paths have an
asymptotic direction by Theorem 4.1. This implies that there exists a random
integer $N$ such that $(\Gamma_{n})_{n\geq N}$ does not visit any line or
column visited by $(\Gamma_{n})_{0\leq n\leq k}$. Lines and columns being
independent, the dynamical system seems to have the mixing property, which
would imply its ergodicity, but we did not manage to prove it.

######  Conjecture 5.3.

Assume that $p\neq\frac{1}{2}$ and $q\neq\frac{1}{2}$. Let us denote by $X$ a
process among $(\mathrm{sgn}(E_{2n}))_{n\in\mathbb{Z}}$ and
$(\mathrm{sgn}(E_{2n+1}))_{n\in\mathbb{Z}}$. Conditionally on the event
“$\Gamma$ is simple”, the process $X$ is _ergodic_ , that is:
$(\\{-1,1\\}^{\mathbb{N}},\mathbb{Q}_{X},\theta)$ is ergodic, where $\theta$
is the shift map, $\mathbb{Q}=\mathbb{P}(.|\Gamma\text{ is simple)}$ and
$\mathbb{Q}_{X}$ is the law of $X$ under $\mathbb{Q}$.

We denote by $A$ the event “$\Gamma\text{ is simple}$”. This conjecture would
imply that

$\lim\limits_{n\to+\infty}\frac{1}{n}\sum\limits_{i=0}^{n-1}\mathrm{sgn}(E_{2i})=\frac{2p-1}{\mathbb{P}(A)}\mathds{1}_{A}\text{\quad
and\quad}\lim\limits_{n\to+\infty}\frac{1}{n}\sum\limits_{i=0}^{n-1}\mathrm{sgn}(E_{2i+1})=\frac{2q-1}{\mathbb{P}(A)}\mathds{1}_{A},$

and the same result for the backward path. We would also obtain the following
improvement of Remark 4.2:

$\mathbb{P}_{A}\left(\lim\limits_{n\to+\infty}\frac{\Gamma_{n}^{+}}{|\Gamma_{n}^{+}|}=e^{i\tilde{\theta}_{p,q}}\text{\quad
and\quad}\lim\limits_{n\to+\infty}\frac{\Gamma_{n}^{-}}{|\Gamma_{n}^{-}|}=e^{-i\tilde{\theta}_{p,q}}\right)=1.$

###  5.3 Distribution of the sequences $\xi$ and $\eta$

We assumed the sequences $\xi$ and $\eta$ to be i.i.d. But in fact, we only
used the three following properties:

  * •

the corner percolation is ergodic for the translations of Proposition 2.2;

  * •

one of the limits of $(\frac{X_{n}}{n})_{n\in\mathbb{N}}$ and
$(\frac{Y_{n}}{n})_{n\in\mathbb{N}}$ is non-zero;

  * •

$\mu^{p,q}$ has a finite energy property.

So we could take more general distributions for $\xi$ and $\eta$, as long as
these properties are still satisfied. In particular, all the results could be
extended to two independent ergodic stationary Markov chains on $\\{-1,1\\}$.

## References

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