The height problem in first passage percolation

TL;DR

This paper addresses the height problem in first passage percolation on Z2, providing a solution under certain distribution conditions, advancing understanding of the model's geometric properties.

Contribution

It offers a novel solution to the height problem in first passage percolation when the distribution F satisfies 0 < F(0) < pc.

Findings

Solved the height problem in first passage percolation on Z2.

Established results for distributions with 0 < F(0) < pc.

Enhanced understanding of geometric aspects of the model.

Abstract

We consider the first passage percolation model in Z2 with a distribution F for 0 < F (0) < pc. In this paper, we solve the height problem.

Equations

$$T(\gamma )=\sum _{e_i\in \gamma }t(e_i).$$

$$T(A,B)=\inf\{T(\gamma):\gamma\mbox{ is a path from $A$ to $B$}\},$$

$$d(A,B)=\min \{d(\mathbf{x},\mathbf{y}):\mathbf{x}\in A\text{mbox}and\mathbf{y}\in B\}$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}T(0,n\mathbf{x})=\underset{n}{\inf }\frac{1}{n}ET(0,n\mathbf{x})=\underset{n\to \infty }{\lim }\frac{1}{n}ET(0,n\mathbf{x})={\mu }_F(\mathbf{x})\text{mbox}a.s.andin{L}_1.$$

$$\mu_{F}({\bf x})\mbox{ is continuous in }{\bf x}\mbox{ and }\mu_{F}({\bf x})>0\mbox{ for ${\bf x}\neq 0$ iff }F(0)<1/2.$$

$$a_{0,n}=T(0,n\mathbf{x})=T(0,(n,0)).$$

$$\underset{n\to \infty }{\lim }{a}_{0,n}/n={\mu }_F((1,0))\text{mbox}a.s.andin{L}_1.$$

$${\mu }_F(1,0)={\mu }_F.$$

$$b_{0,n}=\inf\{T(\gamma):\gamma\mbox{ is a path from the origin to ${\bf L}[n]$}\}.$$

$$\underset{n\to \infty }{\lim }{b}_{0,n}/n={\mu }_F\text{mbox}a.s.andin{L}_1.$$

$${\int }_0^{\infty }{e}^{\iota x}dF(x)<\infty \text{mbox}forsome\iota >0.$$

$$\mathbf{B}:=\mathbf{B}(F)=\{\mathbf{u}:{\mu }_F(\mathbf{u})\le 1\}$$

$$\partial \mathbf{B}:=\{\mathbf{u}\in {\mathbf{R}}^2:\mu (\mathbf{u})=1\}.$$

$$\mathbf{B}(t):=\{\mathbf{u}\in {\mathbf{Z}}^d,T(0,\mathbf{u})\le t\}.$$

$$t(1-ϵ)\mathbf{B}\subset \mathbf{B}(t)\subset t(1+ϵ)\mathbf{B}\text{mbox}eventuallyw.p.1.$$

$$\displaystyle{\bf B}\mbox{ lies between the cube $[-1/\mu_{F},1/\mu_{F}]^{2}$ and the diamond with the four corners }$$

$$\displaystyle\mbox{$(\pm 1/\mu_{F},0)$ and $(0,\pm 1/\mu_{F})$}.\hskip 327.20668pt{(1.7)}$$

$$P(t(1-ϵ)\mathbf{B}\subset \mathbf{B}(t)\subset t(1+ϵ)\mathbf{B})\ge 1-\exp (-O(ϵt)).$$

$$P(|b_{0,n}-n{\mu }_F|\ge ϵn)\le \exp (-O(ϵn)).$$

$$P(|T(0,\mathbf{x})-ET(0,\mathbf{x})|\ge z\Vert \mathbf{x}{\Vert }^{1/2})\le C_1\exp (-C_2{z}^2).$$

$$P(\exists\,\mbox{ an optimal path $\gamma_{n}({\bf x})$ with }|\gamma_{n}({\bf x})|\geq\lambda n)\leq C_{1}\exp(-C_{2}n).$$

$$h_{n}^{\alpha}=\max\{d({\bf u},\mbox{the $X$-axis}):{\bf u}\in\alpha_{n}\}\mbox{ for }\alpha=A\mbox{ or }\alpha=B.$$

$$\underset{n\to \infty }{\lim }\frac{h_n^{\alpha }}{n}=0\text{mbox}insomesensefor\alpha =A\text{mbox}or\alpha =B\text{mbox}ifF(0)<1/2.$$

$$\lim_{n\rightarrow\infty}{h_{n}^{\alpha}\over n}=0\mbox{ in probability and in $L_{1}$ for }\alpha=A\mbox{ or }\alpha=B.$$

$$\xi (d)=\inf \left\{\xi :\underset{n}{\mathrm{limsup}}\frac{E{h}_n^A}{n^{\xi }}<\infty \right\}.$$

$$\underset{n\to \infty }{\lim }\frac{h_n^A}{n^{1-\delta }}=0\text{mbox}inprobability$$

$$\displaystyle s_{0,n}=\inf\{T(\gamma):\gamma\mbox{ is a path from the origin to ${\bf L}[n]$ with vertices in ${\bf L}(0,n)$}$$

$$\text{mbox}exceptitsinitialandterminatevertices\},$$

$$\underset{n\to \infty }{\lim }{s}_{0,n}/n={\mu }_F\text{mbox}a.s.andin{L}_1.$$

$$P(|s_{0,n}-n{\mu }_F|\ge ϵn)\le \exp (-O(ϵn)).$$

$$\displaystyle s_{a,n}((a,b))=\inf\{T(\gamma):\gamma\mbox{ is a path from $(a,b)$ to ${\bf L}[n]$ with vertices in ${\bf L}(a,n)$}$$

Full text

1 Introduction of the model and results.

We consider the first passage percolation model on the $({\bf Z}^{2},{\bf E}^{2})$ lattice, a graph, with the vertices in ${\bf Z}^{2}$, and with the edges in ${\bf E}^{2}$ connecting each pair of vertices one unit apart. We assign independently to each edge a non-negative passage time $t(e)$ with a distribution $F$ in $[0,\infty)$. More formally, we take $\Omega=[0,\infty)^{{\bf E}^{2}}$ as a sample space, whose points are called configurations. Let $P$ be the corresponding product measure on $\Omega$. The expectation and variance with respect to $P$ are denoted by $E(\cdot)$ and $\sigma^{2}(\cdot)$. For any two vertices ${\bf u}$ and ${\bf v}$, a path $\gamma$ from ${\bf u}$ to ${\bf v}$ is an alternating sequence $({\bf v}_{0},e_{1},{\bf v}_{1},...,{\bf v}_{i-1},e_{i},{\bf v}_{i},...,{\bf v}_{n-1},e_{n},{\bf v}_{n})$ of vertices ${\bf v}_{i}$ in ${\bf Z}^{2}$ and edges $e_{i}$ between ${\bf v}_{i-1}$ and ${\bf v}_{i}$ in ${\bf E}^{2}$ with ${\bf v}_{0}={\bf u}$ and ${\bf v}_{n}={\bf v}$. A path is called disjoint if ${\bf v}_{i}\not={\bf v}_{j}$ for $i\neq j$. In this paper, we always consider a path to be disjoint. Given such a path $\gamma$, we define its passage time as

[TABLE]

For any two sets $A$ and $B$, we define the passage time from $A$ to $B$ as

[TABLE]

where the infimum is taken over all possible finite paths from some vertex in $A$ to some vertex in $B$. A path $\gamma$ from $A$ to $B$ with $T(\gamma)=T(A,B)$ is called an optimal path of $T(A,B)$.

The existence of such an optimal path has been proven (see Kesten (1986)). If $t(e)=0$, the edge is called a zero edge or an open edge; otherwise, it is called a closed edge. We also want to point out that the optimal path may not be unique. If all edges in a path are in passage time zero, the path is called a zero path or an open path. If we focus on a special configuration $\omega$, we may write $T(A,B)(\omega)$ instead of $T(A,B)$. When $A=\{\bf u\}$ and $B=\{\bf v\}$ are single vertex sets, $T({\bf u},{\bf v})$ is the passage time from ${\bf u}$ to ${\bf v}$. We may extend the passage time over ${\bf R}^{2}$. More precisely, if ${\bf u}$ and ${\bf v}$ are in ${\bf R}^{2}$, we define $T({\bf u},{\bf v})=T({\bf u}^{\prime},{\bf v}^{\prime})$, where ${\bf u}^{\prime}$ (resp., ${\bf v}^{\prime}$) is the nearest neighbor of ${\bf u}$ (resp., ${\bf v}$) in ${\bf Z}^{2}$. Possible indetermination can be eliminated by choosing an order on the vertices of ${\bf Z}^{2}$ and taking the smallest nearest neighbor for this order. In this paper, for any ${\bf x},{\bf y}\in{\bf R}^{2}$, $\|{\bf x}\|$ is denoted by the Euclidean norm and $d({\bf x},{\bf y})=\|{\bf x}-{\bf y}\|$ is the distance between ${\bf x}$ and ${\bf y}$. For any two sets ${A}$ and ${B}$ of ${\bf R}^{2}$,

[TABLE]

and is denoted as the distance between ${A}$ and ${B}$. For a vertex set $A$, we denote by $|A|$ the number of vertices in $A$. Similarly, if $A$ is an edge set, we also denote by $|A|$ the number of edges in $A$.

Given ${\bf 0}=(0,0)$ and ${\bf x}=(x_{1},x_{2})\in{\bf R}^{2}$, if $Et(e)<\infty$, by Kingman’s sub-additive ergodic theorem, it is well known that

[TABLE]

It is also known (see Kesten (1986)) that

[TABLE]

In particular, Hammersley and Welsh (1965), in their pioneering paper, investigated a special case ${\bf x}=(1,0)$:

[TABLE]

They showed that

[TABLE]

For simplicity’s sake, we denote

[TABLE]

Now let ${\bf L}[n]$ be the vertices in the vertical line passing through $(n,0)$. We look again at Hammersley and Welsh (1965) and their introduction of point-line passage time. Let

[TABLE]

They proved in their paper

[TABLE]

Since we need the rate of convergence in (1.1), we assume in this paper that $t(e)$ is not a constant and satisfies the following:

[TABLE]

When $F(0)<1/2$, the map ${\bf x}\rightarrow\mu_{F}({\bf x})$ induces a norm on ${\bf R}^{2}$. The unit radius ball for this norm is denoted by

[TABLE]

and is called the asymptotic shape. The boundary of ${\bf B}$ is

[TABLE]

By (1.2), if $F(0)<1/2$, ${\bf B}$ is a compact convex deterministic set. Define for all $t>0$ a random shape

[TABLE]

The shape theorem (see Cox and Durrett (1981)) is a well-known result stating that for any $\epsilon>0$,

[TABLE]

It is easy to show (see Fig. 6.1 in Kesten (1986)) that

[TABLE]

By using a rate estimate (see Chow and Zhang (2003)), if $F(0)<p_{c}$ and (1.5) holds, for any $\epsilon>0$, then

[TABLE]

By Zhang’s (2010) Theorem 3, if $F(0)<1/2$ and (1.5) holds, then

[TABLE]

Kesten (1996) also showed the following concentration inequality. If $F(0)<1/2$ and (1.5) holds, then for any $z>0$, there exist $C_{i}=C_{i}(F)$ for $i=1,2$ independent of ${\bf x}$ such that

[TABLE]

In this paper, $C_{i}$ denotes a constant with $0<C_{i}<\infty$ whose precise value is of no importance; its value may change from appearance to appearance, but $C_{i}$ will always be independent of $n$ and $t$, $k$, and $m$, although it may depend on $F$. For simplicity’s sake, we sometimes use $O(n)$ for $C_{1}n\leq O(n)\leq C_{2}n$ if we do not need the precise value of $C_{i}$. Regarding the length of optimal paths, if $F(0)<1/2$, it is well known (see Prop. (5.8) in Kesten (1986)) that there exists $\lambda=\lambda(F)$ and $C_{i}=C_{i}(F)$ for $i=1,2$ such that

[TABLE]

We know that optimal paths of $a_{0,n}$ exist, but they are not unique. When $F(0)<1/2$, each optimal path of $a_{0,n}$ is finite. Similarly, optimal paths of $b_{0,n}$ exist, but they also are not unique. When $F(0)<1/2$, each optimal path of $b_{0,n}$ is finite. We denote by $A_{n}$ and $B_{n}$ the unions of all optimal paths with passage times $a_{0,n}$ and $b_{0,n}$, respectively. Moreover, $\bar{A}_{n}$ and $\bar{B}_{n}$ are the set of all the optimal paths with passage times $a_{0,n}$ and $b_{0,n}$, respectively. $|\bar{A}_{n}|$ and $|\bar{B}_{n}|$ are the numbers of paths in $\bar{A}_{n}$ and $\bar{B}_{n}$, respectively. We denote the *height * of the optimal paths of $a_{0,n}$ or $b_{0,n}$ by

[TABLE]

It is widely believed (see Hammersley and Welsh (1965); Smythe and Wierman (1978); and Kesten (1986)) that

[TABLE]

The conjecture of (1.12) is called the height problem. Note that (1.12) does not hold when $F(0)\geq 1/2$. In this paper, we answer this conjecture affirmatively.

Theorem. If $F$ is a distribution with $0<F(0)<1/2$ and (1.5) holds, then

[TABLE]

Remarks 1-4. 1. In this paper, we only show the theorem for $a_{0,n}$ since the geometry is much easier to handle. In other words, we solve the height conjecture in the horizontal direction. But the proof for the theorem might be carried over to solve the height conjecture in any direction.

2. A distribution F is said to be useful (see van den Berg and Kesten (1993)) if $F(0)<1/2$ and $F^{-}=0$ or $F(F^{-})<\vec{p_{c}}$, where $\vec{p}_{c}$ is the critical probability of two-dimensionally oriented percolation and $F^{-}$ is the infimum of the support of $F$. The condition that $0<F(0)<1/2$ in the theorem is a special case of useful distribution. The same proof of the theorem might show the theorem when $F$ is useful and (1.5) holds. We also want to point out that there is a counter example (see Durrett and Liggett (1981)) that the theorem does not hold in some direction when $F$ is not useful.

3. We may define the height fluctuation exponent to be

[TABLE]

It is believed that $\xi(2)=2/3$ if $F(0)<1/2$. We proved in the theorem that the ratio of the height goes to zero. But we are unable to show that

[TABLE]

for some $\delta>0$.

4. Our proof for the theorem only works for the two-dimensional lattice.

2 The number of pivotal edges for optimal paths.

Among most percolation problems, people have to focus on some specific edges, called pivotal edges, which optimal paths have to use. We first introduce the definition of pivotal edges. We denote by $L[i]$ the vertical line passing $(i,0)$. We also denote by $L[i,j]$ ($L(i,j)$) the space between $L[i]$ and $L[j]$, called a cylinder, including $L[i]$ and $L[j]$ (but not including $L[i]$ and $L[j]$) for $i<j$. Let ${\bf L}[i,j]$ be the vertex set of $L[i,j]$. When $i=j$, ${\bf L}[i]$ is the vertex set of $L[i]$ defined before. Hammersley and Welsh (1965) introduced the cylinder point-line passage time:

[TABLE]

and they proved

[TABLE]

By the same proof of Zhang’s (2010) Theorem 3, if $F(0)<p_{c}$ and (1.5) holds, then

[TABLE]

For convenience, we give a general definition of cylinder point-line passage time starting at any vertex as follows. For $a<n$,

[TABLE]

We need to discuss a few properties regarding optimal path $\gamma_{n}$ of $s_{0,n}$. It follows from Proposition (5.8) in Kesten (1986) that there exists $\lambda=\lambda(F)$ and $C_{i}=C_{i}(F)$ for $i=1,2$ such that

[TABLE]

Let $S_{n}$ be the union of all the optimal paths with passage time $s_{0,n}$. We also denote by $\bar{S}_{n}$ the set of all optimal paths with passage time $s_{0,n}$. Furthermore, let $D_{n}$ be the intersection of all the optimal paths in $\bar{S}_{n}$. For $e\in D_{n}$, each optimal path in $\bar{S}_{n}$ has to go through $e$, so the edges in $D_{n}$ are called pivotal edges. By using Theorem 2 in Zhang (2006), if $F$ is a Bernoulli distribution with $F(0)<1/2$, then there exists $M$ such that

[TABLE]

Later, Nakajima (2019) showed that (2.4) holds for a point-point passage time and for a useful distribution. Furthermore, he also showed that for a point-point passage time,

[TABLE]

Now we need to account for the number of pivotal edges in a slab. We first investigate the behavior of optimal paths for $s_{0,n}$ in a slab. We consider an optimal path $\gamma_{n}\in\bar{S}_{n}$. Let $0<m_{1}<m_{2}<n$ with $m_{1}=O(n)$, $m_{2}-m_{1}=\kappa n$ and $n-m_{2}=O(n)$ for a small, positive constant $\kappa$. $\gamma_{n}$ first meets ${\bf u}^{\prime}$ at $L[{m_{1}}]$ and last meets ${\bf u}$ at $L[{m_{1}}]$. Furthermore, $\gamma_{n}$ continues to first meet ${\bf v}$ at $L[m_{2}]$ and last meet ${\bf v}^{\prime}$ at $L[m_{2}]$. Finally, $\gamma_{n}$ meets $L[n]$ at ${\bf z}$. We denote by $\gamma({\bf 0},{\bf u^{\prime}})$ the sub-path from ${\bf 0}$ to ${\bf u}$. Similarly, we denote the sub-paths $\gamma({\bf u}^{\prime},{\bf u})$, $\gamma({\bf u},{\bf v})$, $\gamma({\bf v},{\bf v}^{\prime})$, as well as $\gamma({\bf v}^{\prime},{\bf z})$ the corresponding point-point sub-paths. We sometimes need to assume that ${\bf v}={\bf z}$. We now show the following lemma.

Lemma 2.1. If $F$ is a distribution with $F(0)<1/2$, $0<\kappa<1$ is a constant, and (1.5) holds, then for any large $n$, $0<m_{1}<m_{2}<n$ with $m_{1}=O(n)$, $m_{2}-m_{1}=\kappa n$, and $n-m_{2}=O(n)$, and for any $\gamma_{n}\in\bar{S}_{n}$ and for any $0<\delta<\mu_{F}/2$, there exist $C_{j}=C_{j}(F,\delta)$ for $j=1,2$ such that

[TABLE]

Proof. We denote the three sub-paths of $\gamma_{n}$ from the origin to ${\bf u}^{\prime}$, from ${\bf u}^{\prime}$ to ${\bf v}^{\prime}$, and from ${\bf v}^{\prime}$ to ${\bf z}$ by

[TABLE]

Note that these sub-paths are also optimal paths from ${\bf 0}$ to ${\bf u}^{\prime}$, from ${\bf u}^{\prime}$ to ${\bf v}^{\prime}$, and from ${\bf v}^{\prime}$ to ${\bf z}$, respectively. Thus, for any $0<\delta\leq\mu_{F}/2$, by (2.3) and (1.8), there exist $C_{l}=C_{l}(F(0))$ for $l=1,2,$

[TABLE]

where the sum takes all possible ${\bf w}\in{\bf L}[m_{1}]$ with $\|{\bf w}\|\leq 2\lambda n$, and $\gamma({\bf 0},{\bf w})$ is an optimal path from ${\bf 0}$ to ${\bf w}$. Thus, by (2.6),

[TABLE]

Similarly, we have

[TABLE]

On the other hand, for ${\bf u}$ and ${\bf v}$, by (2.2),

[TABLE]

Substituting (2.7) and (2.8) into (2.9), note that $m_{2}-m_{1}=\kappa n$, so

[TABLE]

Note that $d({\bf u},{\bf v})\geq n\kappa$, so by the same proof of (2.7),

[TABLE]

Lemma 2.1 follows from (2.10) and (2.11). $\Box$

Regarding the length of $\gamma({\bf u}^{\prime},{\bf v}^{\prime})$, by using Proposition (5.8) in Kesten (1986) and the inequality in Lemma 2.1, we have the following lemma.

Lemma 2.2. If $F$ is a distribution with $F(0)<1/2$, $0<\kappa<1$ is a constant, and (1.5) holds, then for any large $n$, $0<m_{1}<m_{2}<n$ with $m_{1}=O(n)$, $m_{2}-m_{1}=\kappa n$, and $n-m_{2}=O(n)$, there exist $\lambda=\lambda(F)$ and $C_{j}=C_{j}(F)$ for $j=1,2$ such that for all $m\geq n$,

[TABLE]

We want to remark that the same proofs of Lemmas 2.1 and 2.2 also work on the case when ${\bf v}={\bf z}$. For each optimal path $\gamma_{n}\in\bar{S}_{n}$, we can take $m_{1}=n-\kappa n$ and $m_{2}=n$ and the corresponding ${\bf u}^{\prime}$, ${\bf u}$, and ${\bf v}={\bf z}$ defined in Lemma 2.1 for a small $0<\kappa<1$. In this special case, Lemma 2.1 and Lemma 2.2 hold by the same proofs directly. Recall that $D_{n}$ is the number of pivotal edges of $S_{n}$. We denote by $D_{n}(\kappa)$ the number of the pivotal edges of $S_{n}$ in $L[n-m_{1},n]$. We select any optimal path $\gamma_{n}$ and consider the pivotal edges of $S_{n}$ in $\gamma({\bf u},{\bf v})$. We may select another optimal path $\bar{\gamma}_{n}$ and consider the pivotal edges of $S_{n}$ in $\bar{\gamma}(\bar{\bf u},\bar{\bf v})$. Since all optimal paths have to pass all the pivotal edges, the pivotal edges in $\gamma({\bf u},{\bf v})$ and the pivotal edges in $\bar{\gamma}(\bar{\bf u},\bar{\bf v})$ are the same. Thus, we may study the pivotal edges of $S_{n}$ in $L[n-m_{1},n]$ in $\gamma({\bf u},{\bf v})$ by using a particular optimal path $\gamma_{n}$. Now we use Lemma 2.1, the method of Lemma 8 in Nakajima (2019), and the Peierls argument to estimate the pivotal edges in the sub-optimal path $\gamma({\bf u},{\bf v})$ of $\gamma_{n}\in\bar{S}_{n}$.

Lemma 2.3. If $F$ is a distribution with $0<F(0)<p_{c}$ and (1.5) holds, then for any small $0<\kappa<1$, there exist constants $m$ and $M$ independent of $\kappa$ and $n$ such that

[TABLE]

Proof. In the proof of Lemma 2.3, we use the same notations $\kappa$, ${\bf u}$, and ${\bf v}={\bf z}$ as we did in Lemma 2.1. Let $\tau(e)$ be an independent copy of $t(e)$ for each edge $e$. Given an edge $f$, we set $\beta^{f}(e)$:

[TABLE]

For an edge $f\in{\bf L}[0,n]$, let $\bar{S}_{n}^{f}$ be the set of all optimal paths from the origin to ${\bf L}[n]$ in ${\bf L}(0,n)$ except their initial and terminate vertices with edge weights $\{\beta^{f}(e)\}$ for $e\in{\bf L}[0,n]$. In other words, $\bar{S}_{n}^{f}$ is the set of all optimal paths from the origin to ${\bf L}[n]$ with edge weights $t(e)$ for $e\neq f$ and edge weight $\tau(f)$ for $e=f$. Since $t(e)$ and $\beta^{f}(e)$ have the same distribution for each edge $e$, for a fixed edge $f\in{\bf L}[0,n]$,

[TABLE]

Thus, the event in the right side of (2.12) implies that $f$ is a pivotal edge of $\gamma_{n}$. For each $\gamma_{n}\in\bar{S}_{n}$, let

[TABLE]

By (2.12), if we denote by

[TABLE]

then

[TABLE]

Now we estimate $EN_{n}(\kappa)$. Note that for a small $\delta>0$,

[TABLE]

Note that the number of optimal paths in $\bar{S}_{n}$ is finite, so on $\{N_{n}(\gamma_{n},{\bf u},{\bf v})\leq\delta\kappa n\}$, there is a $\gamma_{n}\in\bar{S}_{n}$ such that

[TABLE]

If there are many optimal paths satisfying above inequality, we simply select one in a unique way. For ${\bf z}=(z_{1},z_{2})\in{\bf Z}^{2}$ and $M$ large independent of $n$, we denote $M$-squares by

[TABLE]

For the selected optimal path $\gamma_{n}\in\bar{S}_{n}$, if $B_{M}({\bf z})\cap\gamma({\bf u},{\bf v})\neq\emptyset$, then $B_{M}({\bf z})$ is called a good square. Given a good square, we consider its $3M$-squares, called 3$M$-good squares, each containing $B_{M}({\bf z})$ as its center square and eight $M$-neighbor squares of $B_{M}({\bf z})$. Since $\gamma({\bf u},{\bf v})$ crosses the annulus between an $M$-good square and the boundary of its $3M$-square,

[TABLE]

We say two squares are adjacent if they have a common vertex. Since $\gamma({\bf u},{\bf v})$ is a path, it is easy to verify that all $M$-good squares are adjacent. Let $\gamma({\bf u},{\bf v},M)$ be these $M$-good squares, and let $|\gamma({\bf u},{\bf v},M)|$ be the number of $M$-squares in $\gamma({\bf u},{\bf v},M)$. Note that there are nine many $M$-squares in each $3M$-square, so for each $M$-good square in $\gamma({\bf u},{\bf v},M)$, its good $3M$-square corresponds to $M$ disjoint vertices in $\gamma({\bf u},{\bf v})$. Thus,

[TABLE]

For the selected $\gamma_{n}\in\bar{S}_{n}$, there are many choices for different locations of $\gamma({\bf u},{\bf v})$. On $|\gamma_{n}|\leq\lambda n$, there are at most $(\lambda n)^{2}$ many choices for the locations for vertex ${\bf u}$. If ${\bf u}$ is fixed with ${\bf u}\in B_{M}({\bf z})$ for some ${\bf z}$, and if $|\gamma({\bf u},{\bf v},M)|=k$, by (4.24) in Grimmett (1999), we note that $\gamma({\bf u},{\bf v},M)$ is adjacent, so

[TABLE]

We say a good $M$-square is regular if its $3M$-square contains an edge $e\in\gamma({\bf u},{\bf v})$ with $t(e)>0$; otherwise, we say it is irregular. Thus, if an $M$-square is irregular, by (2.15), there is an open path in its $3M$-square with at least $M$ vertices. Since $F(0)<1/2$, by Theorem 5.4 in Grimmett (1999), there exist $C_{j}=C_{j}(F(0))$ for $j=1,2$ such that

[TABLE]

With the above observations, we use a standard Peierls argument (see Theorem 3 in Zhang (2006) or Theorem 2.13 in van den Berg and Kesten (1993)) to estimate the probability of the event for the number of closed edges in $\gamma({\bf u},{\bf v})$. By (2.13) and (2.15), if $|\gamma({\bf u},{\bf v},M)|=k\geq 9\kappa n/M$, then for $0<\eta$, there exist $C_{j}=C_{j}(F)$ for $j=3,4$,

[TABLE]

where $\Gamma_{M}(k)$ is a fixed, adjacent $M$-square set containing ${\bf u}$ and ${\bf v}$ with $k$ many $M$-squares. If $|\gamma({\bf u},{\bf v},M)|=k\geq 9\kappa n/M$, then we choose $0<\eta=\eta(M)\leq 9/(2M)$ small, independent of $n$ and $\kappa$, such that the number of irregular $M$-squares is larger than $k/2$. On $\{\gamma({\bf u},{\bf v},M)=\Gamma_{M}(k)\}$, note that there are nine many $M$-squares in each $3M$-square, so we select disjoint, adjacent $k/18$ many disjoint $3M$-squares such that each $3M$-square contains an irregular $M$-square selected above. Thus, by (2.18) and (2.19),

[TABLE]

By (2.19), note that $C_{1}$ and $C_{2}$ are independent of $M$, so by choosing $M$ large, there exist $C_{j}=C_{j}(F(0),\eta)$ for $j=3,4$ such that

[TABLE]

By (2.21), there exists $C_{5}$ independent of $\kappa$ and $n$ such that

[TABLE]

By (2.13) and (2.22),

[TABLE]

On the other hand, by Lemma 2.2,

[TABLE]

Lemma 2.3 follows from (2.23) and (2.24). $\Box$

Remark 5. Lemmas 2.1, 2.2, and 2.3 also hold for any $d$-dimensional lattice.

3 Proof of theorem.

In section 3, we show two lemmas for a point-line cylinder height. With these two lemmas, we show the theorem. These proofs are much easier to understand with graphs than words. We suggest that readers use the following three graph proofs for help in the formal proofs of the two lemmas and the theorem.

In their paper, Hammersley and Welsh (1965) introduced another height:

[TABLE]

They pointed out that the knowledge of the behavior of $H_{n}$ would give information about the rate of convergence of $a_{0.n}$. We show the following lemma.

Lemma 3.1. If $F$ is a distribution with $0<F(0)<1/2$ and (1.5) holds, then

[TABLE]

Proof. It is known (see Theorem 8.15 in Smythe and Wierman (1978)) that if $F(0)<1/2$, then

[TABLE]

We suppose that Lemma 3.1 does not hold. By symmetry and (3.1), there is a subsequence $\{n_{s}\}$ such that

[TABLE]

For a small $\epsilon$ with $\delta\geq 8\epsilon>0$, by (3.2), there exists $l_{s}$ with $1\geq l_{s}\geq\delta$ such that for each $s$,

[TABLE]

Since $\epsilon\delta/2$ is uniformly bounded away from zero, it is easy to find a subsequence $\{n_{s_{t}}\}$ from $\{n_{s}\}$ independent of configurations such that

[TABLE]

We set $\{i\}=\{s_{t}\}$ for simplicity. Let $I(0,0)$ be the indicator of the event: $\cap_{i}\{(l_{i}-\epsilon)\leq H_{n_{i}}/n_{i}\leq(l_{i}+\epsilon)\}$. For each $\omega\in\cap_{i}\{(l_{i}-\epsilon)\leq H_{i}/n_{i}\leq(l_{i}+\epsilon)\}$, let $\tau_{q}(\omega)$ be the new configuration by moving $\omega$ vertically up to $q$ units. Let $I(0,q)$ be the indicator of the event that

[TABLE]

We note that $\{I(0,q)\}$ is a stationary indicator sequence for $q=0,1,\cdots$. Since $\{t(e)\}$ is an independent and identically distributed sequence in ${\bf E}^{2}$, $\{I(0,q)\}$ is an ergodic stationary sequence. By Birkhoff’s ergodic theorem and (3.3),

[TABLE]

Thus, for any $\epsilon>0$ with $\delta>8\epsilon$, by (3.4), there exists $N$ such that

[TABLE]

For $\omega\in\{\exists\,j\mbox{ with }I(0,j)=1\mbox{ for }1\leq j\leq N\}$, there exists $j$, depending on $\omega$, such that there is an optimal path of $s_{0,n_{i}}(0,j)$ from $(0,j)$ to $\{n_{i}\}\times(l(1-\epsilon)n_{i}+j,l(1+\epsilon)n_{i}+j)$ for each $i$.

Let ${\cal E}_{n_{i}}=\{\delta n_{i}\leq H_{n_{i}}\leq n_{i}\}$. On ${\cal E}_{n_{i}}$, there is an optimal path $\gamma_{n_{i}}\in\bar{S}_{n_{i}}$ from the origin to ${\bf v}\in\{n_{i}\}\times(\delta n_{i},n_{i})$ (see Fig. 1, left). Now we choose $n_{i}$ large with $l_{i}n_{i}\geq\epsilon n_{i}\geq 4N$ for the $l_{i}$ defined in (3.3) and the $N$ defined in (3.5). Let ${\cal E}_{n_{i}}(l_{i}n_{i},N)$ be the event that there exists an optimal path $\gamma_{n_{i}}(l_{i}n_{i},j)$ from $(0,l_{i}n_{i}+j)$ to ${\bf o}$ for ${\bf o}\in\{n_{i}\}\times(-2n_{i}\epsilon,2n_{i}\epsilon)$ and for some $0\leq j\leq N$ (see Fig. 1, left) with passage time $s_{0,n_{i}}(0,l_{i}n_{i}+j)$. We want to remark that for each $\omega\in{\cal E}_{n_{i}}(l_{i}n_{i},N)$, $j$ depends on $\omega$, so $(0,n_{i}j)$ and ${\bf o}$ may change in different configurations, but $0\leq j\leq N$ and ${\bf o}\in\{n_{i}\}\times(-2n_{i}\epsilon,2n_{i}\epsilon)$. For each $\omega\in\{\exists\,j\mbox{ with }I(0,j)=1\mbox{ for }1\leq j\leq N\}$ and $n_{i}$ selected above, we move up $\omega$ in $l_{i}n_{i}$ units vertically and rotate the repositioned configuration $180^{\circ}$ around the horizontal line $y=l_{i}n_{i}$ to have a new configuration in ${\cal E}_{n_{i}}(l_{i}n_{i},N)$. By (3.5), translation invariance, and symmetry,

[TABLE]

On $\{{\cal E}_{n_{i}}\cap{\cal E}_{n_{i}}(l_{i}n_{i},N)\}$, $\gamma_{n_{i}}$ and $\gamma_{n_{i}}(l_{i}n_{i},j)$ have to meet at ${\bf w}$ for some vertex in $L(0,n_{i})$. Since both $\gamma_{n_{i}}$ and $\gamma_{n_{i}}(l_{i}n_{i},j)$ are optimal paths defined above, the sub-paths from ${\bf w}$ to ${\bf o}\in\{n_{i}\}\times(-2n_{i}\epsilon,2n_{i}\epsilon)$ along $\gamma_{n_{i}}(l_{i}n_{i},j)$ and from ${\bf w}$ to ${\bf v}\in\{n_{i}\}\times(\delta n_{i},n_{i})$ along $\gamma_{n_{i}}$ have the same passage time (see Fig. 1, left). Thus, on $\{{\cal E}_{n_{i}}\cap{\cal E}_{n_{i}}(l_{i}n_{i},N)\}$, we can construct another optimal path $\bar{\gamma}_{n_{i}}$ from the origin along $\gamma_{n_{i}}$ to meet ${\bf w}$, and along $\gamma_{n_{i}}(l_{i}n_{i},j)$ from ${\bf w}$ to ${\bf o}\in\{n_{i}\}\times(-2n_{i}\epsilon,2n_{i}\epsilon)$ (see Fig. 1, left) with passage time $s_{0,{n_{i}}}$. We want to remark that ${\bf w}$ may change in different configurations since $j$ depends on configurations. Let $\bar{\cal E}_{n_{i}}$ be the event that there are two optimal paths of $s_{0,n}$ from the origin to $\{n_{i}\}\times(\delta n_{i},n_{i})$ and to $\{n_{i}\}\times(-2n_{i}\epsilon,2n_{i}\epsilon)$, respectively. By (3.2) and (3.6), by choosing a small $\epsilon$ with $\delta>8\epsilon$, for the $n_{i}$ selected above,

[TABLE]

On $\bar{\cal E}_{n_{i}}$, let $\gamma_{n_{i}}\in\bar{S}_{n_{i}}$ be the optimal path from the origin to $\{n_{i}\}\times(\delta n_{i},n_{i})$, and let $\gamma({\bf u},{\bf v})$ be the corresponding sub-path defined in Lemma 2.1. For the $\kappa>0$, by (2.13) in Lemma 2.3,

[TABLE]

By (3.7), (3.8), and (2.21), for all large $n_{i}$ and $\eta$ in (2.21), there exists $C_{1}=C_{1}(F,\eta)$ such that

[TABLE]

By (3.9) and Lemma 2.2, note that $D_{n_{i}}(\kappa)\leq|\gamma({\bf u},{\bf v})|$, so there exists $C_{j}=C_{j}(F)$ for $j=2,3$ such that

[TABLE]

By (3.10), if $n_{i}$ is selected to be large, then there exists $C_{4}=C_{4}(F,\delta)>0$ independent of $\kappa$ and $n_{i}$ such that

[TABLE]

On $\{\exists\mbox{ a pivotal edge }e\in L(n_{i}-\kappa n_{i},n_{i})\}$, we assume that ${\bf p}\in\gamma({\bf u},{\bf v})$ is a vertex of $e$ and every optimal path in $\bar{S}_{n_{i}}$ has to pass through ${\bf p}$ (see Fig. 1, left). By Lemma 2.2, by taking $n_{i}$ large, there exists $C_{5}=C_{5}(F)$ such that

[TABLE]

By the triangular inequality, with a positive probability,

[TABLE]

Since $\lambda$ and $\delta$ are assumed to be positive constants independent of $\kappa$, (3.13) will contradict if $\kappa$ is small. The contradiction tells us that (3.1) cannot hold. Therefore, Lemma 3.1 follows. $\Box$

Let

[TABLE]

We show the following lemma.

Lemma 3.2. If $F$ is a distribution with $0<F(0)<p_{c}$ and (1.5) holds, then

[TABLE]

Proof. If we suppose that Lemma 3.2 does not hold, then by symmetry, then there exists a subsequence $\{n_{i}\}$ such that

[TABLE]

On $h^{S}_{n_{i}}/n_{i}\geq\delta$ for all $i$, there exist $\gamma_{n_{i}}\in\bar{S}_{n_{i}}$ and $(m_{1},m_{2})\in\gamma_{n_{i}}$ such that

[TABLE]

If there are many $\{(m_{1},m_{2})\}$ such that $h^{S}_{n_{i}}=m_{2}\geq\delta n_{i}$, then we simply select $(m_{1},m_{2})$ among all the optimal paths $\{\gamma_{n_{i}}\}$ in $\bar{S}_{n_{i}}$ with the largest $x$-coordinate when we go along optimal paths starting from the origin. We also select $\gamma_{n_{i}}$ containing $(m_{1},m_{2})$. Thus, by Lemma 3.1 and the assumption of (3.14), for a small $\epsilon$ with $\delta>8\epsilon>0$,

[TABLE]

On $\{\exists\,\,\,(m_{1},m_{2})\mbox{ such that }m_{2}=h^{S}_{n_{i}}\geq\delta n_{i},H_{n_{i}}/n_{i}\leq\epsilon\}$, we know that $0<m_{1}<n_{i}$. We assume that $n_{i}-m_{1}=\delta_{1}n_{i}$ for $\delta_{1}>0$. We will show that $\delta_{1}=O(\delta)$ by using Lemma 2.2. By Lemma 2.2, with a probability larger than $1-\exp(-O(\delta n_{i}))$, ${\bf w}=(w_{1},w_{2})\in L[n_{i}]\cap\gamma_{n_{i}}$, the terminate vertex of $\gamma_{n}$, such that

[TABLE]

Note that $m_{2}>\delta n_{i}$ and $H_{n_{i}}\leq\epsilon n_{i}$, so

[TABLE]

By (3.17),

[TABLE]

Similarly, we show that

[TABLE]

By (3.19), we choose $\delta_{2}=O(\delta)$ small, independent of $n_{i}$, such that

[TABLE]

Let $\gamma_{n_{i}}$, starting from the origin, first meet $L[m_{1}-\delta_{2}n_{i}]$ at ${\bf u}^{\prime}$ and last meet $L[m_{1}+n_{i}\delta_{2}]$ at ${\bf v}^{\prime}$ (see Fig. 1, right). Let $\gamma({\bf u}^{\prime},{\bf v}^{\prime})$ be the sub-path of $\gamma_{n_{i}}$ from ${\bf u}^{\prime}$ and ${\bf v}^{\prime}$ as we defined in Lemma 2.1. By Lemma 2.2 again, for all large $n_{i}$,

[TABLE]

By (3.15), (3.20), and (3.21), for large $n_{i}$, we select $\delta_{2}=O(\delta)$ small independent of $n_{i}$ such that

[TABLE]

Note that $(m_{1},m_{2})\in\gamma({\bf u}^{\prime},{\bf v}^{\prime})$, so if $\delta_{2}$ is selected to be small but independent of $n_{i}$, by (3.22),

[TABLE]

Now we divide ${\bf Z}^{2}$ into equal disjoint squares, except their boundaries, with length $\delta_{2}n$ for small $\delta_{2}<\delta/4$ with $\delta_{2}=O(\delta)$. More precisely, for ${\bf z}=(z_{1},z_{2})\in{\bf Z}^{2}$,

[TABLE]

If $\delta_{2}$ is small, then on $\{\forall(x_{1},x_{2})\in\gamma({\bf u},{\bf v}),0\leq x_{1}\leq n_{i}(1-\delta/(4\lambda),x_{2}>n_{i}\delta/2\}$, there exists a square $B_{\delta_{2}n_{i}}({\bf z})$ below $\gamma_{n_{i}}$ with ${\bf z}=(z_{1},z_{2})$ such that (see Fig. 1, right)

[TABLE]

A square below a curve means that any vertical line from $-\infty$ to $\infty$ always meets the square first before the curve. For a ${\bf z}$ satisfying (3.24), let ${\cal E}_{n_{i}}(\delta,{\bf z})$ be the event of all optimal paths $\gamma({\bf z})$ from ${\bf z}=(z_{1},z_{2})$ to $L[n_{i}]$ with $z_{1}<n_{i}(1-\delta/(4\lambda))$ and $z_{2}\geq\delta n_{i}/4$, and with a passage time $s_{z_{1},n_{i}}({\bf z})$ from ${\bf z}$ to $\{n_{i}\}\times[z_{2}-\epsilon n_{i},z_{2}+\epsilon n_{i}]$ in $L(z_{1},n_{i})$ (see Fig. 1, right). By Lemma 3.1, for any small $\epsilon>0$ and for a fixed ${\bf z}$, if $n_{i}$ is large, then

[TABLE]

Now we need to choose $B_{\delta_{2}n_{i}}({\bf z})$ from $\{B_{\delta_{2}n_{i}}({\bf u})\}$. On $\gamma_{n}\subset[0,n_{i}]\times[-\lambda n_{i},\lambda n_{i}]$, there are at most $(\lambda/\delta_{2})^{2}$ many choices for $\{B_{\delta_{2}n_{i}}({\bf z})\}$. When $B_{\delta_{2}n_{i}}({\bf z})$ is fixed, so is ${\bf z}$. On ${\cal E}_{n_{i}}(\delta_{2},{\bf z})\cap\{H_{n_{i}}\leq\epsilon n_{i}\}$, $\gamma({\bf z})$ and $\gamma_{n_{i}}$ have to meet ${\bf w}$ in $L(0,n_{i})$ (see Fig. 1, right). So we can construct another optimal path from the origin along $\gamma_{n_{i}}$ to ${\bf w}$ and along $\gamma({\bf z})$ from ${\bf w}$ to $\{n_{i}\}\times[z_{2}-\epsilon n_{i},z_{2}+\epsilon n_{i}]$ with passage time $s_{0,n}$. Since $z_{2}\geq n_{i}\delta/4$,

[TABLE]

This will contradict that $H_{n_{i}}\leq\epsilon n_{i}$ (Fig. 1, right). So for a fixed ${\bf z}$,

[TABLE]

By (3.18), (3.25), and (3.27), for $\delta_{2}$, $\delta$, and $\epsilon$ defined above,

[TABLE]

Since $\delta$, $\delta_{2}$, and $\lambda$ are independent from $n_{i}$, (3.28) cannot hold if $\epsilon$ is small for a large $n_{i}$. The contradiction tells that (3.14) cannot hold. Lemma 3.2 follows. $\Box$

Proof of theorem.

For the $\lambda$ in (1.8) and (1.9), let

[TABLE]

By (1.8) and (1.9) with the $\lambda$,

[TABLE]

We denote by (see Fig. 2) ${\cal E}_{n}^{+}(\epsilon)$ the event that optimal paths of $s_{-\lambda n,n}(-\lambda n,\epsilon n)$ stay inside $[-\lambda n,n]\times[0,2\epsilon n]$. Moreover, we also denote by (see Fig. 2) ${\cal E}_{n}^{-}(\epsilon)$ the event that optimal paths of $s_{-\lambda n,n}(-\lambda n,-\epsilon n)$ stay inside $[-\lambda n,n]\times[-2\epsilon n,0]$. If $n$ is large, then by Lemma 3.2,

[TABLE]

On ${\cal E}_{n}\cap{\cal E}^{+}_{n}(\epsilon)\cap{\cal E}^{-}_{n}(\epsilon)$, by the graph proof in Fig. 2, optimal paths of $a_{0,n}$ or $b_{0,n}$ have to stay inside $[-\lambda n,n]\times[-2\epsilon n,2\epsilon n]$, otherwise either ${\cal E}^{+}_{n}(\epsilon)$ or ${\cal E}^{-}_{n}(\epsilon)$ cannot occur. By (3.29) and (3.30), for large $n$,

[TABLE]

Thus, by (3.31), $h_{n}^{\alpha}/n$ converges to zero in probability in the theorem for $\alpha=A$ or $B$.

It remains to show that $h_{n}^{\alpha}/n$ converges to zero in $L_{1}$. By (3.31), (1.8) and (1.9), for $\epsilon>0$,

[TABLE]

By (3.32), $h_{n}^{\alpha}/n$ converges to zero in $L_{1}$ for $\alpha=A$ or $B$. $\Box$

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Yu Zhang

Department of Mathematics

University of Colorado, Colorado Springs, CO 80933

email: yzhang3@uccs.edu