A phase transformation for the number of optimal paths in first passage percolation

TL;DR

This paper investigates the behavior of the number of optimal paths in first passage percolation on a square lattice, revealing a phase transition at critical and sub-critical regimes.

Contribution

It introduces a phase transition framework for the number of optimal paths in first passage percolation, highlighting new critical phenomena.

Findings

Existence of a phase transition in the number of optimal paths

Different behaviors in sub-critical and critical regimes

Insights into the structure of optimal paths at large distances

Abstract

We consider the first passage percolation model on the square lattice with an edge weight distribution F. In this paper, we consider the number of optimal paths for two points separated by a long distance. We show that there is a phase transition in the sub-criticality and the criticality.

Equations

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

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

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

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

$${\mu }_F(\mathbf{x})\text{mbox}iscontinuousin\mathbf{x}\text{mbox}and{\mu }_F(\mathbf{x})=0\text{mbox}iffF(0)\ge p_c,$$

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

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

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

$$\mu <Et(e).$$

$$0<\underset{n\to \infty }{\mathrm{liminf}}{n}^{-1}\log N_n\le \underset{n\to \infty }{\mathrm{limsup}}{n}^{-1}\log N_n<\infty .\text{mbox}a.s.$$

$$P(|{\gamma }_n|\ge \kappa n)\le \exp (-n)$$

$$\underset{n\to \infty }{\mathrm{limsup}}{n}^{-1}\log N_n\le \kappa (2d)\text{mbox}a.s.$$

$$\underset{n}{\mathrm{limsup}}E{n}^{-1}\log N_n\le E\underset{n}{\mathrm{limsup}}{n}^{-1}\log N_n\le 2\kappa d.$$

$$\underset{n\to \infty }{\lim }{n}^{-1}\log N_n\text{mbox}existsinsomesense,$$

$$P(N_n=\infty )>0.$$

$$1-n^{\delta }\le P\left(\exp (n^{c_1})\le N_n\le \exp (n^{c_2})\right).$$

$$P\left(\log N_{n}=O(n^{\beta})\right)=1\mbox{ when $F(0)=p_{c}$}.$$

$$\underset{n}{\mathrm{liminf}}{n}^{-1}\log E{N}_n\ge O((p_c-F(0){)}^{-c_1}).$$

$$P\left(|{\gamma }_n|\ge n^{1+c_1}\right)\ge 1-n^{-c_2}.$$

$$E|{\gamma }_n|=n^{4/3+o(1)}.$$

$$P\left(n^{4/3+o(1)}\le |{\gamma }_n|\le n^{4/3+o(1)}\right)\ge 1-n^{-c_2}.$$

$$A_1=[-2n^{1-{\delta }_1},2n^{1-{\delta }_1}{]}^2\setminus [-n^{1-{\delta }_1},n^{1-{\delta }_1}{]}^2,\cdots ,A_k=[-2^k{n}^{1-{\delta }_1},2^k{n}^{1-{\delta }_1}{]}^2\setminus [-2^{k-1}{n}^{1-{\delta }_1},2^{k-1}{n}^{1-{\delta }_1}{]}^2,$$

$$k=O(\log n).$$

$$P({\mathcal{E}}_i)\ge c_3.$$

$$P(\mathcal{E})\ge 1-n^{-{\delta }_2}.$$

$$1-n^{-{\delta }_2}\le P(\mathcal{E})=\sum _{m}P({\mathcal{D}}_m).$$

$${\mathcal{A}}_{m-1}={\Gamma }_{m-1}\text{mbox}and{\mathcal{A}}_{m+1}={\Gamma }_{m+1}\text{mbox}onlydependontheconfigurationsin{\bar{\Gamma }}_{m-1}\cup {\mathbf{Z}}^d\setminus {\Gamma }_{m+1},$$

$$P({\mathcal{D}}_m)=\sum _{{\Gamma }_{m-1},{\Gamma }_{m+1}}\sum _{\Gamma }P({\mathcal{A}}_{m-1}={\Gamma }_{m-1},{\mathcal{A}}_{m+1}={\Gamma }_{m+1},{\beta }_m=\Gamma ),$$

$$P(|{\beta }_m|\ge n^{1+{\delta }_3})\ge 1-n^{-{\delta }_4}.$$

$$1-n^{-{\delta }_4}\le P({\mathcal{B}}_m)=\sum _{{\Gamma }_{m-1},{\Gamma }_{m+1}}\sum _{\Gamma }P({\mathcal{A}}_{m-1}={\Gamma }_{m-1},{\mathcal{A}}_{m+1}={\Gamma }_{m+1},{\beta }_m=\Gamma ,|\Gamma |\ge n^{1+{\delta }_3}).$$

Full text

1 Introduction of the model and results.

We consider the ${\bf Z}^{2}$ lattice as a graph with edges ${\bf E}^{2}$ connecting each pair of vertices, which are 1 unit apart. We assign independently to each edge a non-negative passage time $t(e)$ with a common distribution $F$. More formally, we consider the following probability space. As the sample space, we take $\Omega=[0,\infty)^{{\bf E}^{2}}$, whose points are called configurations. Let $P=\prod_{e\in{\bf Z}^{2}}\mu_{e}$ be the corresponding product measure on $\Omega$ for the measure $\mu_{e}$ with a common distribution $F$. The expectation and variance with respect to $P$ are denoted by $E(\cdot)$ and $\sigma^{2}(\cdot)$, respectively. For any two vertices ${\bf u}$ and ${\bf v}$, a path $\gamma$ from ${\bf u}$ to ${\bf v}$ is an alternating sequence $({\bf v}_{0},e_{1},{\bf v}_{1},...,{\bf v}_{i},e_{i+1},{\bf v}_{i+1},...,{\bf v}_{n-1},e_{n},{\bf v}_{n})$ of vertices $\{{\bf v}_{i}\}$ and edges $\{e_{i}\}$ between ${\bf v}_{i}$ and ${\bf v}_{i+1}$ in ${\bf Z}^{2}$ 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 $i\neq j$. A path is called a circuit only if ${\bf v}_{0}={\bf v}_{n}$. Given a disjoint 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 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 (1986a)). If $t(e)=0$, the edge is called a zero edge or 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 the edges in a path are in passage time zero, the path is called a zero 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 ${\bf A}$ and ${\bf B}$ of ${\bf R}^{2}$,

[TABLE]

is denoted by the distance between ${\bf A}$ and ${\bf B}$.

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

[TABLE]

With the limit in (1.1), it is also known (see Kesten (1986a)) that

[TABLE]

where $p_{c}$ is the critical probability in two-dimensional percolation. It is well known that $p_{c}=1/2$. In particular, Hammersley and Welsh (1965), in their pioneering paper, investigated

[TABLE]

They showed that

[TABLE]

For simplicity’s sake, we denote by

[TABLE]

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

[TABLE]

By (1.5), one might guess that there should be many optimal paths. It should be interesting to ask how many optimal paths of $T({\bf 0},n{\bf x})$ there are. Let $N_{n}({\bf x})$ be the number of optimal paths with a passage time $T({\bf 0},n{\bf x})$. In this paper, we will focus on the passage time $a_{0,n}$, and the result can be directly generalized to $T({\bf 0},n{\bf x})$. We denote by $N_{n}$ the number of the optimal paths with a passage time $a_{0,n}$. Nakajima (2017) showed that if $F(0)<p_{c}$ for any $F$, then

[TABLE]

In fact, the upper bound in (1.6) is a direct application of Kesten’s Proposition 5.8 (1986a). It shows that if $F(0)<p_{c}$, then there exists $\kappa=\kappa(F)$ such that

[TABLE]

for any optimal path $\gamma_{n}$ of $a_{0,n}$. Note that there are at most $(2d)^{\kappa n}$ many optimal paths if $\gamma_{n}\leq\kappa n$. By (1.7) and the Borel-Cantelli lemma, we have

[TABLE]

Moreover, by Fatou’s lemma,

[TABLE]

It is believed that if $F(0)<p_{c}$, then

[TABLE]

but no one is able to show it.

There is an infinite open cluster at the origin with a positive probability when $F(0)>p_{c}$. Thus, for any $n\geq 1$,

[TABLE]

However, it has been proved there is no infinite open cluster at $p_{c}$. Thus, it is more interesting to ask what the behavior of $N_{n}$ is when $F(0)=p_{c}$. We show the following theorem.

Theorem. If $F(0)=p_{c}$, then there are $1<c_{1}\leq c_{2}<2$ and $\delta>0$ 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 $c_{1}$ and $c_{2}$. If we want to denote some small numbers, we often use $\delta_{i}$ to denote them, whose precise value is of no importance; its value may change from appearance to appearance, but $\delta_{i}$ will always be independent of $n$ and $t$, $k$, $i$, and $m$, although it may depend on $F$.

Remarks. 1. We believe that there is a critical exponent $1<\beta<2$ such that

[TABLE]

We believe that $\beta=7/4$, the two-arm exponent. If we assume that the three-arm exponent is $2/3$, then by using the proof of the theorem, we might show that $4/3\leq c_{2}$ for the $c_{2}$ in the Theorem.

2. By using the proof of the theorem, one can show that there exists a positive constant $c_{1}>0$ such that for $F(0)<p_{c}$,

[TABLE]

Thus, by (1.13), if the limit in (1.10) exists, then the limit will diverge when $F(0)\uparrow p_{c}$.

2 Proof of the lower bound of the Theorem.

If there is an open crossing in $[-n,n]\times[-m,m]$, we may select the lowest crossing (see Grimmett (1999)). Let $\gamma_{n}$ be the lowest crossing from the left to the right in $[-n,n]\times[-m,m]$ for $m=O(n)$. Kesten and Zhang (1993) showed that there exist positive numbers $c_{1}$ and $c_{2}$ such that

[TABLE]

By using a standard $SLE_{6}$ estimate, if $\gamma_{n}$ is the lowest crossing as described above, we can show that for the triangular lattice,

[TABLE]

If we use the estimate from Kesten and Zhang’s method (1993) together with (2.2), it might show that for the triangular lattice,

[TABLE]

For a small $\delta_{1}>0$, we construct the annuli

[TABLE]

for $n/2\leq 2^{k}n^{1-\delta_{1}}\leq n.$ Thus,

[TABLE]

Let ${\cal E}_{i}$ be the event that there exist open circuits in both $A_{i-1}$ and $A_{i+1}$, and there exists a left-right open crossing $\beta_{i}$ in $[2^{i-2}n^{1-\delta_{1}},2^{i+1}n^{1-\delta_{1}}]\times[-2^{i-2}n^{1-\delta_{1}},2^{i-2}n^{1-\delta_{1}}]$ for $1\leq i-1\leq i+1\leq k$ (see Fig. 1). By the RSW lemma and the FKG inequality (see Grimmett (1999)), there exists $c_{3}>0$ such that

[TABLE]

Note that $\{{\cal E}_{2i}\}$ are independent for $i=1,2,\cdots,k/2$. By (2.3)–(2.4) and a simple computation, if ${\cal E}=\cup_{i}{\cal E}_{2i}$, then there exists $\delta_{2}>0$ such that

[TABLE]

On ${\cal E}$, let ${\cal D}_{m}$ be the event that ${\cal E}_{m}$ first occurs for an even number $m$. Note that $\{{\cal D}_{m}\}$ are disjoint, so

[TABLE]

On ${\cal D}_{m}$, let ${\cal A}_{m-1}$ be the innermost open circuit in $A_{m-1}$ and ${\cal A}_{m+1}$ be the outermost open circuit in $A_{m}$ (see the definitions of the innermost and the outermost open circuits in Kesten and Zhang (1993)). In addition, let $\beta_{m}$ be the lowest open path from ${\cal A}_{m-1}$ to ${\cal A}_{m+1}$ inside $[2^{m-1}n^{1-\delta_{1}},2^{m+1}n^{1-\delta_{1}}]\times[-2^{m-2}n^{1-\delta_{1}},2^{m-2}n^{1-\delta_{1}}]$ (see Fig. 1). By Proposition 2.3 of Kesten (1982), if ${\cal A}_{m-1}=\Gamma_{m-1}$ and ${\cal A}_{m+1}=\Gamma_{m+1}$ for fixed circuits $\Gamma_{m-1}$ and $\Gamma_{m+1}$, then

[TABLE]

where $\bar{\Gamma}_{m}$ is all the vertices of $\Gamma_{m}$ and the vertices enclosed by $\Gamma_{m}$. Thus,

[TABLE]

where the sums in (2.8) take all possible fixed circuits $\Gamma_{m-1}$, $\Gamma_{m+1}$, and fixed paths $\Gamma$ from $\Gamma_{m-1}$ to $\Gamma_{m+1}$.

If $0<\delta_{1}$ is much smaller than $c_{1}$, by (2.1), there exist $\delta_{3}$ and $\delta_{4}$ such that

[TABLE]

Let ${\cal B}_{m}$ be the sub-event of ${\cal D}_{m}$ with $|\beta_{m}|\geq n^{1+\delta_{3}}$. Thus, by (2.8)-(2.9),

[TABLE]

If $\beta_{m}=\Gamma$, $\Gamma$ divides $[2^{m-2}n^{1-\delta_{1}},2^{m+1}n^{1-\delta_{1}}]\times[-2^{m-2}n^{1-\delta_{1}},2^{m-2}n^{1-\delta_{1}}]$ between ${\cal A}_{m-1}$ and ${\cal A}_{m+1}$ into two parts (see Fig. 1): the vertices upper $\Gamma$ and lower $\Gamma$. We assume that the lower part includes $\Gamma$. For each $e\in\beta_{m}$, let $e^{*}$ be the edge bisected of $e$. If $\beta_{m}$ is open and for each $e\in\beta_{m}$, there is a closed dual path from $e^{*}$ to $y=2^{m-1}n^{1-\delta_{1}}$, we say $\beta_{m}$ has a three-arm property. By Proposition 2.3 of Kesten (1982),

[TABLE]

and

[TABLE]

Let $S$ be a unit square with four edges of ${\bf E}^{d}$ and four vertices of ${\bf Z}^{2}$. We say $S$ is a good square for path $\beta_{m}$ if $S$ is in the upper part of $\beta_{m}$ and

[TABLE]

We say that two unit squares $S_{i}$ and $S_{j}$ are $k$-disjoint if

[TABLE]

We now consider 3-disjoint good squares for $\beta_{m}$ to show the following lemma.

Lemma 2.1. If $F(0)=p_{c}$, then there exist $\eta>0$ and $\delta_{4}>0$ such that

[TABLE]

Remark. 3. We divide ${\bf Z}^{2}$ into equal squares with side length $M$, called $M$-squares. More precisely, for ${\bf u}=(u_{1},u_{2})\in{\bf Z}^{2}$, an $M$-square is defined to be

[TABLE]

We say $Q_{\bf u}(M)$ is an $M$-good square for path $\beta_{m}$ if $Q_{\bf u}(M)$ is in the upper part of $\beta_{m}$ and

[TABLE]

By using the same proof of Lemma 2.1, we can show that there exists $\eta=\eta(M)>0$ such that

[TABLE]

This result is independently interesting and it might be used for other estimates in critical percolation. Before the proof of Lemma 2.1, we first show that Lemma 2.1 implies the following bound of the theorem.

Proof of the lower bound of the theorem. On $\{\beta_{m}=\Gamma\}$, we list all the good squares for $\Gamma$ to be $\{S_{1},\cdots,S_{i},\cdots,S_{j}\}$. Since $S_{i}$ is a good square, there exists a unit square $S_{i}^{\prime}$ (see Fig. 2) with two edges: one is of $S_{i}$ and the other is an edge of $\Gamma$. If there is more than one such $S^{\prime}$’, we simply select one in a unique way. On $\{\beta_{m}=\Gamma\}$, let ${\cal G}_{i}$ be the event that the edges, except the edge in $\Gamma$, of $S^{\prime}_{i}$ are open. If ${\cal G}_{i}$ occurs, we call the square $S_{i}$ is accessible. Thus, for a fixed $\Gamma$ and a fixed $S_{i}^{\prime}$,

[TABLE]

By the independent properties in (2.7) and (2.11), for a fixed $\Gamma_{m-1}$, $\Gamma_{m}$, and $\Gamma$,

[TABLE]

On the other hand, since $S_{i}$ and $S_{m}$ are 3-disjoint, ${\cal G}_{i}$ and ${\cal G}_{m}$ are also independent if $i\neq m$ (see Fig. 2). By the independent properties, if ${\cal G}$ is the event that there are more than $\eta|\Gamma|F^{3}(0)/6$ many accessible squares, then by (2.13) and a standard large deviation estimate on ${\cal A}_{m-1}=\Gamma_{m-1},{\cal A}_{m+1}=\Gamma_{m+1},\beta_{m}=\Gamma$, and on the event in the probability of Lemma 2.1, there exists $c_{1}>0$ such that

[TABLE]

By Lemma 2.1, and by (2.9) and (2.15), there exists $\delta_{5}<\min\{\delta_{4},\delta_{2}\}$ such that

[TABLE]

On ${\cal G}$, we list all the accessible squares as $\{S_{1},\cdots,S_{\kappa}\}$. Furthermore, on $\{{\cal A}_{m-1}=\Gamma_{m-1},{\cal A}_{m+1}=\Gamma_{m+1},\beta_{m}=\Gamma,{\cal G}\}$, note that any path from the origin to $(n,0)$ has to pass through ${\cal A}_{m-1}$ and ${\cal A}_{m+1}$, and note also that $\beta_{m}$ is open, so $\beta_{m}=\Gamma$ is a sub-piece of an optimal path. On ${\cal G}$, we go along $\Gamma$, or go along the three open edges in $S_{\kappa}^{\prime}$ to find another optimal path. Thus, there are at least two optimal paths if we choose to go these two ways (see Fig. 2). If we continue to use the three edges of $S_{\kappa-1}^{\prime}$, note that $S_{\kappa-1}$ and $S_{\kappa}$ are 3-disjoint, so together with the preview choices in $S_{\kappa}^{\prime}$, there are at least $2^{2}$ optimal paths if we choose to go these four ways. We continue this way for each accessible edge so we have at least $2^{\kappa}$ many optimal paths. By this observation, note that

[TABLE]

so by (2.16),

[TABLE]

Thus, (2.18) implies the lower bound of the theorem. $\Box$

Now it remains to show Lemma 2.1.

Proof of Lemma 2.1. We suggest that readers use Fig. 1 as an aid to understanding the proof. We divide ${\bf Z}^{2}$ into equal squares with a side length of 3 units, called 3-squares. More precisely, for ${\bf u}=(u_{1},u_{2})\in{\bf Z}^{2}$, a $3$-square is defined to be

[TABLE]

A 3-square consists on 9 unit squares: the center unit square and other 8 unit squares surrounding the center one (see Fig. 1). We sometimes need to use 5-squares or 7-squares. Let $\beta_{m}$ be the open path from the left to the right in $[2^{m-2}n^{1-\delta_{1}},2^{m+1}n^{1-\delta_{1}}]\times[-2^{m-2}n^{1-\delta_{1}},2^{m-2}n^{1-\delta_{1}}]$ between ${\cal A}_{m-1}$ and ${\cal A}_{m+1}$ (see Fig. 1). We may assume that $|\beta_{m}|=\zeta$. We then consider the disjoint $3$-squares intersecting with $\beta_{m}$. There might be many ways to select these 3-squares. We just use a unique way, for example, starting from the beginning of $\beta_{m}$ to select the first, the second, …, and the last 3-squares. We denote them by $\beta_{m}(3)$. Let $\{Q_{i}(3)\}$ be these 3-squares in $\beta_{m}(3)$ if we go along $\beta_{m}$ from ${\cal A}_{m-1}$ to ${\cal A}_{m+1}$. In particular, we denote by $Q_{i}(1)$ the center unit square of $Q_{i}(3)$, and denote by $Q_{i}(5)$ the 25 unit squares with the same center $Q_{i}(1)$, and denote by $Q_{i}(7)$ the 49 unit squares with the same center $Q_{i}(1)$. Note that $\{Q_{i}(7)\}$ may not be disjoint, so we select a sub-sequence $\{Q_{i_{j}}(7)\}$, starting at $Q_{1}(3)$, such that they do not have a common unit square, but they are connected . In other words, these vertices contained by the boundaries of these 7-squares are connected. For a simple notation, we denote $\{Q_{i_{j}}(3)\}$ by $\{Q_{i}(3)\}=\{Q_{1}(3),Q_{2}(3),\cdots,Q_{s}(3)\}$ without confusion. Since these 7-squares are connected and each 7-square contains 64 vertices, on ${\cal B}_{m}$,

[TABLE]

We fixed all $\beta_{m}(3)$ such that $\beta_{m}(3)=\Gamma_{m}(3)$ for a fixed $4$-square set $\Gamma_{m}(3)$. After $\beta_{m}(3)$ is fixed, these 7-squares are fixed, so $\{Q_{i}(3)\}$ is also fixed. If $Q_{i}(3)$ contains a good unit square, $Q_{i}(3)$ is called a good 3-square, otherwise it is a bad 3-square. Thus, for a small $\eta$, there exists $\eta^{\prime}$ ($\eta^{\prime}\rightarrow 0$ as $\eta\rightarrow 0$) such that

[TABLE]

where the first sum takes over all open circuits $\Gamma_{m-1}$ and $\Gamma_{m+1}$ in $A_{m-1}$ and $A_{m+1}$, and the second sum takes all possible $\Gamma_{m}(3)$ in the right side of (2.20).

Let us analyze the situation of how $\beta_{m}$ meets $Q_{i}(3)$ (see Fig. 1). $\beta_{m}$ first meets $\partial Q_{i}(5)$ at ${\bf v}_{i}(1)$, uses an edge $e_{i}(1)$ between $\partial Q_{i}(5)$ and $\partial Q_{i}(3)$ to meet $\partial Q_{i}(3)$ at ${\bf u}_{i}(1)$, leaves $\partial Q_{i}(3)$ at ${\bf u}_{i}^{\prime}(1)$, and uses an edge $e_{i}^{\prime}(1)$ between $\partial Q_{i}(5)$ and $\partial Q_{i}(3)$ to meet $\partial Q_{i}(5)$ at ${\bf v}_{i}^{\prime}(1)$. It then re-meets $\partial Q_{i}(5)$ at ${\bf v}_{i}(2)$, and uses an edge $e_{i}(2)$ between $\partial Q_{i}(5)$ and $\partial Q_{i}(3)$ to meet $\partial Q_{i}(3)$ at ${\bf u}_{i}(2)$, leaves $\partial Q_{i}(3)$ at ${\bf u}_{i}^{\prime}(2)$, and uses an edge $e_{i}^{\prime}(2)$ between $\partial Q_{i}(5)$ and $\partial Q_{i}(3)$ to meet $\partial Q_{i}(5)$ at ${\bf v}_{i}^{\prime}(2).\cdots.$ It finally re-meets $\partial Q_{i}(5)$ at ${\bf v}_{i}(j_{i})$, uses an edge $e_{i}(j_{i})$ between $\partial Q_{i}(5)$ and $\partial Q_{i}(3)$ to meet $\partial Q_{i}(3)$ at ${\bf u}_{i}(j_{i})$, leaves $\partial Q_{i}(3)$ at ${\bf u}_{i}^{\prime}(j_{i})$, and uses an edge $e_{i}^{\prime}(j_{i})$ between $\partial Q_{i}(5)$ and $\partial Q_{i}(3)$ to meet $\partial Q_{i}(5)$ at ${\bf v}_{i}^{\prime}(j_{i})$ for $j_{i}\leq 6$. We want to remark that $j_{i}$ may be equal to 1. Let ${\cal Q}(Q_{i}(3),{\bf v}_{i}(l),{\bf v}_{i}^{\prime}(l),l\leq j_{i})$ be the event such that $\beta_{m}$ passes through the above vertices and edges for a fixed $Q_{i}(3)$ and fixed vertices $\{{\bf v}_{i}(l),{\bf v}_{i}^{\prime}(l)\}$ for $l\leq j_{i}$. Let $r_{i}(l)$ be the open path from ${\bf v}_{i}(l)$ to ${\bf v}_{i}^{\prime}(l)$ inside $Q_{i}(5)$ (see Fig. 1). Thus, $r_{i}(l)$ divides $Q_{i}(5)$ into two parts. They belong to the lower and upper parts of $\beta_{m}$, respectively. Let ${\cal Q}^{\prime}(Q_{i}(3),{\bf v}_{i}(l),{\bf v}_{i}^{\prime}(l),l\leq j_{i})$ and ${\cal Q}^{\prime\prime}(Q_{i}(3),{\bf v}_{i}(l),{\bf v}_{i}^{\prime}(l),1\leq l\leq j_{i})$ be the sub-events of ${\cal Q}(Q_{i}(3),{\bf v}_{i}(l),{\bf v}_{i}^{\prime}(l),l\leq j_{i})$ such that $Q_{i}(3)$ is bad, and good, respectively. Thus,

[TABLE]

We first fix $\Gamma_{m-1}$ and $\Gamma_{m+1}$, then fix $\{Q_{1}(3),\cdots,Q_{s}(3)\}$, and fix these $\{Q_{1}^{\prime}(3),\cdots,Q_{t}^{\prime}(3),\cdots,Q^{\prime}_{i}(3)\}$ from $\{Q_{1}(3),\cdots,Q_{s}(3)\}$ for these bad squares. Finally, we fix these ${\bf v}_{t}$ and ${\bf v}_{t}^{\prime}$ in $\partial Q_{t}^{\prime}(5)$. Note that

[TABLE]

so if $q={n^{1+\delta_{2}}\over 64}$ and ${\cal E}_{s,\eta^{\prime}}$ is the event that there are $s$ many bad 3-squares for $s\geq i\geq(1-\eta^{\prime})s\geq(1-\eta^{\prime})q$ among these $i$ many 3-squares for the $\eta^{\prime}$ in (2.20), then

[TABLE]

where the fourth sum in the right side of (2.23) takes over all possible disjoint $\{Q_{t}^{\prime}(3)\}$ for $t=1,\cdots,i$ with $i\geq(1-\eta^{\prime})s$, for disjoint events $\{{\cal Q}^{\prime}(Q_{t}^{\prime}(3),{\bf v}_{t}(l),{\bf v}^{\prime}_{t}(l),l\leq j_{t})\}$ for $l\leq j_{t}$, and the fifth and the sixth sums above take over all possible ${\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l)$ in $Q_{t}^{\prime}(5)$ for $l\leq j_{t}\leq 6$. Thus, for fixed $\Gamma_{m-1}$, $\Gamma_{m+1}$, and $\{Q_{1}^{\prime}(3),\cdots,Q_{i}^{\prime}(3)\}$,

[TABLE]

Now we analyze the situation on ${\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),l\leq j_{t})$ for a fixed $Q_{t}(3)$ and fixed ${\bf v}_{t}(l)$ and ${\bf v}_{t}^{\prime}(l)$ in $\partial Q_{t}(5)$ for $l\leq j_{t}$. By (2.12), we know that $r_{t}(l)$ is open for each $e\in r_{t}(l)$. In addition, there are a closed dual path from one vertex of $e^{*}$ to $\partial Q_{t}(5)$, and another disjoint closed dual path $\iota_{l}^{*}$ from $\partial Q_{t}(5)$ to the horizontal line $y=-2^{m-1}n^{1-\delta_{1}}$ outside of $Q_{t}(5)$ (see Fig. 1). In particular, the center unit square is in the lower parts of one of $\{r_{t}(l)\}$.

On ${\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),l\leq j_{t})$, we will show that we can change a few configurations in $Q_{t}(5)$ to make $Q_{t}(3)$ be good. Suppose that ${\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),l\leq j_{t})$ occurs. We keep all the configurations as the same as ${\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),l\leq j_{t})$ outside $Q_{t}(5)$ (see Fig. 1). As we mentioned above, there is one $r_{t}({l})$ such that its lower part contains the center unit square for some $l$. We call the path $\bar{r}_{t}(\bar{l})$. We also call the corresponding vertices and the edges in $\bar{r}_{t}(\bar{l})$ by $\bar{\bf v}_{t}(\bar{l}),\bar{\bf u}_{t}(\bar{l})$ and $\bar{\bf v}^{\prime}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l})$, and $\bar{e}_{t}(\bar{l})$ and $\bar{e}_{t}^{\prime}(\bar{l})$. We keep the configurations in $r_{t}(l)$ (not $\bar{r}_{t}(\bar{l})$) and in its lower part the same. Now we focus on $\bar{r}_{t}(\bar{l})$. We keep $\bar{e}_{t}(\bar{l})$ and $\bar{e}_{t}^{\prime}(\bar{l})$ open as before. There are two paths, a clockwise path $\bar{r}_{t}(l)$ and a counterclockwise path from $\bar{\bf u}_{t}(\bar{l})$ to $\bar{\bf u}_{t}^{\prime}(\bar{l})$ along $\partial Q_{t}^{\prime}(3)$ (see Fig. 1). We only focus on the the counterclockwise path, denoted by $\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$, such that the center unit square is above the path if we go along $\beta_{m}$ to $\bar{\bf u}_{t}(\bar{l})$, and go along $\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$ to $\bar{\bf u}_{t}^{\prime}(\bar{l})$, and go along $\beta_{m}$ again to reach ${\cal A}_{m+1}$ (see Fig. 1). We also keep the edges $\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$ open if they were open before, but force the other edges in $\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$ open if they were closed (see Fig. 1). $\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$ is open path after changing. We force all the edges between $\partial Q_{t}^{\prime}(3)$ and $\partial Q_{t}^{\prime}(5)$ with common vertices of $\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$ to be closed if they are not closed (see Fig. 1), but keep the others remaining closed (see Fig. 1). Note that the new constructed closed edges is a closed dual path (see Fig. 1), called $\bar{\gamma}^{*}(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$.

After the changes, the newly constructed open path from $\Gamma_{m-1}$ along $\beta_{m}$ first reaches $\bar{\bf u}_{t}(\bar{l})$, goes along $\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$ to $\bar{\bf u}_{t}^{\prime}(\bar{l})$, then goes along $\beta_{m}$ and back to $\Gamma_{m+1}$. The new open path is called $\bar{\beta}_{m}$. Note that $\bar{\beta}_{m}$ also has a lower part and an upper part. It follows from our construction that $Q_{t}(1)$ is on the upper part. Thus, $Q_{t}(3)$ is a good unit square for $\bar{\beta}_{m}$. However, for each $e\in\bar{\beta}_{m}$, if $e\in\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$, note that there is a closed dual path from $\bar{e}^{*}_{t}(\bar{l})$ to $y=-2^{m-2}n^{1-\delta_{1}}$ outside $Q_{t}(5)$, so by our construction, we can use the closed dual path $\bar{\gamma}^{*}(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$ and the outside of the $Q_{t}(5)$ dual closed path above to construct a closed dual path from $e^{*}$ to $y=-2^{m-2}n^{1-\delta_{1}}$. On the other hand, if $e\in\beta_{m}$ but $e\not\in\gamma(\bar{\bf u}_{t}(\bar{l}),\bar{\bf u}_{t}^{\prime}(\bar{l}))$, there has already been a closed dual path from $e^{*}$ to $y=-2^{m-2}n^{1-\delta_{1}}$. By (2.11), $\bar{\beta}_{m}$ is the lowest open path. Thus, the constructed configuration belongs to ${\cal Q}^{\prime\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),l\leq j_{t})$.

Let $\tau({\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),l\leq j_{t})$ be all the configurations after the changes. Note that we only change at most 52 edges of $Q_{t}(5)$ in the configurations of ${\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{i}^{\prime}(l),l\leq j_{t})$, without changing the configurations outside of $Q_{t}(5)$ such that ${\cal Q}^{\prime\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{i}^{\prime}(j_{i}),l\leq j_{t})$ occurs. For the configurations in ${\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),1\leq l\leq j_{t})$, after changing, they may not be disjoint. Note that we only change the configurations inside $Q_{t}(5)$, so there are at most $2^{52}$ many configurations of ${\cal Q}^{\prime}(Q_{t}(3),{\bf v}_{t}(l),{\bf v}_{t}^{\prime}(l),l\leq j_{t})$ merging into one configuration. Thus, there exists $\alpha_{1}<\infty$ dependent on $F(0)$, but independent of $n$, $i$, and $t$ such that

[TABLE]

Thus,

[TABLE]

By (2.25),

[TABLE]

We iterate (2.26) to have

[TABLE]

Together with (2.27) and (2.23), note that $i\geq(1-\eta^{\prime})s\geq(1-\eta^{\prime})q$, so

[TABLE]

By taking $\eta>0$ small, then $\eta^{\prime}$ small in (2.28), note that by (2.24) we sum all disjoint events in (2.28), so there exists $c_{1}>0$ such that

[TABLE]

Thus, by (2.6), (2.9), and (2.29), there exists $\delta>0$ such that

[TABLE]

Lemma 2.1 follows from (2.30). $\Box$

3 Proof of the upper bound of the theorem.

To show the upper bound, we only need to show that for each optimal path, $\gamma_{n}$, there exist $\delta_{1}$ and $\delta_{2}$ such that

[TABLE]

If fact, if (3.1) holds, then by the same estimate in (2.5), on $|\gamma_{n}|\leq n^{2-\delta_{1}}$ for each $\gamma_{n}$, we have $N_{n}\leq(2d)^{2-\delta_{1}}$. Thus, by (3.1),

[TABLE]

By (3.2), the upper bound in the theorem holds. It remains to show (3.1).

If (3.1) will not occur, then for any $\delta_{1}>0$ and $\delta_{2}>0$,

[TABLE]

Chayes, Chayes, and Durrett (1986) showed that if $F(0)=p_{c}$, then

[TABLE]

By Markov’s inequality and (3.4), for any $\delta_{3}$

[TABLE]

By the RSW lemma and the FKG inequality, for any $\delta_{4}>0$, there exists an open circuit in $[-n^{1+\delta_{4}},n^{1+\delta_{4}}]^{2}\setminus[-n,n]^{2}$ with a probability larger than $1-n^{-\delta_{5}}$ for $\delta_{5}>0$. By the RSW lemma and the FKG inequality again, for any $\delta_{6}>\delta_{4}$, there exists a closed dual circuit in $[-n^{1+\delta_{6}},n^{1+\delta_{6}}]^{2}\setminus[-n^{1+\delta_{4}},n^{1+\delta_{4}}]^{2}$ with a probability larger than $1-n^{-\delta_{7}}$ for $\delta_{7}>0$. Therefore, there exist an open circuit in $[-n^{\delta_{4}},n^{\delta_{4}}]^{2}\setminus[-n,n]^{2}$ and a closed dual circuit in $[-n^{1+\delta_{6}},n^{1+\delta_{6}}]^{2}\setminus[-n^{1+\delta_{4}},n^{1+\delta_{4}}]^{2}$ with a probability larger than $(1-n^{-\delta_{6}})$ for $\delta_{6}>0$. With these two circuits, any optimal path from the origin to $(n,0)$ has to stay inside $[-n^{1+\delta_{6}},n^{1+\delta_{6}}]^{2}$. Thus,

[TABLE]

There might be many open clusters inside $[-n^{1+\delta_{6}},n^{1+\delta_{6}}]^{2}$. We select the one with the largest number of vertices. If there are two such clusters with the same size, we simply select one. Let $C_{\max}(\delta_{6},n)$ be the number of this largest open cluster in $[-n^{1+\delta_{6}},n^{1+\delta_{6}}]^{2}$. We show the following lemma.

Lemma 3.1. If $F(0)=p_{c}$, for a small, but fixed $\delta_{6}>0$, there exists $\eta=\eta(\delta_{6})>0$ such that

[TABLE]

Proof. We divide $[-n^{1+\delta_{6}},n^{1+\delta_{6}}]^{2}$ into smaller equal squares with side length $n^{1-\delta_{7}}$. There are at most $n^{2\delta_{6}+2\delta_{7}}$ many such sub-squares. We divide the proof into the following two cases: the case that $C_{\max}(\delta_{6},n)$ will touch a sub-square boundary, denoted by event ${\cal A}$, or the case ${\cal A}^{C}$. Note that on ${\cal A}$, $C_{\max}(\delta_{6},n)$ will stay in a sub-square, so

[TABLE]

We now estimate the second term in the right side of (3.7). We may assume that $C_{\max}(\delta_{6},n)$ meets a sub-square ${\bf S}_{i}$ denoted by event ${\cal S}_{i}$. For each ${\bf S}_{i}$, let $C_{{\bf S}_{i}}$ be all the vertices in ${\bf S}_{i}$ connected by open paths from them to $\partial{\bf S}$. By using Theorem 8 in Kesten (1986b), there exists $\eta_{1}$ independent of $\delta_{i}$ for $i=6,7$ such that

[TABLE]

Note that there are at most $2\delta_{6}+2\delta_{7}$ many sub-squares.

[TABLE]

We select $2\delta_{6}+\delta_{7}=\eta_{1}/4$ to show that

[TABLE]

If we substitute (3.10) into (3.7), Lemma 3.1 follows.$\Box$

Now we show that Lemma 3.1 implies (3.1). By Markov’s inequality and Lemma 3.2, if $\delta_{1}-\delta_{3}$ are small, then there exists $\delta_{8}$ independent of $\delta_{2}$ and $\delta_{3}$ such that

[TABLE]

If (3.3) holds, then there exists an optimal path $\gamma_{n}$ of $a_{0,n}$ such that $|\gamma_{n}|\geq n^{2-\delta_{1}}$ with a probability larger than $n^{-\delta_{2}}$. Note that on

[TABLE]

there is an open cluster larger than $n^{2-\delta_{1}-\delta_{3}}$. By (3.5) and the assumption of (3.3), if $\delta_{2}<\delta_{3}/4$, then for large $n$

[TABLE]

Thus, (3.3) and (3.11) cannot hold at the same time if $\delta_{3}$ is selected to be small. The contradiction tells us that (3.1) should hold. With (3.1), (3.2) holds, so we have the upper bound estimate in the theorem. Together with the lower bound and the upper bound estimates, the theorem follows. $\Box$

References

Chayes, J., Chayes L., and Durrett R. (1986). Critical behavior of the two-dimensional first passage time. J. Stat. Phys. 45 933–948.

Grimmett, G. (1999). Percolation. Springer, Berlin.

Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In Bernoulli, Bayes, Laplace Anniversary Volume (J. Neyman and L. LeCam, eds.) 61–110. Springer, Berlin.

Kesten, H. (1982). Percolation theory for mathematicians. Birkhauser. Boston.

Kesten (1986a). Aspects of first-passage percolation. Lecture Notes in Math. 1180 125–264. Springer, Berlin.

Kesten, H. (1986b). The incipient infinite cluster in two-dimensional percolation. Probab. Related Field 73 369–394.

Kesten, H. and Zhang, Y. (1993). The tortuosity of occupied crossings of a box in critical percolation. J. Stat. Phys. 70 599–611.

Nakajima, S. (2017). On properties of optimal paths in first passage percolation. arXiv: 1709.03647v3 [math.PR].

Smythe, R. T. and Wierman, J. C. (1978). First passage percolation on the square lattice. Lecture Notes in Math. 671. Springer, Berlin.

Yu Zhang

Department of Mathematics

University of Colorado

Colorado Springs, CO 80933

email: yzhang3@uccs.edu