Power Lower Bounds for the Central Moments of the Last Passage Time for Directed Percolation in a Thin Rectangle

TL;DR

This paper establishes lower bounds on the central moments of the last passage time in directed percolation models on thin rectangular grids, revealing how these moments scale with grid size for certain parameters.

Contribution

It provides the first rigorous lower bounds on the moments of the last passage time in directed percolation within thin rectangles, extending understanding of fluctuation behavior.

Findings

Lower bounds on the $r$-th central moment scale as $n^{r(1-eta)/2}$ for $0<eta<1/3$.

Results apply to i.i.d. weights with finite support on $n imes n^eta$ grids.

The bounds hold for sufficiently large $n$ and for all $r eq 0$.

Abstract

In directed last passage site percolation with i.i.d.~random weights with finite support over a $n\times\lfloor n^{\alpha}\rfloor$ grid, we prove that for $n$ large enough, the order of the $r$-th central moment, $1\le r<+\infty$, of the last passage time is lower bounded by $n^{r(1-\alpha)/2}$, $0<\alpha<1/3$.

Equations

$$L_n=\underset{v\in \Pi }{\max }\sum _{v\in \pi }w(v),$$

$${\mathbb{M}}_r\left(L_n\right)$$

$$M_n=\underset{v\in \Pi }{\max }\sum _{v\in \pi }1\left(w(v)>m\right),$$

$$\mathbb{P}(M_n\ge c_{1}n)\le \exp (-c_{2}n),$$

$$\mathbb{P}(M_n-\mathbb{E}{M}_n\ge t\sqrt{n+n^{\alpha }-1})\le \exp (-c_3{t}^2).$$

$$\psi (\lambda )\le c(n+n^{\alpha }-1){\lambda }^2.$$

$$\mathbb{P}\left(M_n-\mathbb{E}{M}_n\ge \sqrt{n+n^{\alpha }-1}t\right)$$

$$EntX=\mathbb{E}X\log X-\mathbb{E}X\log \mathbb{E}X.$$

$$\frac{d}{d\lambda }\left(\frac{\psi (\lambda )}{\lambda }\right)$$

$$Ent\exp (\lambda M_n)\le c(n+n^{\alpha }-1){\lambda }^2\mathbb{E}\exp (\lambda M_n),$$

$$\frac{d}{d\lambda }\left(\frac{\psi (\lambda )}{\lambda }\right)$$

$$\le c(n+n^{\alpha }-1),$$

$$Ent\exp (\lambda M)\le \sum _{i=1}^{n^{1+\alpha }}\mathbb{E}En{t}_i\exp (\lambda M),$$

$$Ent\exp (tX)\le \mathbb{E}\left(\exp (tX)q(-t(X-X^{\prime }{)}_{+})\right),$$

$$Ent\exp (\lambda M)\le \sum _{i=1}^{n^{1+\alpha }}\mathbb{E}\left(\exp (\lambda M)q(-\lambda (M-M_i^{\prime }{)}_{+})\right).$$

$$(M-M_i^{\prime }{)}_{+}\le 1-w^{\prime }(v_i),$$

$$-\lambda (M-M_i^{\prime }{)}_{+}\ge -\lambda (1-w^{\prime }(v_i)).$$

$$q(-\lambda (M-M_i^{\prime }{)}_{+})\le q(-\lambda (1-w^{\prime }(v_i)).$$

$$\mathbb{E}\left(\exp (\lambda M)q(-\lambda (M-M_i^{\prime }{)}_{+})\right)=\mathbb{E}\left(\exp (\lambda M)q(-\lambda (M-M_i^{\prime }{)}_{+})1(v_i\in \mathcal{G})\right).$$

$$Ent\exp (\lambda M)$$

$$\le \sum _{i=1}^{n^{1+\alpha }}\mathbb{E}\left(\exp (\lambda M)q(-\lambda (1-w^{\prime }(v_i))1(v_i\in \mathcal{G})\right)$$

$$=\sum _{i=1}^{n^{1+\alpha }}\mathbb{E}\left(\exp (\lambda M)1(v_i\in \mathcal{G})\right)\mathbb{E}q(-\lambda (1-w(v_i))$$

$$=Card(\mathcal{G})\mathbb{E}q(-\lambda (1-w(v_1))\mathbb{E}\left(\exp (\lambda M)\right).$$

$$Ent\exp (\lambda M)\le (n+n^{\alpha }-1)\mathbb{E}q(-\lambda (1-w(e_1))\mathbb{E}\exp (\lambda M).$$

$$\underset{\lambda \searrow 0}{\lim }\frac{\mathbb{E}q(-\lambda (1-w(v_1))}{{\lambda }^2}$$

$$=\mathbb{E}(1-w(v_1){)}^2=1-p.$$

$$Ent\exp (\lambda M)\le (n+n^{\alpha }-1){\lambda }^2\mathbb{E}\exp (\lambda M),$$

$$W^k{=}_d(W|N=k),$$

$$W^{N_1}{=}_{d}W,$$

$$\mathbb{P}\left(W^{k}=(\omega_{i})_{i=1}^{n^{1+\alpha}}\right)=\left(\begin{array}[]{c}n^{1+\alpha}\\ k\end{array}\right)^{-1}.$$

$$\mathbb{P}\left(W^{k+1}=({\omega }_i{)}_{i=1}^{n^{1+\alpha }}\right)=\sum _{j=1}^{k+1}\mathbb{P}\left(W^{k+1}=({\omega }_i{)}_{i=1}^{n^{1+\alpha }}|B_j^{k+1}\right)\mathbb{P}(B_j^{k+1}),$$

$$\mathbb{P}\left(W^{k+1}=({\omega }_i{)}_{i=1}^{n^{1+\alpha }}\right)$$

$$\mathbb{E}\left(\exp (i⟨t,W⟩)\right)$$

Full text

00footnotetext: MSC2010: Primary 60K35, 82B43.00footnotetext: Keywords: Last-passage percolation, longitudinal fluctuation.

1 Introduction and statements of results

Longitudinal/shape fluctuations, i.e., the standard deviation of first/last passage time, has attracted a lot of attention in the study of percolation systems. It is conjectured that, on a two dimensional $n\times n$ grid, the fluctuation should be of order $n^{1/3}$ in undirected/directed first/last passage percolation, with various weight distributions satisfying moment conditions. However, this result has only been proved under exponential or geometric weights, e.g., see [13, 14, 3]. For general weight distributions satisfying moments conditions, to date, only a upper bound of sublinear order $O(\sqrt{n/\ln n})$ (see [15, 16, 4, 5, 9]) and a lower bound of order $o(\sqrt{\ln n})$ (see [20, 19, 21, 1]) have been proved in first passage percolation in various dimensions. More is known for the directed last passage time ($DLPP$) in a thin rectangular lattice where, via a coupling to Brownian directed percolation, it has been shown, in [6], with proper renormalization, to converge to the Tracy-Widom distribution. Recently, a general method to prove lower bounds for variances is devised in [8]. It is applicable to first passage percolation with continuous weights, providing a lower bound of order $o(\sqrt{\ln n})$ for the fluctuation. For a list of other results on these topics, we refer the interested reader to the recent comprehensive survey [2].

In a related topic, i.e., the study of the length of the longest common subsequences ($LCSs$) in random words, this fluctuation has also been longed for. It is well known that $LCSs$ can be viewed as a directed last passage percolation problem with random but dependent weights. In [17], the variance of the length of $LCSs$ is shown to be linear when the letters are drawn from a highly concentrated Bernoulli distribution. This method is further developed in [11] to show that the $r$-th moment of $LCSs$ is of order $\Theta(n^{r/2})$ under a similarly concentrated distribution over some finite dictionary. This power lower bound on the fluctuation is essential in proving a Gaussian limiting law for the length of $LCSs$. (See [10])

The present paper aims at studying the $r$-th, $1\leq r<+\infty$, central moments of $DLPP$ in a thin rectangular $n\times\lfloor n^{\alpha}\rfloor$ grid. These are shown to be lower-bounded by $n^{r(1-\alpha)/2}$, for $0<\alpha<1/3$, when $n$ is large enough. (For $r=1$, results on the first order central moments are very sparse in the percolation literature.) Moreover, our methodology is also applicable to first passage time for directed site/edge percolation.

Hereafter, for convenience, $n^{\alpha}$ will be short for $\lfloor n^{\alpha}\rfloor$. Next, the model under study is specified as follows: we consider a $n\times n^{\alpha}$ grid having $n^{1+\alpha}$ vertices, each of which is associated with i.i.d. random weights $w$. We require the weight distribution to be non-degenerate and to have finite non-negative support, i.e., its $c.d.f.$ $F$ is such that $F(0-)=0$ and such that there exists $C>0$ with $F(C)=1$. Then, in this setting, the last passage time $L_{n}$ is the maximum of the sums over all the weights, along all the unit-step up-right paths on the grid, from $(1,1)$ to $(n,n^{\alpha})$. Namely,

[TABLE]

where $\Pi$ is the set of all unit-step up-right paths from $(1,1)$ to $(n,n^{\alpha})$, and where any path $\pi\in\Pi$ is an ordered set of vertices, i.e., $\pi=\{v_{1}=(1,1),v_{2},...,v_{n+n^{\alpha}-1}=(n,n^{\alpha})\}$ such that $v_{i+1}-v_{i}$, $i\in[n_{1}+n_{2}-1]=\{1,2,...,n_{1}+n_{2}-1\}$, is either $\textbf{e}_{1}:=(1,0)$ or $\textbf{e}_{2}:=(0,1)$ and where $w:\ v\rightarrow w(v)\in\mathbb{R}$ is the random weight associated with the vertex $v\in[n]\times[n^{\alpha}]$, where $[n]:=\{1,2,...,n\}$. Hereafter, directed path is short for such type of path. Further, any directed path realizing the last passage time is called a geodesic. Within this framework, our main result is:

Theorem 1.1**.**

The $r$-th central moment of the directed last passage time in site percolation over a $n\times n^{\alpha}$, $0<\alpha<1/3$, grid is lower-bounded, of order $n^{r(1-\alpha)/2}$, i.e., for $1\leq r<+\infty$,

[TABLE]

where $c_{0}>0$ is a constant which depends on $r$ but is independent of $n$.

The remaining of this paper is dedicated to the proof of the above theorem and is organized as follows: at the beginning of the next section, we show that with high probability the number of $hi$-mode weights (to be defined) on any geodesic grows at most linearly in $n$. More importantly, this indicates that there exist at least linearly many $lo$-mode weights on any geodesic. In turn, this helps showing that if $L_{n}$ is represented as a random function of the number of $lo$-mode weights over the grid, then with high probability it locally satisfies a reversed Lipschitz condition. In Section 3, the proof of the main theorem is completed by showing how such a local reversed Lipschitz condition ensures a power lower bound for any central moment. In the concluding section, we briefly discuss the potential extension of our proof to the case of the second order central moment, i.e., the variance over a square grid, i.e., $\alpha=1$.

2 Preliminaries

We start by introducing the notions of $hi/lo$ mode of site weights: since the weight distribution is non-degenerate and non-negative, there exists $m>0$ such that $\mathbb{P}(w>m)=p>0$ and $\mathbb{P}(w\leq m)=1-p>0$. Then, $w$ is said to be in $hi$ mode if $w>m$; otherwise, $w$ is in $lo$ mode. In addition, let $M_{n}$ be the maximum of the number of weights in $hi$ mode over all directed paths:

[TABLE]

which is the same as the last passage time for the same grid with Bernoulli weights $\boldsymbol{1}(w(v)>m)$. In this section, on an explicitly constructed event of very high probability, $L_{n}$ is shown to locally satisfy a reversed Lipschitz condition, where now $L_{n}$ is considered as a function of the number of $hi$ mode weights over the grid.

2.1 Linear Growth of $M_{n}$

First, we show that there exists an absolute constant $0<c_{1}<1$ such that the probability that $M_{n}$ is larger than $c_{1}n$ is exponentially small.

Proposition 2.1**.**

There exist constants $0<c_{1}<1$ and $0<c_{2}<+\infty$, independent of $n$, such that

[TABLE]

for $n$ large enough.

To prove Proposition 2.1, we start by showing a concentration inequality for $M_{n}$. The proof, via the entropy method is akin to the proof of Theorem $3.12$ described in [2].

Proposition 2.2**.**

There exists $0<c_{3},c_{4}<\infty$ such that for $t\in(0,c_{4}\sqrt{n+n^{\alpha}-1})$,

[TABLE]

Proof.

Let $\psi(\lambda)=\log\mathbb{E}\exp(\lambda(M_{n}-\mathbb{E}M_{n}))$. Then, as shown next, it suffices to show that for some $c>0$ and $\lambda\in(0,c)$,

[TABLE]

Indeed, for any $\lambda>0$,

[TABLE]

Letting $\lambda=t\sqrt{n+n^{\alpha}-1}/2c$ will complete the proof, wherever (2.1), which we proceed to prove next, holds true. For any non-negative random variable $X$ (and the convention $0\ln 0=0$), let

[TABLE]

Then,

[TABLE]

If

[TABLE]

for $\lambda\in(0,c)$, then we would have

[TABLE]

for which, it would follow that $\psi(\lambda)\leq c(n+n^{\alpha}-1)\lambda^{2}$. Let us therefore prove (2.2). First, enumerate the $n^{1+\alpha}$ vertices as $v_{1},v_{2},...,v_{n^{1+\alpha}}$ and denote the associated Bernoulli weights as $w(v_{i})$, i.e., the indicator function of whether $v_{i}$ is in $hi$-mode. By the tensorization property of the entropy,

[TABLE]

where $Ent_{i}(\cdot)$ is the entropy taken only relative to the random weight $w(v_{i})$. Now, recall (see [7, Theorem 6.15]): for all $t\in\mathbb{R}$,

[TABLE]

where $q(x)=x(e^{x}-1)$ and where $X^{\prime}$ is an independent copy of $X$ . Therefore, (2.3) and (2.4) lead to

[TABLE]

However, it is clear that $M-M_{i}^{\prime}\leq 1$ with equality if and only if $w(v_{i})=1$ and, its independent copy, $w^{\prime}(v_{i})=0$, for $v_{i}\in\mathcal{G}$, where $\mathcal{G}$ is the set of vertices in the intersection of all the geodesics, i.e., $\mathcal{G}=\cap_{geodesics}\{v\in geodesic\}$. So it follows that

[TABLE]

which in turn yields that

[TABLE]

On the other hand, $q^{\prime}(x)=xe^{x}+e^{x}-1<0$, when $x<0$, and so

[TABLE]

Moreover, $q(0)=0$ gives us

[TABLE]

Thus,

[TABLE]

Since any geodesic covers exactly $n+n^{\alpha}-1$ vertices, $Card(\mathcal{G})\leq n+n^{\alpha}-1$, and

[TABLE]

Now, by dominated convergence,

[TABLE]

Hence, there exists $c$ such that when $\lambda\in(0,c)$, $\mathbb{E}q(-\lambda(1-w(v_{1}))\leq\lambda^{2}$. Combining (2.6) with (2.5), it finally follows that

[TABLE]

for $\lambda\in(0,c)$. ∎

Remark 2.3*.*

Note that in Proposition 2.2, and in contrast to [9, Theorem 1.1], the subcritical condition, i.e., $p<p_{c}$, where $p_{c}$ is the critical probability in directed bond percolation, in two dimensions, is not required. This is mainly due to the fact that the subcritical condition is needed there to bound the length of the geodesics in undirected percolation; however, in our directed case, any directed path is naturally of length $n+n^{\alpha}-1$.

Proof of Proposition 2.1:.

Let $g$ be the shape function, i.e., let $g((1,a))=\lim_{n\rightarrow+\infty}\allowbreak\mathbb{E}M(n,na)/n$, where $M(n,na)$ is the last passage time over a $n\times na$ grid. It is shown in [18] that $g((1,a))=p+2\sqrt{p(1-p)a}+o(\sqrt{a})$, as $a\rightarrow 0$. Hence, there exists $N$ such that for $n>N$, $\mathbb{E}M(n,n^{\alpha})\leq(p+1)n/2$, which, when combined with Proposition 2.2, gives $\mathbb{P}(M\geq(p+1)n/2+t\sqrt{n+n^{\alpha}-1})\leq\exp(-c_{1}t^{2})$, for any $t\in(0,c_{4}\sqrt{n+n^{\alpha}-1})$.

Further, let 0<\text{\varepsilon}<(1-p)/2. Then there exists a constant $0<\varepsilon^{\prime}<c_{4}$, independent of $n$, such that if $t=\varepsilon^{\prime}\sqrt{n+n^{\alpha}-1}\in(0,c_{4}\sqrt{n+n^{\alpha}-1})$, then $t\sqrt{n+n^{\alpha}-1}\leq\varepsilon n$ and $t^{2}=(\varepsilon^{\prime})^{2}(n+n^{\alpha}-1)>(\varepsilon^{\prime})^{2}n$. Hence, for this particular $t$, $\mathbb{P}(M\geq(\varepsilon+(p+1)/2)n)\leq\exp(-c_{3}(\varepsilon^{\prime})^{2}n)$. Setting $c_{1}=(\varepsilon+(p+1)/2)<1$ and $c_{2}=c_{3}(\varepsilon^{\prime})^{2}>0$, finishes the proof. ∎

2.2 Local Reversed Lipschitz Condition

To begin with, let us set the underlying probability space as $\Omega_{n}=\mathbb{R}^{n^{1+\alpha}}$ associated with the product measure $\bigotimes_{i=1}^{n^{1+\alpha}}F$ and let $W=(w(v_{i}))_{i=1}^{n^{1+\alpha}}$ be the random vector of weights under an arbitrary but deterministic enumeration of weights over all the $n^{1+\alpha}$ vertices. Let $N$ be the total number of $v_{i}$ such that $w(v_{i})$ is in $hi$ mode and so, clearly, $N$ is a binomial variable with parameters $n^{1+\alpha}$ and $p$. In addition, any weight $w$ can be decided in a two-step way: it is first fixed to be in $hi/lo$ mode by flipping a Bernoulli random variable with parameter $p$; then it is further associated with a non-negative real value by drawing from $F$ conditioned on the fixed $hi/lo$ mode in the first step. Based on this point of view, one can construct an iterative scheme to decide $W$ by starting from a grid with all the weights in $lo$ mode and changing them into $hi$ mode one by one until after some deliberate random steps.

To be more precise, a (finite) sequence of random vectors of weights $\{W^{k}=(w^{k}(v_{i}))_{i=1}^{n^{1+\alpha}}\}_{k=0}^{n^{1+\alpha}}$ is iteratively defined as follows: First, let $W^{0}=\{w^{0}(v_{i})\}_{i=1}^{n^{1+\alpha}}$, where $w^{0}(v_{i})$ has distribution $F$ conditioned on being in $lo$ mode. Then, $W^{0}$ is clearly identical, in distribution, to $W$ conditioned on $N=0$. Second, once $W^{k}$ is defined, one vertex $v_{i_{0}}$ is uniformly chosen at random from the set $\{v_{i}:\ w^{k}(v_{i})\ in\ lo\ mode\}$ and then $W^{k+1}$ is defined such that $w^{k+1}(v_{i_{0}})$ is sampled from $F(\cdot)$ conditioning on being in $hi$ mode and $w^{k+1}(v_{i})=w^{k}(v_{i})$ for $i\neq i_{0}$, i.e., $W^{k+1}$ is defined by changing one uniformly chosen $lo$-mode weight in $W^{k}$ to a $hi$-mode weight. The second step is repeated $n^{1+\alpha}$ times until all $lo$-mode weights, in $W^{0}$, are changed to only $hi$-mode weights in $W^{n^{1+\alpha}}$.

By the very definition, for $0\leq k\leq n^{1+\alpha}$, there are $k$ $lo$-mode weights in $W^{k}$. Moreover, $\{W^{k}\}_{k=0}^{n^{1+\alpha}}$ are dependent but independent of both $W$ and $N$. Next, we show that $W^{k}$ has the same law as $W$ conditioned on $N=k$.

Lemma 2.4**.**

For any $k=0,1,...,n^{1+\alpha}$,

[TABLE]

and moreover,

[TABLE]

where $=_{d}$ denotes equality in distribution.

Proof.

The proof is by induction on $k$. By definition, $W^{0}=_{d}W$ conditioned on $N=0$. Assume now that (2.7) is true for $k$, i.e., that for any $(\omega_{i})_{i=1}^{n^{1+\alpha}}\in\Omega_{n}$ such that $Card\left(\{\omega_{i}\ in\ lo\ mode\}_{i=1}^{n^{1+\alpha}}\right)=k$,

[TABLE]

Then, for any $(\omega_{i})_{i=1}^{n^{1+\alpha}}\in\Omega$ such that $Card\left(\{\omega_{i}\ in\ lo\ mode\}_{i=1}^{n^{1+\alpha}}\right)=k+1$,

[TABLE]

where $B_{j}^{k+1}$, $1\leq j\leq k+1$, denotes the event that the $jth$ weight $1$ in $\{\omega_{i}^{k+1}:\omega_{i}^{k+1}=1,\ 1\leq i\leq n^{1+\alpha}\}$ is the one which has been flipped uniformly at random from the weight [math] in $W^{k}$. Combining (2.9) and (2.10) gives

[TABLE]

Next, (2.7) and the independence of $N$ and $\{W^{k}\}_{k=0}^{n^{1+\alpha}}$ give

[TABLE]

∎

This particular way of iterative sampling provides a new perspective on $L_{n}$. Letting $L_{n}(k):=L_{n}(W^{k})$ and $L_{n}:=L_{n}(W)$ be respectively the last passage times under weights settings $W^{k}$ and $W$, it is clear from Lemma 2.4, that $L_{n}(N)=_{d}L_{n}$ and so it is equivalent to study $\mathbb{M}_{r}(L_{n}(N))$ or $\mathbb{M}_{r}(L_{n})$. We finish this section by showing that on an event of probability exponentially close to $1$, $\{L_{n}(k)\}_{i=1}^{n^{1+\alpha}}$ satisfies locally a reversed Lipschitz condition.

Lemma 2.5**.**

There exist positive constants $c_{2}$, $c_{5}$ and $c_{6}$ not depending on $n$ such that, when $n$ is large enough,

[TABLE]

where $I=\left(n^{1+\alpha}p-\sqrt{(1-p)pn^{1+\alpha}},\ n^{1+\alpha}p+\sqrt{(1-p)pn^{1+\alpha}}\right)$.

Proof.

Define a set $B_{n}=\{w:\ w\in\Omega_{n},\ M_{n}(w)<c_{1}n\}$ and so, by Proposition 2.1, $\mathbb{P}(B_{n})\geq 1-\exp(-c_{2}n)$, when $n$ is large enough. Further, let $A_{n}:=\{W\in B_{n}\}$ and $A_{n}^{k}:=\{W^{k}\in B_{n}\}$. Then, by Lemma 2.4,

[TABLE]

Meanwhile, for any $k\in I$, $\mathbb{P}(N_{1}=k)\geq 1/(6\sqrt{n^{1+\alpha}p(1-p)})$. Indeed,

[TABLE]

and, by de Moivre–Laplace Theorem,

[TABLE]

when $n$ is large enough. Similarly, this lower bound also holds for $\mathbb{P}(N_{1}=pn^{1+\alpha}+\lfloor\sqrt{n^{2+\alpha-\delta}}\rfloor)$ and therefore

[TABLE]

for any $k\in I$. Combining (2.13) and (2.14) gives:

[TABLE]

Next, before building a martingale difference sequence, we show that, with high probability, the difference between $L_{n}(k+1)$ and $L_{n}(k)$ conditioned on $W^{k}$ can be lower bounded by a fractional polynomial in $n$. Indeed, it always holds true that

[TABLE]

since $L_{n}(k+1)$ increases if and only if the chosen $lo$-mode weight is on any geodesic under $W^{k}$. Note that there are at least $(n+n^{\alpha}-M_{n}(k))$ many $lo$-mode weights on any geodesic and $(n^{1+\alpha}-k)$ many $lo$-mode weights over the grid under $W^{k}$, so the probability that any $lo$-mode weight on some geodesic is chosen is at least $\left(n+n^{\alpha}-M_{n}(k)\right)/\left(n^{1+\alpha}-k\right)$. In addition, the expected increment of a single flipping should be $\left(\mathbb{E}(w|hi)-m\right)>0$. Hence, by conditioning on $A_{n}^{k}=\{M_{n}(k)<c_{1}n\}$,

[TABLE]

Based on this lower bound, a martingale difference sequence is built as follows: for each $k\geq 0$, letting

[TABLE]

Therefore, letting $c_{5}:=(1-c_{1})\left(\mathbb{E}(w|hi)-m\right)$,

[TABLE]

Now, for each $k=0,1,...,n^{1+\alpha}$, let $\mathcal{F}_{k}:=\sigma(W^{0},W^{1},..,W^{k})$, be the $\sigma$-field generated by $W^{0}$, $W^{1}$,…,$W^{k}$. Clearly, $\{\Delta_{k}-\mathbb{E}(\Delta_{k}|\mathcal{F}_{k-1}),\mathcal{\mathcal{F}}_{k}\}_{1\leq k\leq n^{1+\alpha}}$ forms a martingale differences sequence and since 0\leq\text{\Delta}_{k}\leq C and thus $-C\leq\Delta_{k}-\mathbb{E}(\Delta_{k}|\mathcal{F}_{k-1})\leq C$, Hoeffding’s martingale inequality gives, for any $i<j$,

[TABLE]

Moreover, from (2.17),

[TABLE]

and therefore,

[TABLE]

For each $n\geq 1$, set

[TABLE]

where $\ell(n)\geq 0$ will be fixed later. Then, by (2.18),

[TABLE]

Next, by $\sum_{k=i+1}^{j}\Delta_{k}=L_{n}(j)-L_{n}(i)$ conditioned on $A_{n}^{k}$ for $k\in[i,j]$ and the very definitions of $\Delta_{k}$ and $O_{n}$,

[TABLE]

Therefore, combining (2.15) and (2.19) and letting $\ell(n)=c_{6}\sqrt{n^{2+\alpha-\delta}}$ gives

[TABLE]

Clearly, the right hand side of (2.20)converges, to [math], exponentially fast, as $n\rightarrow+\infty$, when $\alpha<2/3-\delta/2$ for any $\delta>0$. ∎

3 Proof of Theorem 1.1

The beginning of the proof is similar to a corresponding proof in [11]. For a random variable $U$ with finite $r$-th moment and for a random vector $V$, let $\mathbb{M}_{r}(U|V):=\mathbb{E}(|U-\mathbb{E}(U|V)^{r}|V$). Clearly, by convexity and the conditional Jensen’s inequality,

[TABLE]

and so, for any $n\geq 1$,

[TABLE]

Moreover, since $N$ is independent of $\left(L_{n}(k)\right)_{0\leq k\leq n^{1+\alpha}}$, and from (3.1), for each $\omega\in\Omega_{n}$,

[TABLE]

In addition (see [11, Lemma 2.2]), note that if $f:D\rightarrow\mathbb{Z}$ satisfies a local reversed Lipschitz condition, i.e., $f$ is such that for any $i,j\in D$ with $j>i+\ell$, $\ell\geq 0$, $f(j)-f(i)\geq c(j-i)$ for some $c>0$ and if $T$ is a $D$-valued random variable with $\mathbb{E}|f(T\}|^{r}<+\infty$, $r\geq 1$, then

[TABLE]

So, for each $\omega\in O_{n}$, since $N$ is independent of $\left(L_{n}(k)\right)_{0\leq k\leq n^{1+\alpha}}$,

[TABLE]

Next, (3.2), (3.3) and (3.4) lead to

[TABLE]

and it remains to estimate the first two terms on the right side of (3.5). By the Berry-Esséen Theorem, and for all $n\geq 1$,

[TABLE]

On the other hand,

[TABLE]

and when $n$ is large enough,

[TABLE]

where $F_{n}$ is the distribution function of $(N-n^{1+\alpha}p)/\sqrt{n^{1+\alpha}p(1-p)}$, while $\Phi$ is the one of the standard Gaussian. Likewise,

[TABLE]

Next, (3.7), (3.8) and (3.9) give

[TABLE]

For $\mathbb{M}_{r}\left(N|\boldsymbol{1}_{N\in I}=1\right)$ to dominate the first term $\mathbb{M}_{r}\left(N|\boldsymbol{1}_{N\in I}=1\right)-\ell(n)^{r}$ in (3.5), the constant $c_{1}$ (which depends on $r$ and $p$ but not $n$) is chosen such that:

[TABLE]

(Recall that $\ell(n)=c_{1}n^{(1+\alpha)/2}$). So,

[TABLE]

This last estimate combined with (3.6) and Lemma 2.5 gives

[TABLE]

4 Conclusions and Remarks

The major limitation of our method is the upper bound $1/3$ on $\alpha$, which stems from application of Hoeffding’s classical martingale exponential inequality. Specifically, we note there is some discrepancy between the orders of the upper and lower bounds for the martingale differences in (2.16) conditioned on the event $O_{n}$, i.e., the conditional lower bound is of order $o(n^{-\alpha})$ compared to the upper bound $o(1)$. With the existence of this discrepancy, it takes exactly $\alpha<1/3$ to have the exponential concentration hold. But a more sophisticated way of flipping weights from $lo$ mode to $hi$ mode in the construction of the martingale might be produced to mitigate this so as to relieve the $1/3$ bound. Or even better, a more powerful concentration inequality can be used to replace Hoeffding’s.

However, even if our method is generalizable to the case when $\alpha=1$, i.e., the grid is perfect square, the corresponding lower bound for the variance will be $O(n^{1-\alpha=1})=O(1)$ and thus not useful. Nevertheless, a well-known fact that geodesics in $DLPP$ are confined to a cylinder centered on the main diagonal of the grid and of width of order strictly smaller than $o(n)$ will help producing a non-trivial lower bound. The typical order of the width of the cylinder is the transversal fluctuation, which is believed to be $n^{2/3}$. Further, it is also believed that there is exponentially high probability that geodesics are confined to such kind of cylinder of width $o(n^{2/3+\epsilon})$, for $\epsilon>0$. Actually it has been proved that the transversal fluctuation exponent can be upper bounded by $3/4$ in the setting of undirected first passage percolation in [19] and an exponential concentration holds for all the geodesics in a cylinder of width $O(n^{(2\kappa+2)/(2\kappa+3)}\sqrt{\ln n})$ in [12] in the current setting, both of which assume the finite curvature exponent $\kappa$. This is equivalently to say that if let $\tilde{L}_{n}$ be the last passage time within the cylinder, then $\tilde{L}_{n}\geq L_{n}$ holds with exponentially high probability. So

[TABLE]

Meanwhile, it is trivial that $\tilde{L}_{n}\leq L_{n}$. So $\mathbb{E}\tilde{L}_{n}-\mathbb{E}L_{n}\rightarrow 0$ exponentially fast. This shows the potential of bounding the variance of $L_{n}$ by that of $\tilde{L}_{n}$. Indeed,

[TABLE]

Symmetrically, it is also true that $VarL_{n}\leq Var\tilde{L}_{n}+8n^{2}\mathbb{P}(\tilde{L}_{n}<L_{n})-\left(\mathbb{E}L_{n}-\mathbb{E}\tilde{L}_{n}\right)^{2}.$ So the variances of $L_{n}$ and $\tilde{L}_{n}$ share the same asymptotic order. On the other hand, our method here for the thin rectangle applies to the cylinder of the length $O(n)$ and the width $O(n^{\alpha})$ with slight modification. This will produce a power lower bound $n^{1-\alpha}$. Considering the best the scenario, if exponential concentration for the width $n^{2/3+\epsilon}$ for any $\epsilon>0$ can be proved, the corresponding power lower bound for longitudinal fluctuation will be $n^{1-2/3-\epsilon}=n^{1/3-\epsilon}$. Although this is still not the tight conjectured bound $n^{2/3}$, it still serves as a good power lower bound.