Biased Random Walk Conditioned on Survival Among Bernoulli Obstacles: Subcritical Phase

Abstract

We consider a discrete time biased random walk conditioned to avoid Bernoulli obstacles on ${\mathbb {Z}}^d$ ($d\ge 2$) up to time N. This model is known to undergo a phase transition: for a large bias, the walk is ballistic whereas for a small bias, it is sub-ballistic. We prove that in the sub-ballistic phase, the random walk is contained in a ball of radius $O(N^{1/(d+2)})$, which is the same scale as for the unbiased case. As an intermediate step, we also prove large deviation principles for the endpoint distribution for the unbiased random walk at scales between $N^{1/(d+2)}$ and $o(N^{d/(d+2)})$. These results improve and complement earlier work by Sznitman (Ann Sci École Norm Sup (4), 28(3):345–370, 371–390, 1995).

A. Proof of Lemmas 5.2 and 5.4 by the Method of Enlargement of Obstacles

If the reader is familiar with the method of enlargement of obstacles developed in [30, 31], or its discrete version in [1], then it is rather easy to prove Lemmas 5.2 and 5.4 by adapting the proofs of [24, Proposition 2] and [24, Lemma 1], respectively. In fact, a statement similar to Lemma 5.2 appeared in [29, Remark 1.4].

In [24], the method of enlargement of obstacles is used to construct a certain set $\mathscr {U}_\mathrm{cl}$ which is almost free from obstacles and also any point outside is well surrounded by obstacles. We shall recall these properties more precisely in the proofs.

In this alternative argument, we need to change the exponent 1 / 5 in (3.1) to another $\chi >0$ depending only on d and p, to be determined later:

$$\begin{aligned} \delta _{N,x}=\varrho _N^{-\chi }\vee ({|x|}/{\varrho _N^{d}}). \end{aligned}$$

(A.1)

Remark A.1

When one compares the following argument with that in [24], it is important to keep in mind that space is scaled by the factor $\varrho _N^{-1}$ in [24].

Proof of Lemma 5.2

It is shown in [24, (46), (49), (52) and (56)] that there exist $c_4>1$, $\alpha _1,\alpha _2>0$ and a random set $\mathscr {U}_\mathrm{cl}\subset B\left( 0;2N\right)$ such that

$$\begin{aligned} \begin{aligned}&{\mathbb {P}}\otimes \mathbf{P } \left( |\mathscr {U}_\mathrm{cl}|\log \tfrac{1}{p}+N\lambda _{\mathscr {U}_\mathrm{cl}}>N^{\frac{d}{d+2}}\left( c(d,p)+c_4N^{-\frac{\alpha _1}{d+2}}\right) , \tau _\mathcal{O}>N\right) \\&\quad \le \exp \left\{ -N^{\frac{d}{d+2}}\left( c(d,p)+{c_4}N^{-\frac{\alpha _1}{d+2}}\right) +N^{\frac{d-\alpha _1}{d+2}}\right\} , \end{aligned} \end{aligned}$$

(A.2)

and

$$\begin{aligned} {\mathbb {P}}\otimes \mathbf{P } \left( |\mathcal{O}\cap \mathscr {U}_\mathrm{cl}| \ge t^{\frac{d-\alpha _2}{d+2}}\right) \le \exp \left\{ -\tfrac{3}{2} c(d,p) N^{\frac{d}{d+2}}\right\} . \end{aligned}$$

(A.3)

(The parameter $\alpha _2$ corresponds to $\kappa -d\alpha _0$ in [24].) In (A.2), the term $c_4N^{-\frac{\alpha _1}{d+2}}$ in fact directly inherits from the first line to the second, and this bound remains valid if we replace it by $2\beta \left( \mathbf{e } _x\right) \delta _{N,x}$, and $\tau _\mathcal{O}>N$ by $\tau _\mathcal{O}>\tau _x^N$, if we choose $\chi <\alpha _1$ in (A.1). Then we can repeat the argument in [24, Proposition 2] to see that on the event

$$\begin{aligned} \left\{ |\mathscr {U}_\mathrm{cl}|\log \tfrac{1}{p}+N\lambda _{\mathscr {U}_\mathrm{cl}} \le N^{\frac{d}{d+2}}\left( c(d,p)+2\beta \left( \mathbf{e } _x\right) \delta _{N,x}\right) \right\} , \end{aligned}$$

(A.4)

there exists a ball and $c_5>0$ such that

(A.5)

by the quantitative Faber–Krahn inequality in [4].

Remark A.2

As in Sect. 5.1, we need to enlarge $\mathscr {U}_\mathrm{cl}$ to $\varvec{\mathscr {U}}_\mathrm{cl}^+$ in order to apply the quantitative Faber–Krahn inequality in [4]. But we do not need to worry about the increase of volume since $\mathscr {U}_\mathrm{cl}$ is defined as a union of cubes of the form $\varrho _N^{1-\gamma } (q+[0,1)^d)$ with $\gamma \in (0,1)$ and $q\in {\mathbb {Z}}^d$.

Combining the above consideration with (4.2) and (A.3), we conclude that if $c<\frac{1}{32}\wedge \frac{\alpha _2}{\chi }$ so that

$$\begin{aligned} \delta _{N,x}^c\varrho _N^d \ge c_5(2\beta \left( \mathbf{e } _x\right) \delta _{N,x})^{\frac{1}{32}}\varrho _N^d +N^{\frac{d-\alpha _2}{d+2}} \end{aligned}$$

(A.6)

for sufficiently large N, then

(A.7)

which implies (5.3). $\quad \square$

Proof of Lemma 5.4

By (A.7), it suffices to prove that on the event (A.5), there exists $c>0$ such that

$$\begin{aligned} \lambda _{B\left( 0;2N\right) {\setminus } (B^-\cup \mathcal{O})} \ge \delta _{N,x}^{-c}\varrho _N^{-2}. \end{aligned}$$

(A.8)

Indeed, this and a well-known semigroup bound [17, (2.21)] imply that when N is sufficiently large, for any $y\in {\mathbb {Z}}^d$ and $\delta _{N,x}^{-c/3} \varrho _N^{-2}\le t\le 2N$,

$$\begin{aligned} \begin{aligned} \mathbf{P } _y\left( S_{[0,t]}\cap \left( \mathcal{O}\cup B^-\right) =\emptyset \right)&\le |B\left( 0;2N\right) |^{1/2}\left( 1-\delta _{N,x}^{-c} \varrho _N^{-2}\right) ^t\\&\le \exp \left\{ -\delta _{N,x}^{-c/2} \varrho _N^{-2} t\right\} . \end{aligned} \end{aligned}$$

(A.9)

The proof of (A.8) is almost identical to that of [24, Lemma 1]. We only recall a brief outline of the argument. The key element is [31, Proposition 2.4 on pp.160–161], which roughly says that for a set $U\subset {\mathbb {Z}}^d$, the eigenvalue $\lambda _{U{\setminus }\mathcal{O}}$ is large if the clearing set $\mathscr {U}_\mathrm{cl}$ is locally thin in the following sense:

$$\begin{aligned} \left| U\cap \mathscr {U}_\mathrm{cl} \cap \varrho _N(q+[0,1]^d)\right| =o(\varrho _N^d)\text { for any }q\in {\mathbb {Z}}^d. \end{aligned}$$

(A.10)

On (A.5), we know that $\mathscr {U}_\mathrm{cl}$ largely coincides with $B^-$ and hence $B\left( 0;2N\right) {\setminus } (B^-\cup \mathcal{O})$ is locally thin in the above sense. $\quad \square$