Boundary of the range of transient random walk

Abstract

We study the boundary of the range of simple random walk on $\mathbb {Z}^d$ in the transient case $d\ge 3$. We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain an upper bound on the variance of order $n\log n$ in dimension three. We also establish a Central Limit Theorem in dimension four and larger.

Appendix: Estimates on ranges

In this section, we prove Proposition 1.5. We first introduce some other range-like sets allowing us to use the approach of Jain and Pruitt [9]. Recall that the sets $\mathcal {R}_{n,V}$ are disjoint, and for $U\subset V_0$, define

$$\begin{aligned} \overline{\mathcal {R}}_{n,U}:=\bigcup _{V\supset U} \mathcal {R}_{n,V}=\{S_k\ :\ S_k \notin {\mathcal R}_{k-1} \text { and }S_i\notin (S_k+U),\, i\le k-1,\, 1\le k\le n\}.\nonumber \\ \end{aligned}$$

(5.15)

Next for $U\subset V_0$, define

$$\begin{aligned} \alpha (U)=| \overline{\mathcal {R}}_{n,U}| - \mathbb {E}(| \overline{\mathcal {R}}_{n,U}|) \qquad \text {and}\qquad \beta (U)=| \mathcal {R}_{n,U}| -\mathbb {E}(| \mathcal {R}_{n,U}|). \end{aligned}$$

The definition (5.15) yields

$$\begin{aligned} \alpha (U)=\sum _{V\supset U} \beta (V), \end{aligned}$$

and this relation is inverted as follows:

$$\begin{aligned} \beta (V)=\sum _{U\supset V} (-1)^{|U\backslash V|}\, \alpha (U). \end{aligned}$$

As a consequence, for $V\subset V_0$,

$$\begin{aligned} \text {Var}(|\mathcal {R}_{n,V} |)=\mathbb {E}(\beta ^2(V))\le 2^{|V_0\backslash V|}\, \sum _{U\supset V} \mathbb {E}(\alpha ^2(U)). \end{aligned}$$

We will see below that each $\overline{\mathcal {R}}_{n,V}$ has the same law as a range-like functional that Jain and Pruitt analyze by using a last passage decomposition, after introducing some new variables. But let us give more details now. So first, we fix some $V\subset V_0$, and for $n\in \mathbb {N}$, set $Z_n^n=1$, and

$$\begin{aligned} \begin{array}{llll} Z_i &{}=&{} \mathbf{1}(\{S_{i+k}\not \in (S_i+\overline{V} )\quad \forall k\ge 1\}) &{}\quad \forall i\in \mathbb {N},\\ Z_i^n &{}=&{} \mathbf{1}(\{S_{i+k}\not \in (S_i+\overline{V}) \quad \forall k=1,\dots ,n-i\}) &{}\quad \forall i<n \\ W_i^n &{}=&{}Z^n_i-Z_i &{}\quad \forall i\le n, \end{array} \end{aligned}$$

where

$$\begin{aligned} \overline{V}= V\cup \{0\}. \end{aligned}$$

A key point in this decomposition is that $Z_n$ and $Z^n_i$ are independent. Now, define

$$\begin{aligned} \underline{\mathcal {R}}_{n,V}=\{S_k:\ S_i\not \in S_k+\overline{V},\ n\ge i>k,\ 0\le k< n\}, \quad \text {and}\quad |\underline{\mathcal {R}}_{n,V}|=\sum _{i=0}^{n-1} Z^n_i.\nonumber \\ \end{aligned}$$

(5.16)

Since the increments are symmetric and independent, $|\overline{\mathcal {R}}_{n,V}|$ and $|\underline{\mathcal {R}}_{n,V}|$ are equal in law. Now, equality (5.16) reads as

$$\begin{aligned} |\underline{\mathcal {R}}_{n,V}|=\sum _{i=0}^{n-1} Z_i+\sum _{i=0}^{n-1} W^n_i. \end{aligned}$$

Now using that Var$(|\overline{\mathcal {R}}_{n,V}|)\le \mathbb {E}[( |\overline{\mathcal {R}}_{n,V}|-\sum _{i\le n-1} \mathbb {E}[Z_i])^2]$, and that $(a+b)^2\le 2(a^2+b^2)$ we obtain

$$\begin{aligned} \text {Var}(|\underline{\mathcal {R}}_{n,V}|) \le 2\sum _{i=1}^{n-1} \text {Var}(Z_i)+ 4\sum _{j=1}^{n-1}\sum _{i=0}^{j-1}\text {Cov}(Z_i,Z_j)+ 4\sum _{j=1}^{n-1}\sum _{i=0}^j \mathbb {E}(W^n_iW^n_j).\nonumber \\ \end{aligned}$$

(5.17)

Next for $i<j< n$, we have (recall the definition (2.1))

$$\begin{aligned} \mathbb {E}\left( W^n_i\, W^n_j\right)= & {} \mathbb {P}\left( n<H^{(i+1)}_{S_i+\overline{V}}<\infty ,\, n<H^{(j+1)}_{S_j+\overline{V}}<\infty \right) \\= & {} \sum _{x\notin \overline{V}}\mathbb {P}(S_{j-i}=x,\ H_{\overline{V}}>j-i)\, \mathbb {P}_x( n-j<H_{\overline{V}}\\&\quad<\infty ,\ n-j<H_{x+\overline{V}}<\infty )\\\le & {} \sum _{x\not \in V}\mathbb {P}(S_{j-i}=x,\ H_{\overline{V}}>j-i)\, \mathbb {P}_x( n-j<H_{\overline{V}}\\&\quad<\infty ,\ n-j<H_{x+\overline{V}}<\infty ), \end{aligned}$$

where for the second equality we just used the Markov property and translation invariance of the walk. The last inequality is written to cover as well the case $i=j$. Therefore,

$$\begin{aligned} \sum _{i=0}^j \mathbb {E}\left( W^n_iW^n_j\right)\le & {} \sum _{x\not \in V} G_j(0,x) \, \mathbb {P}_x(H_{\overline{V}}<\infty ,\ n-j<H_{x+\overline{V}}<\infty ) \\\le & {} \sum _{y,z\in \overline{V}}\, \sum _{x\notin V}\, G_j(0,x)\, \mathbb {P}_x(H_y<\infty ,\ n-j<H_{x+z}<\infty ). \end{aligned}$$

Then Lemma 4 of [9] shows that

$$\begin{aligned} \sum _{i=0}^j \mathbb {E}\left( W^n_iW^n_j\right) = \left\{ \begin{array}{l@{\quad }l} \mathcal {O}\left( \sqrt{\frac{j}{n-j}}\right) &{} \text {if }d=3\\ \mathcal {O}\left( \frac{\log j}{n-j}\right) &{} \text {if }d=4\\ \mathcal {O}\left( (n-j)^{1-d/2}\right) &{} \text {if }d\ge 5, \end{array} \right. \end{aligned}$$

and thus

$$\begin{aligned} \sum _{j=1}^{n-1}\sum _{i=0}^j \mathbb {E}\left( W^n_iW^n_j\right) = \left\{ \begin{array}{l@{\quad }l} \mathcal {O}(n) &{} \text {if }d=3\\ \mathcal {O}((\log n)^2) &{} \text {if }d=4\\ \mathcal {O}(1) &{} \text {if }d\ge 5. \end{array} \right. \end{aligned}$$

(5.18)

Now, for $i<j<n$, by using that $Z_i^j$ and $Z_j$ are independent, we get

$$\begin{aligned} \text {Cov}(Z_i,Z_j)=-\text {Cov}(W^j_i, Z_j). \end{aligned}$$

On the other hand, assuming $i<j\le n$,

$$\begin{aligned} \mathbb {E}(W^j_iZ_j)= & {} \mathbb {P}\left( j<H^{(i+1)}_{S_i+\overline{V}}<\infty ,\, H^{(j+1)}_{S_j+\overline{V}}=\infty \right) \\= & {} \sum _{x\not \in \overline{V}}\, \mathbb {P}(S_{j-i}=x,\, H_{\overline{V}}>j-i)\, \mathbb {P}_x(H_{\overline{V}}<\infty ,\, H_{x+\overline{V}}=\infty ). \end{aligned}$$

Since in addition,

$$\begin{aligned} \mathbb {E}(Z_j)=\mathbb {P}_x(H_{x+\overline{V}}=\infty )\quad \text {for all }x, \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}(W_i^j)= \sum _{x\notin \overline{V}} \, \mathbb {P}( S_{j-i} =x,\, H_{\overline{V}}>j-i)\, \mathbb {P}_x(H_{\overline{V}}<\infty ), \end{aligned}$$

we deduce that

$$\begin{aligned} \text {Cov}(Z_i,Z_j)=\sum _{x\not \in \overline{V}}\mathbb {P}(S_{j-i}=x,\, H_{\overline{V}}>j-i)\, b_V(x), \end{aligned}$$

with

$$\begin{aligned} b_V(x):=\mathbb {P}_x(H_{\overline{V}}<\infty )\, \mathbb {P}_x(H_{x+\overline{V}}=\infty )- \mathbb {P}_x(H_{\overline{V}}<\infty ,\ H_{x+\overline{V}}=\infty ).\nonumber \\ \end{aligned}$$

(5.19)

Now we need the following equivalent of Lemma 5 of [9].

Lemma 5.4

For any $V\subset V_0$, and $x\notin \overline{V}$,

$$\begin{aligned} b_V(x)=\mathbb {P}_x(H_{\overline{V}}<H_{x+\overline{V}}<\infty )\mathbb {P}_x(H_{\overline{V}}=\infty )+\mathcal {E}(x,V), \end{aligned}$$

with

$$\begin{aligned} \mathcal {E}(x,V):=\sum _{z\in x+\overline{V}}\mathbb {P}_x(S_{H_{x+\overline{V}}}=z,\, H_{x+\overline{V}}<H_{\overline{V}}) (\mathbb {P}_z(H_{\overline{V}}<\infty )-\mathbb {P}_x(H_{\overline{V}}<\infty )). \end{aligned}$$

Moreover,

$$\begin{aligned} |\mathcal {E}(x,V)|= \mathcal {O}\left( \frac{1}{\Vert x\Vert ^{d-1}}\right) . \end{aligned}$$

Assuming this lemma for a moment, we get

$$\begin{aligned} a_j= & {} \sum _{i=0}^{j-1}\, \text {Cov}(Z_i,Z_j)=\sum _{i=0}^{j-1} \sum _{x\not \in \overline{V}}\, \mathbb {P}(S_{j-i}=x,\, H_{\overline{V}}>j-i)\, b_V(x)\\= & {} \mathcal {O}\left( \sum _{x\notin \overline{V}} \frac{G_{j}(0,x)}{\Vert x\Vert ^{d-1}}\right) = \mathcal {O}\left( \sum _{1\le \Vert x\Vert \le j} \frac{1}{\Vert x\Vert ^{2d-3}}\right) \\= & {} \left\{ \begin{array}{l@{\quad }l} \mathcal {O}(\log j)&{} \text {if }d=3\\ \mathcal {O}(1) &{} \text {if }d\ge 4, \end{array} \right. \end{aligned}$$

from which we deduce that

$$\begin{aligned} \sum _{j=0}^{n-1} a_j = \left\{ \begin{array}{l@{\quad }l} \mathcal {O}(n \log n)&{} \text {if }d=3\\ \mathcal {O}(n) &{} \text {if }d\ge 4. \end{array} \right. \end{aligned}$$

(5.20)

Then Proposition 1.5 follows from (5.17), (5.18) and (5.20).

Proof of Lemma 5.4

Note first that

$$\begin{aligned} b_V(x)= & {} \mathbb {P}_x(H_{\overline{V}}<\infty ,\ H_{x+\overline{V}}<\infty )- \mathbb {P}_x(H_{\overline{V}}<\infty )\mathbb {P}_x(H_{x+\overline{V}}<\infty )\\= & {} \mathbb {P}_x(H_{\overline{V}}<H_{x+\overline{V}}<\infty )+ \mathbb {P}_x(H_{x+\overline{V}}<H_{\overline{V}}<\infty )\\&-\,\mathbb {P}_x(H_{\overline{V}}<\infty )\mathbb {P}_x(H_{x+\overline{V}}<\infty ). \end{aligned}$$

Moreover,

$$\begin{aligned} \mathbb {P}_x(H_{x+\overline{V}}<H_{\overline{V}}<\infty )= & {} \sum _{z\in x+\overline{V}} \mathbb {P}_x(S_{H_{x+\overline{V}}}=z,\ H_{x+\overline{V}}<H_{\overline{V}})\mathbb {P}_z(H_{\overline{V}}<\infty )\\= & {} \mathbb {P}_x(H_{x+\overline{V}}<H_{\overline{V}})\mathbb {P}_x(H_{\overline{V}}<\infty )+\mathcal {E}(x,V). \end{aligned}$$

The first assertion of the lemma follows. The last assertion is then a direct consequence of standard asymptotics on the gradient of the Green’s function (see for instance [10, Corollary 4.3.3]). $\square$

Remark 5.5

By adapting the argument in [9] we could also prove that in dimension 3, $\text {Var}(|\underline{\mathcal {R}}_{n,V}|) \sim \sigma ^2 n \log n$, for some constant $\sigma >0$, and then obtain a central limit theorem for this modified range. However it is not clear how to deduce from it an analogous result for $|\mathcal {R}_{n,V}|$, which would be useful in view of a potential application to the boundary of the range.