Optimal bounds for self-intersection local times

TL;DR

This paper establishes optimal bounds for the variances of self-intersection local times of random walks in various dimensions, showing they are comparable to those of simple symmetric random walks and deriving implications for jump distributions.

Contribution

It provides the first bounds on variances of local times for general random walks, linking their behavior to simple symmetric cases and identifying conditions on jumps in low dimensions.

Findings

Variances of local times are bounded by those of simple symmetric random walks.

In dimensions d≥4, variances grow linearly with n.

In dimensions d≤3, jumps must have zero mean and finite second moment.

Abstract

For a random walk $S_n, n\geq 0$ in $\mathbb{Z}^d$, let $l(n,x)$ be its local time at the site $x\in \mathbb{Z}^d$. Define the $\alpha$-fold self intersection local time $L_n(\alpha) := \sum_{x} l(n,x)^{\alpha}$, and let $L_n(\alpha|\epsilon, d)$ the corresponding quantity for $d$-dimensional simple random walk. Without imposing any moment conditions, we show that the variances of the local times $\mathop{var}(L_n(\alpha))$ of any genuinely $d$-dimensional random walk are bounded above by the corresponding characteristics of the simple symmetric random walk in $\mathbb{Z}^d$, i.e. $\mathop{var}(L_n(\alpha)) \leq C \mathop{var}[L_n(\alpha|\epsilon, d)]\sim K_{d,\alpha}v_{d,\alpha}(n)$. In particular, variances of local times of all genuinely $d$-dimensional random walks, $d\geq 4$, are similar to the $4$-dimensional symmetric case $\mathop{var}(L_n(\alpha)) = O(n)$. On the other hand, in dimensions $d\leq 3$ the resemblance to the simple random walk $\liminf_{n\to \infty} \mathop{var}(L_n(\alpha))/v_{d,\alpha}(n)>0$ implies that the jumps must have zero mean and finite second moment.