Optimal bounds for self-intersection local times

Abstract

For a random walk $S_n,n\ge 0$ 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 $\mathrm{var}(L_n(\alpha ))$ 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. $\mathrm{var}(L_n(\alpha ))\le C\mathrm{var}[L_n(\alpha |\epsilon ,d)]\sim K_{d,\alpha }{v}_{d,\alpha }(n)$. In particular, variances of local times of all genuinely $d$-dimensional random walks, $d\ge 4$, are similar to the $4$-dimensional symmetric case $\mathrm{var}(L_n(\alpha ))=O(n)$. On the other hand, in dimensions $d\le 3$ the resemblance to the simple random walk ${\mathrm{liminf}}_{n\to \infty }\mathrm{var}(L_n(\alpha ))/v_{d,\alpha }(n)>0$ implies that the jumps must have zero mean and finite second moment.

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