Optimal bounds for self-intersection local times

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

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