Optimal Bounds for the Variance of Self-Intersection Local Times

For a Zd-valued random walk (Sn)n N0, let l(n,x) be its local time at the site x Zd. For α N, define the α-fold self-intersection local time as Ln(α) xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var (LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var (Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var Lnα/var(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.