Thick points for intersections of planar sample paths

Abstract

Let $L_n^X(x)$ denote the number of visits to $x\in {\mathbf{Z}}^2$ of the simple planar random walk $X$, up till step $n$. Let $X^{\prime }$ be another simple planar random walk independent of $X$. We show that for any $0<b<1/(2\pi )$, there are $n^{1-2\pi b+o(1)}$ points $x\in {\mathbf{Z}}^2$ for which $L_n^X(x)L_n^{X^{\prime }}(x)\ge b^2(\log n{)}^4$. This is the discrete counterpart of our main result, that for any $a<1$, the Hausdorff dimension of the set of {\it thick intersection points} $x$ for which ${\mathrm{limsup}}_{r\to 0}\mathcal{I}(x,r)/(r^2|\log r{|}^4)=a^2$, is almost surely $2-2a$. Here $\mathcal{I}(x,r)$ is the projected intersection local time measure of the disc of radius $r$ centered at $x$ for two independent planar Brownian motions run till time 1. The proofs rely on a `multi-scale refinement' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius $r$ centered at $x$ by $x+rK$ for general sets $K$.