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Thick points for intersections of planar sample paths. (English) Zbl 1007.60077
Summary: Let $$L_n^{X}(x)$$ denote the number of visits to $$x \in \mathbf{Z} ^2$$ of the simple planar random walk $$X$$, up to step $$n$$. Let $$X'$$ 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 $$L_n^{X}(x)L_n^{X'}(x)\geq b^2 (\log n)^4$$. This is the discrete counterpart of our main result, that for any $$a<1$$, the Hausdorff dimension of the set of thick intersection points $$x$$ for which $$\limsup_{r \rightarrow 0} \mathcal{I} (x,r)/(r^2|\log r|^4)=a^2$$, 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 until 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 $$K$$.

##### MSC:
 60J55 Local time and additive functionals 60J65 Brownian motion 28A80 Fractals 60G50 Sums of independent random variables; random walks
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##### References:
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