Capacity of the range of random walk: The law of the iterated logarithm

arXiv2208.02184MaRDI QIDQ6506375

Amir Dembo, Izumi Okada

Abstract: We establish both the $limsup$ and the $liminf$ law of the iterated logarithm LIL, for the capacity of the range of a simple random walk in any dimension $dge3$. While for $dge4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk.

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