Moderate Deviations for the Capacity of the Random Walk range in dimension four

TL;DR

This paper establishes a four-dimensional analog of moderate deviation results for the capacity of the random walk range, linking it to a constant in generalized Gagliardo-Nirenberg inequalities.

Contribution

It introduces a novel four-dimensional moderate deviation analysis for the capacity of the random walk, extending previous two-dimensional volume deviation results.

Findings

Capacity deviation statistics relate to Gagliardo-Nirenberg inequality constants.

Provides a new probabilistic interpretation of a functional inequality.

Extends known results from volume to capacity in four dimensions.

Abstract

In this paper, we find a natural four dimensional analog of the moderate deviation results for the capacity of the random walk, which corresponds to Bass, Chen and Rosen \cite{BCR} concerning the volume of the random walk range for $d=2$. We find that the deviation statistics of the capacity of the random walk can be related to the following constant of generalized Gagliardo-Nirenberg inequalities, \begin{equation*} \label{eq:maxineq} \inf_{f: \|\nabla f\|_{L^2}<\infty} \frac{\|f\|^{1/2}_{L^2} \|\nabla f\|^{1/2}_{L^2}}{ [\int_{(\mathbb{R}^4)^2} f^2(x) G(x-y) f^2(y) \text{d}x \text{d}y]^{1/4}}. \end{equation*}