Logarithmic Sobolev inequalities for mollified compactly supported measures

TL;DR

This paper proves that convolving a compactly supported measure with a Gaussian yields a measure satisfying a logarithmic Sobolev inequality, providing new insights into eigenvalue distributions in random matrix theory and exploring extensions to higher dimensions.

Contribution

It establishes that convolution with a Gaussian ensures LSIs for compactly supported measures and offers a new proof for eigenvalue convergence in random matrices, with analysis of optimal constants and higher-dimensional extensions.

Findings

Convolution with Gaussian satisfies LSIs for compactly supported measures.

New proof of eigenvalue distribution convergence in random matrix theory.

Partial results on extending LSIs to higher dimensions.

Abstract

We show that the convolution of a compactly supported measure on $\mathbb{R}$ with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). We use this result to give a new proof of a classical result in random matrix theory that states that, under certain hypotheses, the empirical law of eigenvalues of a sequence of random real symmetric matrices converges weakly in probability to its mean. We then examine the optimal constants in the LSIs for the convolved measures in terms of the variance of the convolving Gaussian. We conclude with partial results on the extension of our main theorem to higher dimensions.