Explicit bounds for a Gaussian decomposition Lemma of Sellke

Tobias Schmidt

Explicit bounds for a Gaussian decomposition Lemma of Sellke

Tobias Schmidt

Abstract

In arXiv:2212.14023 a decomposition of Gaussian measures on finite-dimensional spaces was introduced, which turned out to be a central technical tool to improve currently known bounds on a long standing conjecture in statistical mechanics called the Polaron problem. This note slightly generalizes this decomposition and provides numerical values for all occurring constants.

Explicit bounds for a Gaussian decomposition Lemma of Sellke

Abstract

Paper Structure (1 theorem, 36 equations)

This paper contains 1 theorem, 36 equations.

Key Result

Lemma 1

Let $\mu$ be a centered Gaussian measure on $(\Omega$,$\mathcal{B}(\Omega))$. Furthermore, let $K \in \mathcal{B}(\Omega)$ be a symmetric, closed convex set with $\mu(K^c) \leq \delta < 0.5$ and Then, there exist two probability measures $\nu_{good}, \nu_{bad}$ and $\delta'>0$ such that Moreover, for $n > 1$, $C:= 32 \frac{n^2}{n^2 -1}$ it holds that

Theorems & Definitions (2)

Lemma 1
proof : Proof of Lemma \ref{['31_thm']}