Behavior of the Poincar{\'e} constant along the Polchinski renormalization flow

TL;DR

This paper investigates how the Poincaré constant evolves along the Polchinski renormalization flow using advanced calculus techniques, extending previous methods to higher eigenvalues and broader measures.

Contribution

It introduces a dynamic $ ext{Gamma}$-calculus approach to analyze the Poincaré constant's behavior and generalizes existing methods to higher-order eigenvalues and $ ext{Φ}^4$-measures.

Findings

Controlled the Poincaré constant along the flow

Extended analysis to higher eigenvalues

Applied method to $ ext{Φ}^4$-measures and transport maps

Abstract

We control the behavior of the Poincar{\'e} constant along the Polchinski renormalization flow using a dynamic version of $\Gamma$-calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by B. Klartag and E. Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general $\Phi$ 4-measures and discuss the interpretation in terms of transport maps.