A lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles

TL;DR

This paper establishes a meaningful lower bound for the spectral gap of a three-particle conjugate Kac process, combining theoretical proof with Monte Carlo simulations to provide a comprehensive quantitative estimate.

Contribution

It provides the first rigorous lower bound for the spectral gap of the conjugate Kac process with three particles, complemented by empirical estimates.

Findings

Spectral gap lower bound around 0.02

Monte Carlo simulations improve empirical estimate

Complete quantitative spectral gap estimate achieved

Abstract

In this paper, we proceed as suggested in the final section of arXiv:1812.03874v2 and prove a lower bound for the spectral gap of the conjugate Kac process with 3 interacting particles. This bound turns out to be around $0.02$, which is already physically meaningful, and we perform Monte Carlo simulations to provide a better empirical estimate for this value via entropy production inequalities. This finishes a complete quantitative estimate of the spectral gap of the Kac process.