On the spectral gap of the Kac walk and other binary collision processes

TL;DR

This paper introduces a simple method to compute the spectral gap of the Kac walk and similar binary collision processes, reducing complex N-component analysis to a manageable N=3 case, applicable to various models.

Contribution

The paper presents an elementary approach to analyze the spectral gap, extending to non-homogeneous and disordered binary collision processes.

Findings

Spectral gap of the Kac walk on the N-sphere computed explicitly.

Reduction of spectral gap analysis from N components to N=3 components.

Method applicable to a range of binary collision processes, including disordered systems.

Abstract

We give a new and elementary computation of the spectral gap of the Kac walk on the N-sphere. The result is obtained as a by-product of a more general observation which allows to reduce the analysis of the spectral gap of an N-component system to that of the same system for N=3. The method applies to a number of random 'binary collision' processes with complete-graph structure, including non-homogeneous examples such as exclusion and colored exclusion processes with site disorder.