# The $L^2$-cutoffs for reversible Markov chains

## TL;DR

This paper introduces a new approach to analyze the cutoff phenomenon in reversible Markov chains using Laplace transforms, simplifying criteria and enabling comparison between discrete and continuous time chains.

## Contribution

It provides a novel pathway to study $L^2$-cutoffs, simplifying existing criteria and extending analysis to product chains and chain comparisons.

## Key findings

- Derived simplified criteria for $L^2$-cutoffs
- Established equivalence of cutoffs in product chains
- Compared cutoffs between discrete and continuous time chains

## Abstract

In this article, we considers reversible Markov chains of which $L^2$-distances can be expressed in terms of Laplace transforms. The cutoff of Laplace transforms was first discussed by Chen and Saloff-Coste in [8], while we provide here a completely different pathway to analyze the $L^2$-distance. Consequently, we obtain several considerably simplified criteria and this allows us to proceed advanced theoretical studies, including the comparison of cutoffs between discrete time lazy chains and continuous time chains. For an illustration, we consider product chains, a rather complicated model which could be involved to analyze using the method in [8], and derive the equivalence of their $L^2$-cutoffs.