# Numerical simulation of polynomial-speed convergence phenomenon

## TL;DR

This paper introduces a hybrid numerical method combining coupling and renewal theory to estimate polynomial convergence and mixing rates of Markov processes, validated on heat conduction models.

## Contribution

It presents a novel hybrid approach that replaces difficult analytical estimates with numerical simulations for ergodicity analysis of Markov processes.

## Key findings

- Numerical results match expected polynomial mixing rates.
- Method effectively estimates convergence in complex Markov models.
- Applicable to heat conduction models with challenging analytical proofs.

## Abstract

We provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for Markov processes. The hybrid method that we introduce is based on the coupling technique and renewal theory. We propose to replace some estimates in classical results about the ergodicity of Markov processes by numerical simulations when the corresponding analytical proof is difficult. After that, all remaining conclusions can be derived from rigorous analysis. Then we apply our results to two 1D microscopic heat conduction models. The mixing rate of these two models are expected to be polynomial but very difficult to prove. In both examples, our numerical results match the expected polynomial mixing rate well.