Numerical simulation of polynomial-speed convergence phenomenon
This work provides a practical approach for studying polynomial ergodicity in Markov processes where analytical proofs are difficult, benefiting researchers in statistical physics and stochastic processes.
The authors introduce a hybrid method combining coupling and renewal theory with numerical simulation to analyze polynomial-speed convergence and mixing in Markov processes, applying it to two 1D heat conduction models where polynomial mixing rates are expected but hard to prove analytically, with numerical results matching expectations.
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.