Adiabatic times for Markov chains and applications

TL;DR

This paper establishes a generalized adiabatic theorem for Markov chains, demonstrating how slowly changing Markov dynamics can be approximated, with applications to Glauber dynamics in statistical physics models.

Contribution

It introduces a new adiabatic theorem for time-inhomogeneous Markov chains, extending concepts similar to quantum adiabatic theorems, with practical applications.

Findings

Proves a generalized adiabatic theorem for Markov chains.

Provides examples related to Glauber dynamics of the Ising model.

Shows how adiabatic approximations apply to inhomogeneous Markov processes.

Abstract

We state and prove a generalized adiabatic theorem for Markov chains and provide examples and applications related to Glauber dynamics of Ising model over Z^d/nZ^d. The theorems derived in this paper describe a type of adiabatic dynamics for l^1(R_+^n) norm preserving, time inhomogeneous Markov transformations, while quantum adiabatic theorems deal with l^2(C^n) norm preserving ones, i.e. gradually changing unitary dynamics in C^n.