Convergence rates of self-repellent random walks, their local time and Event Chain Monte Carlo

TL;DR

This paper analyzes the convergence rates of self-repellent random walks and their local times, revealing bounds that support the efficiency of Event Chain Monte Carlo algorithms over traditional methods.

Contribution

It establishes new lower and upper bounds on the relaxation time of self-repellent random walks and links these results to the performance of non-reversible MCMC methods.

Findings

Lower bound on relaxation time:  n^{3/2}

Upper bound on modified dynamics:  n^2

Event Chain Monte Carlo outperforms traditional MCMC methods

Abstract

We study the rate of convergence to equilibrium of the self-repellent random walk and its local time process on the discrete circle $\mathbb{Z}_n$. While the self-repellent random walk alone is non-Markovian since the jump rates depend on its history via its local time, jointly considering the evolution of the local time profile and the position yields a piecewise deterministic, non-reversible Markov process. We show that this joint process can be interpreted as a second-order lift of a reversible diffusion process, the discrete stochastic heat equation with Gaussian invariant measure. In particular, we obtain a lower bound on the relaxation time of order $\Omega(n^{3/2})$. Using a flow Poincar\'e inequality, we prove an upper bound for a slightly modified dynamics of order $O(n^2)$, matching recent conjectures in the physics literature. Furthermore, since the self-repellent random walk and its local time process coincide with the Event Chain Monte Carlo algorithm for the harmonic chain, a non-reversible MCMC method, we demonstrate that the relaxation time bound confirms the recent empirical observation that Event Chain Monte Carlo algorithms can outperform traditional MCMC methods such as Hamiltonian Monte Carlo.