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

2025-11-28

AI Summary

This work studies the convergence rate to equilibrium of self-repelling random walks (SRW) and their local time processes on discrete cycles. Methodologically, it models the joint evolution as a nonreversible piecewise-deterministic Markov process (PDMP), thereby uncovering, for the first time, its structural identity as a second-order lift of a discrete stochastic heat equation and establishing a rigorous theoretical link between nonreversible MCMC and diffusion limits. Combining local time analysis, manifold Poincaré inequalities, and spectral analysis of the Gaussian invariant measure, the authors derive a tight lower bound Ω(n^{3/2}) on the relaxation time and an O(n^2) upper bound for an improved dynamics. Both theoretical analysis and numerical experiments consistently demonstrate that Event Chain Monte Carlo converges significantly faster than conventional reversible MCMC methods (e.g., Hamiltonian Monte Carlo), providing critical convergence guarantees and a novel analytical framework for nonreversible sampling algorithms.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

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 $Ω(n^{3/2})$. Using a flow Poincaré 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.

Problem

• Analyzing convergence rates of self-repellent random walks and local time processes
• Establishing relaxation time bounds for non-reversible Markov processes
• Comparing Event Chain Monte Carlo efficiency with traditional MCMC methods

Innovation

• Joint process modeling for non-Markovian systems
• Second-order lift of reversible diffusion process
• Flow Poincaré inequality for convergence bounds