Irreversible Markov chain Monte Carlo algorithm for self-avoiding walk

TL;DR

This paper introduces an irreversible Markov chain Monte Carlo algorithm for the self-avoiding walk that outperforms traditional methods, especially in higher dimensions, and provides precise estimates of critical points and scaling exponents.

Contribution

The paper develops a novel irreversible MCMC algorithm for SAW that violates detailed balance, leading to significantly improved efficiency over existing algorithms across multiple dimensions.

Findings

Performance increases with dimension, up to 40 times faster in 5D.

Accurate determination of the critical point in 5D SAW.

Finite-size scaling governed by renormalized exponents.

Abstract

We formulate an irreversible Markov chain Monte Carlo algorithm for the self-avoiding walk (SAW), which violates the detailed balance condition and satisfies the balance condition. Its performance improves significantly compared to that of the Berretti-Sokal algorithm, which is a variant of the Metropolis-Hastings method. The gained efficiency increases with the spatial dimension (D), from approximately $10$ times in 2D to approximately $40$ times in 5D. We simulate the SAW on a 5D hypercubic lattice with periodic boundary conditions, for a system with a linear size up to $L=128$, and confirm that as for the 5D Ising model, the finite-size scaling of the SAW is governed by renormalized exponents $\nu^*=2/d$ and $\gamma/\nu^*=d/2$. The critical point is determined, which is approximately $8$ times more precise than the best available estimate.