One-dimensional Monte Carlo dynamics at zero temperature

TL;DR

This paper analyzes a universal zero-temperature Monte Carlo dynamics for a random walker in continuous space, revealing scaling properties and independence from potential landscape details, supported by analytical and numerical results.

Contribution

It introduces a universal zero-temperature Monte Carlo dynamics that is independent of the potential landscape and jump distribution details, with analytical and numerical validation.

Findings

Dynamics is universal and landscape-independent.

Scaling properties of the walker probability density are characterized.

Analytical predictions match Monte Carlo simulations.

Abstract

We investigate, both analytically and with numerical simulations, a Monte Carlo dynamics at zero temperature, where a random walker evolving in continuous space and discrete time seeks to minimize its potential energy, by decreasing this quantity at each jump. The resulting dynamics is universal in the sense that it does not depend on the underlying potential energy landscape, as long as it admits a unique minimum; furthermore, the long time regime does not depend on the details of the jump distribution, but only on its behaviour for small jumps. We work out the scaling properties of this dynamics, as embodied by the walker probability density. Our analytical predictions are in excellent agreement with direct Monte Carlo simulations.