# Distance between configurations in Markov chain Monte Carlo simulations

## TL;DR

This paper introduces a universal distance measure between configurations in Markov chain Monte Carlo algorithms, demonstrating its properties and effects, especially in multimodal distributions, and revealing geometric insights.

## Contribution

It defines a universal distance for local move algorithms in MCMC, explicitly calculates it for Langevin, and explores its implications for multimodal distributions and geometry.

## Key findings

- Distance reduces significantly with tempering in multimodal distributions.
- The distance exhibits expected properties for local move algorithms.
- An anti-de Sitter-like geometry emerges in highly multimodal cases.

## Abstract

For a given Markov chain Monte Carlo algorithm we introduce a distance between two configurations that quantifies the difficulty of transition from one configuration to the other configuration. We argue that the distance takes a universal form for the class of algorithms which generate local moves in the configuration space. We explicitly calculate the distance for the Langevin algorithm, and show that it certainly has desired and expected properties as distance. We further show that the distance for a multimodal distribution gets dramatically reduced from a large value by the introduction of a tempering method. We also argue that, when the original distribution is highly multimodal with large number of degenerate vacua, an anti-de Sitter-like geometry naturally emerges in the extended configuration space.