Emergence of AdS geometry in the simulated tempering algorithm

TL;DR

This paper demonstrates that the simulated tempering algorithm can induce an emergent anti-de Sitter geometry in the extended configuration space of complex stochastic systems with multimodal distributions.

Contribution

It introduces a geometric perspective to Markov chain Monte Carlo algorithms, revealing the emergence of AdS geometry during simulated tempering of multimodal systems.

Findings

Anti-de Sitter geometry emerges in the extended configuration space.

Optimal tempering minimizes distances and reveals geometric structure.

The approach links stochastic dynamics with geometric and holographic concepts.

Abstract

In our previous work [1], we introduced to an arbitrary Markov chain Monte Carlo algorithm a distance between configurations. This measures the difficulty of transition from one configuration to the other, and enables us to investigate the relaxation of probability distribution from a geometrical point of view. In this paper, we investigate the geometry of stochastic systems whose equilibrium distributions are highly multimodal with a large number of degenerate vacua. Implementing the simulated tempering algorithm to such a system, we show that an asymptotically Euclidean anti-de Sitter geometry emerges with a horizon in the extended configuration space when the tempering parameter is optimized such that distances get minimized.