Distance between configurations in Markov chain Monte Carlo simulations

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.

20 pages, 3 figures. v2: typos corrected, references and a figure added, a few additional explanations made. v3: discussions in section 4 improved

References in corpus (6)

• Toward solving the sign problem with path optimization method
• Parallel tempering algorithm for integration over Lefschetz thimbles
• Tempered transitions between thimbles
• Does the complex Langevin method give unbiased results?
• Combining the complex Langevin method and the generalized Lefschetz-thimble method
• Reweighting complex Langevin trajectories

Cited by in corpus (6)

• Applying the tempered Lefschetz thimble method to the Hubbard model away from half-filling
• Worldvolume approach to the tempered Lefschetz thimble method
• Implementation of the HMC algorithm on the tempered Lefschetz thimble method
• Emergence of AdS geometry in the simulated tempering algorithm
• Tempered Lefschetz thimble method and its application to the Hubbard model away from half filling
• Emergent quantum geometry from stochastic random matrices