2017/05/31 by Masafumi Fukuma, Nobuyuki Matsumoto, Naoya Umeda · 1 citation
Physics and Astronomy · #hep-lat #cond-mat.stat-mech
paper · pdf · doi:10.1007/jhep12(2017)001
published as JHEP 1712 (2017) 001 · 20 pages, 3 figures. v2: typos corrected, references and a figure added, a few additional explanations made. v3: discussions in section 4 improved
arxiv created 2018/03/20 · arxiv updated 2018/03/21
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.