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 global geometry of a stochastic system whose equilibrium distribution is highly multimodal with a large number of degenerate vacua. We show that, when the simulated tempering algorithm is implemented to such a system, the extended configuration space has an asymptotically Euclidean anti-de Sitter (AdS) geometry. We further show that this knowledge of geometry enables us to optimize the tempering parameter in a simple, geometrical way.
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Fukuma, M., Matsumoto, N. & Umeda, N. Emergence of AdS geometry in the simulated tempering algorithm. J. High Energ. Phys. 2018, 60 (2018). https://doi.org/10.1007/JHEP11(2018)060
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DOI: https://doi.org/10.1007/JHEP11(2018)060