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A Spectral Independence View on Hard Spheres via Block Dynamics

Authors Tobias Friedrich , Andreas Göbel , Martin S. Krejca , Marcus Pappik

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Author Details

Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany
Andreas Göbel
  • Hasso Plattner Institute, University of Potsdam, Germany
Martin S. Krejca
  • Sorbonne University, CNRS, LIP6, Paris, France
Marcus Pappik
  • Hasso Plattner Institute, University of Potsdam, Germany

Funding

  • Krejca, Martin S.: This work was supported by the Paris Île-de-France Region.

Cite As Get BibTex

Tobias Friedrich, Andreas Göbel, Martin S. Krejca, and Marcus Pappik. A Spectral Independence View on Hard Spheres via Block Dynamics. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.66

Abstract

The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in d dimensions. Up to a fugacity of λ < e/2^d, the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure.
Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Hard-sphere Model
  • Markov Chain
  • Partition Function
  • Gibbs Distribution
  • Approximate Counting
  • Spectral Independence

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