Improved Strong Spatial Mixing for Colorings on Trees

Authors Charilaos Efthymiou, Andreas Galanis, Thomas P. Hayes, Daniel Štefankovič, Eric Vigoda



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Author Details

Charilaos Efthymiou
  • Department of Computer Science, University of Warwick, UK
Andreas Galanis
  • Department of Computer Science, University of Oxford, UK
Thomas P. Hayes
  • Department of Computer Science, University of New Mexico, Albuquerque, NM, USA
Daniel Štefankovič
  • Department of Computer Science, University of Rochester, NY, USA
Eric Vigoda
  • School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA

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Charilaos Efthymiou, Andreas Galanis, Thomas P. Hayes, Daniel Štefankovič, and Eric Vigoda. Improved Strong Spatial Mixing for Colorings on Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 48:1-48:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.48

Abstract

Strong spatial mixing (SSM) is a form of correlation decay that has played an essential role in the design of approximate counting algorithms for spin systems. A notable example is the algorithm of Weitz (2006) for the hard-core model on weighted independent sets. We study SSM for the q-colorings problem on the infinite (d+1)-regular tree. Weak spatial mixing (WSM) captures whether the influence of the leaves on the root vanishes as the height of the tree grows. Jonasson (2002) established WSM when q>d+1. In contrast, in SSM, we first fix a coloring on a subset of internal vertices, and we again ask if the influence of the leaves on the root is vanishing. It was known that SSM holds on the (d+1)-regular tree when q>alpha d where alpha ~~ 1.763... is a constant that has arisen in a variety of results concerning random colorings. Here we improve on this bound by showing SSM for q>1.59d. Our proof establishes an L^2 contraction for the BP operator. For the contraction we bound the norm of the BP Jacobian by exploiting combinatorial properties of the coloring of the tree.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Random walks and Markov chains
Keywords
  • colorings
  • regular tree
  • spatial mixing
  • phase transitions

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