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Improved Strong Spatial Mixing for Colorings on Trees

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

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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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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.

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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

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

Funding

  • Efthymiou, Charilaos: Supported by the Centre of Discrete Mathematics and its Applications (DIMAP), University of Warwick, EPSRC award EP/D063191/1.
  • Galanis, Andreas: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) ERC grant agreement no. 334828. The paper reflects only the authors' views and not the views of the ERC or the European Commission. The European Union is not liable for any use that may be made of the information contained therein.
  • Hayes, Thomas P.: Partially supported by NSF CAREER award CCF-1150281.
  • Štefankovič, Daniel: Research supported in part by NSF grant CCF-1563757.
  • Vigoda, Eric: Research supported in part by NSF grants CCF-1617306 and CCF-1563838.

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