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Glauber Dynamics on Trees: Boundary Conditions and Mixing Time

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Abstract

We give the first comprehensive analysis of the effect of boundary conditions on the mixing time of the Glauber dynamics in the so-called Bethe approximation. Specifically, we show that the spectral gap and the log-Sobolev constant of the Glauber dynamics for the Ising model on an n-vertex regular tree with (+)-boundary are bounded below by a constant independent of n at all temperatures and all external fields. This implies that the mixing time is O(logn) (in contrast to the free boundary case, where it is not bounded by any fixed polynomial at low temperatures). In addition, our methods yield simpler proofs and stronger results for the spectral gap and log-Sobolev constant in the regime where the mixing time is insensitive to the boundary condition. Our techniques also apply to a much wider class of models, including those with hard-core constraints like the antiferromagnetic Potts model at zero temperature (proper colorings) and the hard–core lattice gas (independent sets).

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Authors and Affiliations

  1. Department of Mathematics, University of Roma Tre, Largo San Murialdo 1, 00146, Roma, Italy

    Fabio Martinelli

  2. Computer Science Division, University of California, Berkeley, CA, 94720-1776, USA

    Alistair Sinclair & Dror Weitz

Authors
  1. Fabio Martinelli
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  2. Alistair Sinclair
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  3. Dror Weitz
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Correspondence to Fabio Martinelli.

Additional information

Communicated by H. Spohn

An extended abstract of this paper appeared under the title “The Ising model on trees: Boundary conditions and mixing time” in Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, October 2003, pp. 628–639.

This work was done while this author was visiting the Departments of EECS and Statistics, University of California, Berkeley, supported in part by a Miller Visiting Professorship.

Supported in part by NSF Grant CCR-0121555 and DARPA cooperative agreement F30602-00-2-0601.

Supported in part by NSF Grant CCR-0121555.

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Martinelli, F., Sinclair, A. & Weitz, D. Glauber Dynamics on Trees: Boundary Conditions and Mixing Time. Commun. Math. Phys. 250, 301–334 (2004). https://doi.org/10.1007/s00220-004-1147-y

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