Mixing time of critical ising model on trees is polynomial in the height

Jian Ding, Eyal Lubetzky, Yuval Peres

Research output: Contribution to journalArticle

Abstract

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuous-time chain exhibits the following behavior. For some critical inverse-temperature β c, the inverse-gap is O(1) for β < β c, polynomial in the surface area for β = β c and exponential in it for β > β c. This has been proved for ℤ 2 except at criticality. So far, the only underlying geometry where the critical behavior has been confirmed is the complete graph. Recently, the dynamics for the Ising model on a regular tree, also known as the Bethe lattice, has been intensively studied. The facts that the inverse-gap is bounded for β < β c and exponential for β > β c were established, where β c is the critical spin-glass parameter, and the tree-height h plays the role of the surface area. In this work, we complete the picture for the inverse-gap of the Ising model on the b-ary tree, by showing that it is indeed polynomial in h at criticality. The degree of our polynomial bound does not depend on b, and furthermore, this result holds under any boundary condition. We also obtain analogous bounds for the mixing-time of the chain. In addition, we study the near critical behavior, and show that for β > β c, the inverse-gap and mixing-time are both exp[Θ((β - β c)h)].

Original languageEnglish (US)
Pages (from-to)161-207
Number of pages47
JournalCommunications in Mathematical Physics
Volume295
Issue number1
DOIs
StatePublished - Feb 2010

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Mixing Time
Ising model
Ising Model
polynomials
Polynomial
Criticality
Surface area
Critical Behavior
Bethe Lattice
Glauber Dynamics
Heat Bath
Spectral Gap
Spin Glass
Complete Graph
spin glass
Continuous Time
baths
boundary conditions
Boundary conditions
heat

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Mixing time of critical ising model on trees is polynomial in the height. / Ding, Jian; Lubetzky, Eyal; Peres, Yuval.

In: Communications in Mathematical Physics, Vol. 295, No. 1, 02.2010, p. 161-207.

Research output: Contribution to journalArticle

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