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

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 ${\beta }_c$, the inverse-gap is bounded for $\beta <{\beta }_c$, polynomial in the surface area for $\beta ={\beta }_c$ and exponential in it for $\beta >{\beta }_c$. This has been proved for ${\mathbb{Z}}^2$ 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 $\beta <{\beta }_c$ and exponential for $\beta >{\beta }_c$ were established, where ${\beta }_c$ 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 $\beta >{\beta }_c$, the inverse-gap and mixing-time are both $\exp [\Theta ((\beta -{\beta }_c)h)]$.

53 pages; 3 figures

References in corpus (3)

• Elementary bounds on Poincare and log-Sobolev constants for decomposable Markov chains
• The mixing time evolution of Glauber dynamics for the mean-field Ising model
• The critical Ising model on trees, concave recursions and nonlinear capacity

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• Cutoff and Dynamical Phase Transition for the General Multi-component Ising Model
• Mixing time of a kinetically constrained spin model on trees: power law scaling at criticality
• Rapid mixing of subset Glauber dynamics on graphs of bounded tree-width