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

In the heat-bath Glauber dynamics for the Ising model on the lattice, physicists believe that the spectral gap of the continuoustime chain exhibits the following behavior. For some critical inversetemperature βc, the inverse-gap is O(1) for β < βc, polynomial in the surface area for β = βc and exponential in it for β > βc. This has been proved for Z 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 < βc and exponential for β > βc were established, where βc is the critical spin-glass parameter, and the treeheight 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 > βc, the inverse-gap and mixing-time are both exp[Θ((β − βc)h)].