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 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 {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 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)].
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- April 2010
- DOI:
- 10.1007/s00220-009-0978-y
- arXiv:
- arXiv:0901.4152
- Bibcode:
- 2010CMaPh.295..161D
- Keywords:
-
- Mathematics - Probability;
- Mathematical Physics;
- 60K35;
- 82B20;
- 82B27;
- 82C20
- E-Print:
- 53 pages