Phase transition for Glauber dynamics for independent sets on regular trees

• Authors: Ricardo Restrepo;Daniel Stefankovic;Juan C. Vera;Eric Vigoda;Linji Yang
• Venue: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2011

We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hardcore lattice gas model on the n-vertex regular b-ary tree of height h. The hard-core model is defined on independent sets weighted by an activity (or fugacity) λ on trees. Reconstruction studies the effect of a 'typical' boundary condition, i.e., fixed assignment to the leaves, on the root. The threshold for when reconstruction occurs (and a typical boundary influences the root in the limit h → ∞) has been of considerable recent interest since it appears to be connected to the efficiency of certain local algorithms on locally tree-like graphs. The reconstruction threshold occurs at ω ≈ ln b/b where λ = ω(1 + ω)b is a convenient re-parameterization of the model. We prove that for all boundary conditions, the relaxation time τ in the non-reconstruction region is fast, namely τ = O (n1+ob(1)) for any ω ≤ ln b/b. In the reconstruction region, for all boundary conditions, we prove τ = O (n1+δ+ob(1)) for ω = (1 + ω) ln b/b, for every δ 0. In contrast, we construct a boundary condition, for which the Glauber dynamics slows down in the reconstruction region, namely τ = Ω (n1+δ-ob(1)) for ω = (1 + δ) ln b/b, for every δ 0. The interesting part of our proof is this lower bound result, which uses a general technique that transforms an algorithm to prove reconstruction into a set in the state space of the Glauber dynamics with poor conductance.