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In the heat-bath Glauber dynamics for the Ising model on the lattice,physicists believe that the spectral gap of the continuous-time chain exhibitsthe following behavior. For some critical inverse-temperature $\beta_c$, theinverse-gap is bounded for $\beta < \beta_c$, polynomial in the surface areafor $\beta = \beta_c$ and exponential in it for $\beta > \beta_c$. This hasbeen proved for $\Z^2$ except at criticality. So far, the only underlyinggeometry 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 theBethe lattice, has been intensively studied. The facts that the inverse-gap isbounded for $\beta < \beta_c$ and exponential for $\beta > \beta_c$ wereestablished, where $\beta_c$ is the critical spin-glass parameter, and thetree-height $h$ plays the role of the surface area.  In this work, we complete the picture for the inverse-gap of the Ising modelon the $b$-ary tree, by showing that it is indeed polynomial in $h$ atcriticality. The degree of our polynomial bound does not depend on $b$, andfurthermore, this result holds under any boundary condition. We also obtainanalogous bounds for the mixing-time of the chain. In addition, we study thenear critical behavior, and show that for $\beta > \beta_c$, the inverse-gapand mixing-time are both $\exp[\Theta((\beta-\beta_c) h)]$.

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