  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 $\Z^2$ 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 $\exp[\Theta((\beta-\beta_c) h)]$.
