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- W2111981124 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 $beta_c$, 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 $exp[Theta((beta-beta_c) h)]$." @default.
- W2111981124 created "2016-06-24" @default.
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- W2111981124 date "2010-01-08" @default.
- W2111981124 modified "2023-10-03" @default.
- W2111981124 title "Mixing Time of Critical Ising Model on Trees is Polynomial in the Height" @default.
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- W2111981124 doi "https://doi.org/10.1007/s00220-009-0978-y" @default.
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