Mixing Time of Critical Ising Model on Trees is Polynomial in the Height
Mixing Time of Critical Ising Model on Trees is Polynomial in the Height
复制标题
树木临界隔离模型的混合时间在高度上是多项式
DOI：
--
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
Y. Peres
中科院分区：
文献类型：
--
作者：
Jian Ding;E. Lubetzky;Y. Peres
在晶格上Ising模型的热浴格劳伯动力学中，物理学家认为连续时间链的谱隙表现出以下行为。对于某些临界逆温度βc， β < βc的逆间隙为0 (1),β = βc的逆间隙为多项式，β > βc的逆间隙为指数。这已被证明$${\mathbb{Z}^2}$$除了在临界。到目前为止，唯一确定临界行为的底层几何是完全图。最近，在规则树（也称为Bethe晶格）上的Ising模型的动力学得到了深入的研究。建立了β < βc的逆隙是有界的，β > βc的逆隙是指数的，其中βc是临界自旋玻璃参数，树高h是表面积的作用。在这项工作中，我们通过证明它确实是h临界多项式，完成了b-任树上的Ising模型的逆间隙图。多项式界的阶与b无关，并且在任何边界条件下都成立。我们也得到了链的混合时间的类似界。此外，我们还研究了β > βc的近临界行为，并证明了β > βc的反间隙和混合时间都是exp[Θ（(β−βc)h）]。
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 βc, the inverse-gap is O(1) for β < βc, polynomial in the surface area for β = βc and exponential in it for β > βc. 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 < βc and exponential for β > βc were established, where βc 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 > βc, the inverse-gap and mixing-time are both exp[Θ((β − βc)h)].