Ising动力学在Z²上的快速相序问题
Rapid phase ordering of Ising dynamics on $\mathbb Z^2$
AI总结：
研究低温度Ising动力学在偏向初始条件下的快速相序行为，证明在二维情况下存在阈值p₀，使得初始化为+1的概率p>p₀时，动力学快速收敛到+相位测度。
摘要：
我们考虑低温度Ising动力学在偏向和无序初始条件下的相序问题。Fontes等人(2002)的工作显示，在零温度下，初始化为i.i.d.自旋的Ising Glauber动力学在Z^d(d≥2)上快速吸收至全+配置。我们证明在二维Ising模型的低温度范围内，类似行为成立。即存在p₀<1，使得初始化为+1的概率p>p₀时，动力学在任意低温度β>β_c下快速收敛到+相位测度π^+。该结果通过适用于任何d≥2的时空多尺度耦合证明，将具有+边界条件的Ising动力学的混合时间统一界提升为无边界条件的快速相序行为。
英文摘要：
We consider the phase ordering problem for the low-temperature Ising dynamics initialized from a biased and disordered initialization. Work of Fontes, Schonmann, Sidoravicius (2002) showed that at zero-temperature, Ising Glauber dynamics on $\mathbb Z^d$ for $d\ge 2$ initialized from i.i.d. spins on each vertex that are $+1$ with sufficiently large probability, absorbs into the all-plus configuration quickly. We prove that analogous behavior holds throughout the low-temperature regime of the Ising model in two dimensions. Namely, there exists $p_0 <1$ such that Ising Glauber dynamics initialized from i.i.d. spins that are $+1$ with probability $p>p_0$, run at any low temperature $β>β_c$ converges rapidly to the plus phase measure $π^+$. The result is proved using a spacetime multiscale coupling valid in any $d\ge 2$, that boosts a uniform-in-$β$ quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions, into rapid phase ordering from biased initializations with no boundary conditions.
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