Wasserstein 稳定性、跨体积耦合以及 Edwards--Anderson 模型中的 $1+1/d$ 矩阈值
Wasserstein Stability, Couplings Across Volumes, and the $1+1/d$ Moment Thresholds in the Edwards--Anderson Model
浏览论文内容
中文总结 AI 辅助
本文证明 Edwards--Anderson 模型淬火压力的 Wasserstein 稳定性，给出跨体积耦合下几乎必然收敛的等价条件，并确定 $1+1/d$ 矩阈值的最优性及一维必要性。
中文摘要 AI 辅助
我们研究具有自由和周期边界条件的最近邻 Edwards--Anderson Ising 模型的淬火压力。首先，我们证明在 $1$-Wasserstein 距离下，无限体积压力关于耦合分布是 $\beta d$-Lipschitz 的，从而为空间非均匀无序给出定量的热力学极限。其次，对于周期体积，我们证明在有限体积无序阵列的每个联合耦合下几乎必然收敛等价于单体积压力定律的完全收敛。第三，我们证明 $\mathbb{E}\vert{}J\vert{}^{1+1/d}<\infty$ 保证这一普适耦合结论。我们进一步证明该指数在均匀幂矩假设中是最优的：对于每个 $1\leq q<1+1/d$，存在一个中心对称分布，其 $q$ 阶矩有限，使得典型嵌套体积几乎必然收敛，而具有相同固定体积边缘分布的独立重采样体积依概率收敛但不几乎必然收敛。最后，在一维情形，我们证明一阶矩条件也是有限极限压力的必要条件。
英文摘要
We study the quenched pressure of the nearest-neighbor Edwards--Anderson Ising model with free and periodic boundary conditions. First, we prove that the infinite-volume pressure is $βd$-Lipschitz in the coupling law for the $1$-Wasserstein distance, yielding quantitative thermodynamic limits for spatially inhomogeneous disorder. Second, for periodic volumes, we prove that almost-sure convergence under every joint coupling of the finite-volume disorder arrays is equivalent to complete convergence of the one-volume pressure laws. Third, we prove that $\mathbb{E}\vert{}J\vert{}^{1+1/d}<\infty$ guarantees this universal-coupling conclusion. We further show that this exponent is optimal among uniform power-moment assumptions: for every $1\leq q<1+1/d$, there is a centered symmetric law with finite $q$-th moment for which canonical nested volumes converge almost surely, whereas independently resampled volumes with the same fixed-volume marginals converge in probability but not almost surely. Finally, in dimension one, we prove that the first-moment condition is also necessary for a finite limiting pressure.
