标题： 零温爱德华-安德森模型的谱性质 显示英文标题

标题： Spectral properties of the zero temperature Edwards-Anderson model

摘要： 具有随机耦合的图上的伊辛模型是自旋玻璃模型。 虽然夏里顿-基尔帕特里克模型的平均场情况被广泛研究，但更现实的格点设置，即爱德华兹-安德森（EA）模型，进展相当有限。 在（Chatterjee,'23）中，通过研究两点自旋关联的傅里叶谱，建立了EA模型中基态的混沌性质。 一个自然的研究方向是类似于临界渗流的傅里叶谱的分形性质。 特别是数值结果（Bray, Moore,'87）似乎支持这样一个观点，即根据傅里叶谱绘制的相关谱样本的分形维数严格大于一。 为此，在本文中，我们引入了一种类似渗流的论证方法，依赖于“障碍物”的构造，以获得谱样本大小的新概率下限。 显示英文摘要

摘要： An Ising model with random couplings on a graph is a model of a spin glass. While the mean field case of the Sherrington-Kirkpatrick model is very well studied, the more realistic lattice setting, known as the Edwards-Anderson (EA) model, has witnessed rather limited progress. In (Chatterjee,'23) chaotic properties of the ground state in the EA model were established via the study of the Fourier spectrum of the two-point spin correlation. A natural direction of research concerns fractal properties of the Fourier spectrum in analogy with critical percolation. In particular, numerical findings (Bray, Moore,'87) seem to support the belief that the fractal dimension of the associated spectral sample drawn according to the Fourier spectrum is strictly bigger than one. Towards this, in this note we introduce a percolation-type argument, relying on the construction of ``barriers'', to obtain new probabilistic lower bounds on the size of the spectral sample.