Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L^p-norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, and Yin, FOCS’13). The non-linearity of L^p-norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L^p-analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G (n, d/n), where the previously known algorithms run in time n^O(log d) or applied only to large d. We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d, throughout the uniqueness regime.

谱独立性是最近开发的一个框架，用于获得经典格劳伯动力学收敛时间的尖锐界限。这个新框架在有界度图上产生了最优 O ( n log n ) 采样算法，用于解决整个所谓唯一性机制中的一大类问题，例如，包括独立集采样、匹配和Ising的问题 。模型配置。

我们的主要贡献是放宽了迄今为止在建立和应用谱独立性方面非常重要的有界度假设。以前避免度界限的方法依赖于使用L ^p范数来分析具有有界连接常数的图的收缩（Sinclair、Srivastava、Yin；FOCS'13）。L ^p范数的非线性是应用这些结果来限制谱独立性的障碍。我们的解决方案是通过对用于分析收缩的递归子树进行摊销来递归地捕获L ^p分析。我们的方法概括了先前仅适用于有界度图的分析。

作为我们技术的主要应用，我们考虑随机图G ( n , d / n )，其中先前已知的算法在时间n ^{O (log  d )}中运行或仅应用于大d。我们显着地改进了这些算法界限，并开发了基于格劳伯动力学的快速近线性算法，该算法适用于整个唯一性范围内的所有常数d 。