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, 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 $n^{1+o(1)}$ algorithms based on Glauber dynamics that apply to all $d$, throughout the uniqueness regime.

翻译：谱独立性是近期发展出的一个框架，用于获取经典格劳伯动力学收敛时间的紧界。该新框架已在有界度图上，针对所谓唯一性区域内的广泛问题（例如独立集采样、匹配采样和伊辛模型配置采样）实现了最优的 $O(n \log n)$ 采样算法。我们的主要贡献是放宽了此前在建立和应用谱独立性中至关重要的有界度假设。先前避免度界的方法依赖于使用 $L^p$ 范数分析具有有界连接常数的图上的收缩性（Sinclair, Srivastava, Yin; FOCS'13）。$L^p$ 范数的非线性特性成为将这些结果应用于谱独立性界定的障碍。我们的解决方案是通过在用于分析收缩性的递归子树上进行摊销，以递归方式捕获 $L^p$ 分析。我们的方法推广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 $G(n,d/n)$，此前已知算法的时间复杂度为 $n^{O(\log d)}$ 或仅适用于较大的 $d$。我们显著改进了这些算法界，并开发了基于格劳伯动力学的快速 $n^{1+o(1)}$ 算法，该算法适用于所有 $d$ 且在唯一性区域内有效。