Leslie Goldberg

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.