  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.
