From Coupling to Spectral Independence and Blackbox Comparison with the Down-Up Walk

Abstract

We show that the existence of a "good" coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-toglobal arguments [CLV21], we establish asymptotically optimal lower bounds on the standard and modified log-Sobolev constants for the Glauber dynamics for sampling from spin systems on bounded-degree graphs when a curvature condition [Oll09] is satisfied. To achieve this, we use Stein's method for Markov chains [BN19; RR19] to show that a "good" coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of [ALO20].Our primary application is to sampling proper list-colorings on bounded-degree graphs. In particular, combining the coupling for the flip dynamics given by [Vig00; Che+19] with our techniques, we show optimal O(n log n) mixing for the Glauber dynamics for sampling proper list-colorings on any boundeddegree graph with maximum degree ∆ whenever the size of the color lists are at least 11 6 − ǫ ∆, where ǫ ≈ 10 −5 is small constant. While O(n 2 ) mixing was already known before, our approach additionally yields Chernoff-type concentration bounds for Hamming Lipschitz functions in this regime, which was not known before. Our approach is markedly different from prior works establishing spectral independence for spin systems using spatial mixing [ALO20; CLV20; Che+20; Fen+20], which crucially is still open in this regime for proper list-colorings.

Entropic Independence I: Modified Log-Sobolev Inequalities for Fractionally Log-Concave Distributions and High-Temperature Ising Models

“…Liu [Liu21] conjectured that if µ is an O(1)-spectrally independent distribution, the down-up walk for sampling from µ has modified log-Sobolev constant Ω(1/k) [see Liu21, Conjecture 1]. We refute this conjecture.…”

Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

“…Higher order random walks have also seen an impressive number of applications in recent years, frequently closely tied to analysis of their spectral structure. This has included breakthrough works on approximate sampling [5,4,[6][7][8][9][10][11][12][13], CSP-approximation [14,15], error-correcting codes [16,17], and agreement testing [18][19][20]. In this vein, our work is most closely related to that of Bafna, Barak, Kothari, Schramm, and Steurer [33], and BHKL [15], who used the spectral and combinatorial structure of HD-walks to build new algorithms for unique games.…”

“…Random walks on high dimensional expanders (HDX) have been the object of intense study in theoretical computer science in recent years. Starting with their original formulation by Kaufman and Mass [1], a series of works on the spectral structure of these walks [2][3][4] led to significant breakthroughs in approximate sampling [5,4,[6][7][8][9][10][11][12][13], CSP-approximation [14,15], error-correcting codes [16,17], agreement testing [18][19][20], and more. Most of these works focus on the structure of expansion in hypergraphs (typically called simplicial complexes in the HDX literature).…”

Sampling Colorings and Independent Sets of Random Regular Bipartite Graphs in the Non-Uniqueness Region

“…Current bounds are far from resolving this conjecture but have made considerable progress. On the algorithmic side, recent results establish O(n log n) mixing time of the Glauber dynamics on an n-vertex graph of maximum degree ∆ when q > (11/6 − ε 0 )∆ for a positive constant ε 0 ≈ 10 −5 [3,24,7] and on triangle-free graphs when q > 1.764∆ [10,12,9]. On the negative side, it was shown in [15] that for even q < ∆ it is NP-hard to approximate the number of q-colorings.…”