On the Spectral Expansion of Monotone Subsets of the Hypercube

TL;DR

This paper establishes an optimal lower bound on the spectral gap of monotone subsets of the hypercube, leading to improved mixing time bounds for related random walks and introducing new inequalities of independent interest.

Contribution

It proves the optimal spectral gap lower bound for monotone hypercube subsets and develops two new inequalities, enhancing understanding of spectral properties and mixing times.

Findings

Spectral gap lower bound improved to b3  b1/n

Mixing time for constant-density monotone sets reduced to O(n^2)

Introduces new directed L^2-Poincare9 and approximate FKG inequalities

Abstract

We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset $A\subseteq\{0,1\}^{n}$ of density $\mu(A)$, the previous best lower bound on the spectral gap, due to Cohen, was $\gamma\gtrsim \mu(A)/n^{2}$, improving upon the earlier bound $\gamma\gtrsim \mu(A)^{2}/n^{2}$ established by Ding and Mossel. In this paper, we prove the optimal lower bound $\gamma\gtrsim \mu(A)/n$. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from $O(n^{3})$, as shown by Ding and Mossel, to $O(n^{2})$. Along the way, we develop two new inequalities that may be of independent interest: (1)~a directed $L^{2}$-Poincar\'{e} inequality on the hypercube, and (2)~an ``approximate'' FKG inequality for monotone sets.