On the Spectral Expansion of Monotone Subsets of the Hypercube
Yumou Fei, Renato Ferreira Pinto
TL;DR
This work proves an optimal lower bound on the spectral gap for subgraphs of the hypercube induced by monotone sets, showing γ(H_A) ≥ (1 − √(1 − μ(A)))/n, which implies a mixing time bound t_mix = O(n^2) for constant-density A. The authors develop two new tools: a directed L^2-Poincaré inequality and an approximate FKG inequality tailored to monotone subsets, enabling a direct L^2 analysis that bypasses prior Cheeger-based gaps. They connect directed isoperimetry to undirected spectral gaps through a domain-extension and correlation framework, and they compute the directed hypercube’s dynamical spectral gap as λ^−(H_n) = 1, via a coordinate-decomposed energy argument. The paper also establishes a reverse directed Poincaré inequality and a near-tight converse to the main result, showing that a poor approximate FKG ratio δ(A) ≈ −1 corresponds to torpid mixing, thereby highlighting the central role of δ(A) in fast mixing. Together, these results yield not only the primary bound but also a robust toolkit for directed spectral theory with potential broader applicability to monotone-structure problems and property testing.
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 $μ(A)$, the previous best lower bound on the spectral gap, due to Cohen, was $γ\gtrsim μ(A)/n^{2}$, improving upon the earlier bound $γ\gtrsim μ(A)^{2}/n^{2}$ established by Ding and Mossel. In this paper, we prove the optimal lower bound $γ\gtrsim μ(A)/n$. 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 $L^{2}$-Poincaré inequality on the hypercube, and (2)~an ``approximate'' FKG inequality for monotone sets.