On the Spectral Expansion of Monotone Subsets of the Hypercube
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910928278126592 |
|---|---|
| author | Fei, Yumou Pinto Jr, Renato Ferreira |
| author_facet | Fei, Yumou Pinto Jr, Renato Ferreira |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02685 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Spectral Expansion of Monotone Subsets of the Hypercube Fei, Yumou Pinto Jr, Renato Ferreira Probability Combinatorics 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. |
| title | On the Spectral Expansion of Monotone Subsets of the Hypercube |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2505.02685 |