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Range of random $\mathbb Z$-homomorphisms on weak expanders

Dingding Dong, Jinyoung Park

Abstract

We prove that random $\mathbb{Z}$-homomorphisms on weakly expanding bipartite graphs exhibit a strong "flatness" phenomenon. Extending prior work of Peled, Samotij, and Yehudayoff for expanders, we first show that on any bipartite $(n, d, λ)$-graph with $λ\leq (1-δ)d$, a uniformly chosen $\mathbb{Z}$-homomorphism has a range at most $O(\log \log n)$ with high probability, which is tight up to a constant factor. This provides an affirmative answer to their question in the spectral setting. As a concrete application, we prove that a random $\mathbb{Z}$-homomorphism on the middle layers of the Hamming cube takes at most $5$ values with high probability. This shows that the $O(1)$-flatness for the full Hamming cube, proved by Kahn and Galvin, persists even when the rigid structural properties are relaxed.

Range of random $\mathbb Z$-homomorphisms on weak expanders

Abstract

We prove that random -homomorphisms on weakly expanding bipartite graphs exhibit a strong "flatness" phenomenon. Extending prior work of Peled, Samotij, and Yehudayoff for expanders, we first show that on any bipartite -graph with , a uniformly chosen -homomorphism has a range at most with high probability, which is tight up to a constant factor. This provides an affirmative answer to their question in the spectral setting. As a concrete application, we prove that a random -homomorphism on the middle layers of the Hamming cube takes at most values with high probability. This shows that the -flatness for the full Hamming cube, proved by Kahn and Galvin, persists even when the rigid structural properties are relaxed.

Paper Structure

This paper contains 21 sections, 37 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof overview and key ideas
  4. Preliminaries
  5. Graph theory basics
  6. Isoperimetry in bipartite spectral expanders
  7. Isoperimetry in $Q_{2d-1}^{\mathrm{mid}}$
  8. Graph containers
  9. The $\psi$-approximating quadruples
  10. Container lemmas for approximating quadruples
  11. Reduction: from homomorphisms to partitions
  12. Merge
  13. The merge operation
  14. Proof of \ref{['thm:Z-hom-spectral-expanders']}
  15. Proof of \ref{['thm:Z-hom-middle-layers']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

For every $\delta>0$, there exist constants $d_0=d_0(\delta)>0$ and $C=C(\delta)>0$ such that the following holds. Let $\Gamma$ be a bipartite $(n,d,\lambda)$-graph with vertex bipartition $V(\Gamma)=\mathcal{E} \cup \mathcal{O}$ such that $d \ge d_0$ and $\lambda\le (1-\delta)d$. Fix any $v_0\in \m $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (89)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1: $k$-linked set and component
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4: Lovász Lov75, Stein Ste74
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 79 more