Note on the trace of random walks on pseudorandom graphs

TL;DR

This paper analyzes the trace of simple random walks on deterministic pseudorandom graphs, specifically $(n,d,
lambda)$-graphs. By leveraging spectral expansion and the expander-mixing lemma, it proves that for any $\u0001varepsilon>0$ there exists a constant $C(\u0001varepsilon)$ such that if $d/\u0001lambda\ge C$, the cover time is at most $(1+\u0001varepsilon)n\log n$, and the trace of length $(1+\u0001varepsilon)n\log n$ is Hamiltonian with high probability, regardless of the starting vertex. These results extend to random $d$-regular graphs with sufficiently large $d$, and they address questions posed by Frieze, Krivelevich, Michaeli, and Peled. The authors further show a strong version where all vertices are covered whp within the same length, and the analysis hinges on spectral decompositions, random-walk blanket-time connections, and modern Hamiltonicity results for expanders. The findings are asymptotically optimal and illuminate the deep interplay between expansion, random-walk dynamics, and Hamiltonicity in pseudorandom graphs, with potential implications for related diffusion and network-design problems.

Abstract

We study the graph-theoretic properties of the trace of random walks on pseudorandom graphs. We show that for any $\varepsilon>0$, there exists a constant $C$ such that the cover time of an $(n,d,λ)$-graph $G$ with $d/λ\ge C$ is at most $(1+\varepsilon)n\log n$, meaning the expected number of steps needed to reach all vertices at least once is at most $(1+\varepsilon)n\log n$ regardless of the starting vertex. Furthermore, we prove that with high probability, the trace of a random walk of length $(1+\varepsilon)n\log n$ on $G$ is Hamiltonian, regardless of the starting vertex. These results also hold for random $d$-regular graphs with sufficiently large $d$. These findings answer two questions proposed by Frieze, Krivelevich, Michaeli, and Peled [PLMS, 2018]. Notably, our results imply a bound on a stronger version of the cover time: with high probability, all vertices are covered after $(1+\varepsilon)n\log n$ steps, regardless of the starting vertex. Our proofs rely on the spectral properties of the adjacency matrix and the graph expansion. All results are asymptotically optimal.

Theorems & Definitions (25)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5: Alon-Chung-expander-mixing
• Definition 2.6