Spectral gap and embedded trees for the Laplacian of the Erdős-Rényi graph

For the Erdős-Rényi graph of size N with mean degree (1+o(1)) logN t+1≤d≤(1−o(1)) logN t where t∈N∗, with high probability the smallest non zero eigenvalue of the Laplacian is equal to 2−2cos(π(2t+1)−1)+o(1). This eigenvalue arises from a small subgraph isomorphic to a line of size t linked to the giant connected component by only one edge.

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Publication Details

Journal
Electronic Journal of Probability
Published
2025-01-01
DOI
https://doi.org/10.1214/25-ejp1303
Primary Topic
Spectral Theory in Mathematical Physics
Type
article
Field-Weighted Citation Impact
0.00

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article

Spectral gap and embedded trees for the Laplacian of the Erdős-Rényi graph

Renaud Rivier, Raphaël Ducatez
Electronic Journal of Probability
Spectral Theory in Mathematical Physics
article

Spectral gap and embedded trees for the Laplacian of the Erdős-Rényi graph

Renaud Rivier, Raphaël Ducatez
article en

Abstract

For the Erdős-Rényi graph of size N with mean degree (1+o(1)) logN t+1≤d≤(1−o(1)) logN t where t∈N∗, with high probability the smallest non zero eigenvalue of the Laplacian is equal to 2−2cos(π(2t+1)−1)+o(1). This eigenvalue arises from a small subgraph isomorphic to a line of size t linked to the giant connected component by only one edge.

Electronic Journal of ProbabilityVol. 30(none)
Université Claude Bernard Lyon 1 (FR), University of Geneva (CH), Institut Camille Jordan (FR)
National Science Foundation, National Centres of Competence in Research SwissMAP, European Commission, Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, Staatssekretariat für Bildung, Forschung und Innovation
Openalex Percentile: Top 90%
Spectral Theory in Mathematical Physics
DOIOpen Access PDFOpenAlex
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