May 2021 Extremal eigenvalues of critical Erdős–Rényi graphs
Johannes Alt, Raphaël Ducatez, Antti Knowles
Author Affiliations +
Ann. Probab. 49(3): 1347-1401 (May 2021). DOI: 10.1214/20-AOP1483


We complete the analysis of the extremal eigenvalues of the adjacency matrix A of the Erdős–Rényi graph G(N,d/N) in the critical regime dlogN of the transition uncovered in (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 2141–2161; Ann. Probab. 47 (2019) 1653–1676), where the regimes dlogN and dlogN were studied. We establish a one-to-one correspondence between vertices of degree at least 2d and nontrivial (excluding the trivial top eigenvalue) eigenvalues of A/d outside of the asymptotic bulk [2,2]. This correspondence implies that the transition characterized by the appearance of the eigenvalues outside of the asymptotic bulk takes place at the critical value d=d=1log41logN. For d<d, we obtain rigidity bounds on the locations of all eigenvalues outside the interval [2,2], and for d>d, we show that no such eigenvalues exist. All of our estimates are quantitative with polynomial error probabilities.

Our proof is based on a tridiagonal representation of the adjacency matrix and on a detailed analysis of the geometry of the neighbourhood of the large degree vertices. An important ingredient in our estimates is a matrix inequality obtained via the associated nonbacktracking matrix and an Ihara–Bass formula (Ann. Inst. Henri Poincaré Probab. Stat. 56 (2020) 2141–2161). Our argument also applies to sparse Wigner matrices, defined as the Hadamard product of A and a Wigner matrix, in which case the role of the degrees is replaced by the squares of the 2-norms of the rows.


Download Citation

Johannes Alt. Raphaël Ducatez. Antti Knowles. "Extremal eigenvalues of critical Erdős–Rényi graphs." Ann. Probab. 49 (3) 1347 - 1401, May 2021.


Received: 1 May 2019; Revised: 1 May 2020; Published: May 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/20-AOP1483

Primary: 05C80 , 15B52 , 60B20

Keywords: eigenvalue rigidity , Erdős–Rényi graphs , sparse random matrix

Rights: Copyright © 2021 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.49 • No. 3 • May 2021
Back to Top