The Annals of Probability

The spectral gap of dense random regular graphs

Konstantin Tikhomirov and Pierre Youssef

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For any $\alpha\in(0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda({\mathbf{G}})\leq C_{\alpha}\sqrt{d}$ with probability at least $1-\frac{1}{n}$, where ${\mathbf{G}}$ is the uniform random undirected $d$-regular graph on $n$ vertices, $\lambda({\mathbf{G}})$ denotes its second largest eigenvalue (in absolute value) and $C_{\alpha}$ is a constant depending only on $\alpha$. Combined with earlier results in this direction covering the case of sparse random graphs, this completely settles the problem of estimating the magnitude of $\lambda({\mathbf{G}})$, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of Vu. The result is obtained as a consequence of an estimate for the second largest singular value of adjacency matrices of random directed graphs with predefined degree sequences. As the main technical tool, we prove a concentration inequality for arbitrary linear forms on the space of matrices, where the probability measure is induced by the adjacency matrix of a random directed graph with prescribed degree sequences. The proof is a nontrivial application of the Freedman inequality for martingales, combined with self-bounding and tensorization arguments. Our method bears considerable differences compared to the approach used by Broder et al. [SIAM J. Comput. 28 (1999) 541–573] who established the upper bound for $\lambda({\mathbf{G}})$ for $d=o(\sqrt{n})$, and to the argument of Cook, Goldstein and Johnson [Ann. Probab. 46 (2018) 72–125] who derived a concentration inequality for linear forms and estimated $\lambda({\mathbf{G}})$ in the range $d=O(n^{2/3})$ using size-biased couplings.

Article information

Ann. Probab., Volume 47, Number 1 (2019), 362-419.

Received: March 2017
Revised: November 2017
First available in Project Euclid: 13 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Random regular graphs uniform model spectral gap random matrices


Tikhomirov, Konstantin; Youssef, Pierre. The spectral gap of dense random regular graphs. Ann. Probab. 47 (2019), no. 1, 362--419. doi:10.1214/18-AOP1263.

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