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- W2950207218 abstract "For any $alphain (0,1)$ and any $n^{alpha}leq dleq n/2$, we show that $lambda(G)leq C_alpha sqrt{d}$ with probability at least $1-frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $lambda(G)$ 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 $lambda(G)$, 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 {it 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 non-trivial application of the Freedman inequality for martingales, combined with boots-trapping and tensorization arguments. Our method bears considerable differences compared to the approach used by Broder, Frieze, Suen and Upfal (1999) who established the upper bound for $lambda(G)$ for $d=o(sqrt{n})$, and to the argument of Cook, Goldstein and Johnson (2015) who derived a concentration inequality for linear forms and estimated $lambda(G)$ in the range $d= O(n^{2/3})$ using size-biased couplings." @default.
- W2950207218 created "2019-06-27" @default.
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- W2950207218 date "2016-10-06" @default.
- W2950207218 modified "2023-09-23" @default.
- W2950207218 title "The spectral gap of dense random regular graphs" @default.
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