The spectral gap of dense random regular graphs

@article{Tikhomirov2019TheSG,
  title={The spectral gap of dense random regular graphs},
  author={Konstantin E. Tikhomirov and Pierre Youssef},
  journal={The Annals of Probability},
  year={2019}
}
For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, 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… Expand
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Size biased couplings and the spectral gap for random regular graphs
Let λλ be the second largest eigenvalue in absolute value of a uniform random dd-regular graph on nn vertices. It was famously conjectured by Alon and proved by Friedman that if dd is fixedExpand
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