The spectral gap of random regular graphs

We bound the second eigenvalue of random d$$ d $$ ‐regular graphs, for a wide range of degrees d$$ d $$ , using a novel approach based on Fourier analysis. Let Gn,d$$ {G}_{n,d} $$ be a uniform random d$$ d $$ ‐regular graph on n$$ n $$ vertices, and λ(Gn,d)$$ \lambda \left({G}_{n,d}\right) $$ be its second largest eigenvalue by absolute value. For some constant c>0$$ c>0 $$ and any degree d$$ d $$ with log10n≪d≤cn$$ {\log}^{10}n\ll d\le cn $$ , we show that λ(Gn,d)=(2+o(1))d(n−d)/n$$ \lambda \left({G}_{n,d}\right)=\left(2+o(1)\right)\sqrt{d\left(n-d\right)/n} $$ asymptotically almost surely. Combined with earlier results that cover the case of sparse random graphs, this fully determines the asymptotic value of λ(Gn,d)$$ \lambda \left({G}_{n,d}\right) $$ for all d≤cn$$ d\le cn $$ . To achieve this, we introduce new methods that use mechanisms from discrete Fourier analysis, and combine them with existing tools and estimates on d$$ d $$ ‐regular random graphs—especially those of Liebenau and Wormald.