The Spectral Edge of Constant Degree Erd\H{o}s-R\'{e}nyi Graphs

We show that for an Erd\H{o}s-R\'{e}nyi graph on $N$ vertices with expected degree $d$ satisfying $\log^{-1/9}N\leq d\leq \log^{1/40}N$, the largest eigenvalues can be precisely determined by small neighborhoods around vertices of close to maximal degree. Moreover, under the added condition that $d\geq\log^{-1/15}N$, the corresponding eigenvectors are localized, in that the mass of the eigenvector decays exponentially away from the high degree vertex. This dependence on local neighborhoods implies that the edge eigenvalues converge to a Poisson point process. These theorems extend a result of Alt, Ducatez, and Knowles, who showed the same behavior for $d$ satisfying $(\log\log N)^4\ll d\leq (1-o_{N}(1))\frac{1}{\log 4-1}\log N$. To achieve high accuracy in the constant degree regime, instead of attempting to guess an approximate eigenvector of a local neighborhood, we analyze the true eigenvector of a local neighborhood, and show it must be localized and depend on local geometry.