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We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the ErdH{o}s-Renyi graph $G(N,p)$. Tracy-Widom fluctuations of the extreme eigenvalues for $pgg N^{-2/3}$ was proved in [17,46]. We prove that there is a crossover in the behavior of the extreme eigenvalues at $psim N^{-2/3}$. In the case that $N^{-7/9}ll pll N^{-2/3}$, we prove that the extreme eigenvalues have asymptotically Gaussian fluctuations. Under a mean zero condition and when $p=CN^{-2/3}$, we find that the fluctuations of the extreme eigenvalues are given by a combination of the Gaussian and the Tracy-Widom distribution. These results show that the eigenvalues at the edge of the spectrum of sparse ErdH{o}s-Renyi graphs are less rigid than those of random $d$-regular graphs [4] of the same average degree.
We consider an inhomogeneous ErdH{o}s-Renyi random graph $G_N$ with vertex set $[N] = {1,dots,N}$ for which the pair of vertices $i,j in [N]$, $i eq j$, is connected by an edge with probability $r(tfrac{i}{N},tfrac{j}{N})$, independently of other pai
We consider a dynamic ErdH{o}s-Renyi random graph (ERRG) on $n$ vertices in which each edge switches on at rate $lambda$ and switches off at rate $mu$, independently of other edges. The focus is on the analysis of the evolution of the associated empi
We develop a quantitative large deviations theory for random Bernoulli tensors. The large deviation principles rest on a decomposition theorem for arbitrary tensors outside a set of tiny measure, in terms of a novel family of norms generalizing the c
Let $bY =bR+bX$ be an $Mtimes N$ matrix, where $bR$ is a rectangular diagonal matrix and $bX$ consists of $i.i.d.$ entries. This is a signal-plus-noise type model. Its signal matrix could be full rank, which is rarely studied in literature compared w
We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converg