Abstract
We present a large deviation principle at speed N for the largest eigenvalue of some additively deformed Wigner matrices. In particular this includes Gaussian ensembles with full-rank general deformation. For the non-Gaussian ensembles, the deformation should be diagonal, and we assume that the laws of the entries have sharp sub-Gaussian Laplace transforms and satisfy certain concentration properties. For these latter ensembles we establish the large deviation principle in a restricted range $(-\infty, x_c)$, where $x_c$ depends on the deformation only and can be infinite.
Highlights
1.1 Deformed ensembles: typical behaviorIn this paper, our goal is to prove a large deviation principle (LDP) for the largest eigenvalue of the random matrixXN = √WN + DN
Our goal is to prove a large deviation principle (LDP) for the largest eigenvalue of the random matrix
We assume that DN is a deterministic matrix whose empirical spectral measure tends to a deterministic limit μD and whose extreme eigenvalues tend to the edges of μD
Summary
Our goal is to prove a large deviation principle (LDP) for the largest eigenvalue of the random matrix. We assume that DN is a deterministic matrix whose empirical spectral measure tends to a deterministic limit μD and whose extreme eigenvalues tend to the edges of μD. LDP for deformed Wigner matrices the deformed Gaussian models even when DN is not diagonal. Our model exhibits edge universality for many choices of DN ; that is, the fluctuations of λN (XN ), rescaled appropriately, are known to follow the Tracy-Widom distribution. This was first established by [36] for the deformed GUE, if μDN → μD quickly (d(μDN , μD) = O(N −2/3− ) is enough, where d is defined in Equation (1.5)) and without outliers. The assumption of Gaussianity was removed by [30], under a similar technical assumption on μD
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