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arXiv · 2112.14107

Spherical Sherrington-Kirkpatrick model for deformed Wigner matrix with fast decaying edges

Ji Oon Lee·Yiting Li
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Abstract

We consider the $2$-spin spherical Sherrington--Kirkpatrick model whose disorder is given by a deformed Wigner matrix of the form $W+λV$, where $W$ is a Wigner matrix and $V$ is a random diagonal matrix with i.i.d. entries. Assuming that the density function of the entries of $V$ decays faster than a certain rate near the edges of its spectrum, we prove the sharp phase transition of the limiting free energy and its fluctuation. In the high temperature regime, the fluctuation of $F_N$ converges in distribution to a Gaussian distribution, whereas it converges to a Weibull distribution in the low temperature regime. We also prove several results for deformed Wigner matrices, including a local law for the resolvent entries, a central limit theorem of the linear spectral statistics, and a theorem on the rigidity of eigenvalues.

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Ji Oon Lee, Yiting Li. 2022-10-12. Spherical Sherrington-Kirkpatrick model for deformed Wigner matrix with fast decaying edges. https://arxiv.org/abs/2112.14107

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