Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local law and Tracy-Widom limit for sparse random matrices   We consider spectral properties and the edge universality of sparse random
matrices, the class of random matrices that includes the adjacency matrices of
the Erdos-Renyi graph model $G(N,p)$. We prove a local law for the eigenvalue
density up to the spectral edges. Under a suitable condition on the sparsity,
we also prove that the rescaled extremal eigenvalues exhibit GOE Tracy-Widom
fluctuations if a deterministic shift of the spectral edge due to the sparsity
is included. For the adjacency matrix of the Erdos-Renyi graph this establishes
the Tracy-Widom fluctuations of the second largest eigenvalue for $p\gg
N^{-2/3}$ with a deterministic shift of order $(Np)^{-1}$.

33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
5
The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat