Local law and Tracy–Widom limit for sparse random matrices

J. Lee, K. Schnelli, Probability Theory and Related Fields 171 (2018).


Journal Article | Published | English
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Abstract
We consider spectral properties and the edge universality of sparse random matrices, the class of random matrices that includes the adjacency matrices of the Erdős–Rényi 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 Erdős–Rényi graph this establishes the Tracy–Widom fluctuations of the second largest eigenvalue when p is much larger than N−2/3 with a deterministic shift of order (Np)−1.
Publishing Year
Date Published
2018-06-14
Journal Title
Probability Theory and Related Fields
Volume
171
Issue
1-2
Article Number
543-616
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Cite this

Lee J, Schnelli K. Local law and Tracy–Widom limit for sparse random matrices. Probability Theory and Related Fields. 2018;171(1-2). doi:10.1007/s00440-017-0787-8
Lee, J., & Schnelli, K. (2018). Local law and Tracy–Widom limit for sparse random matrices. Probability Theory and Related Fields, 171(1–2). https://doi.org/10.1007/s00440-017-0787-8
Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” Probability Theory and Related Fields 171, no. 1–2 (2018). https://doi.org/10.1007/s00440-017-0787-8.
J. Lee and K. Schnelli, “Local law and Tracy–Widom limit for sparse random matrices,” Probability Theory and Related Fields, vol. 171, no. 1–2, 2018.
Lee J, Schnelli K. 2018. Local law and Tracy–Widom limit for sparse random matrices. Probability Theory and Related Fields. 171(1–2).
Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random Matrices.” Probability Theory and Related Fields, vol. 171, no. 1–2, 543–616, Springer, 2018, doi:10.1007/s00440-017-0787-8.

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