All of my work thus far is connected in some way with random matrices. Much of my work going back to my PhD centers on using tools from high-dimensional probability and graph theory to understand the spectrum and condition number of various non-Hermitian random matrix models. Recently I've branched in some new directions (see below).

Current interests:

  • Large deviations for nonlinear functionals of random graphs and random matrices, and connections with the naive mean field approximation in statistical physics;
  • Extreme values of logarithmically correlated fields, with connections to random matrices, number theory, branching processes and PDE (the Fisher–KPP equation and related reaction-diffusion equations);
  • Brown's spectral measure for non-normal operators and related topics in random matrix theory and free probability.

See also my arXiv page and Google Scholar citations.

Papers and preprints:

  • Convergence to Brown's spectral measure for quadratic polynomials in Ginibre matrices, slides
    with Alice Guionnet and Jonathan Husson.
    In preparation.
  • Non-Hermitian random matrices with a variance profile (II): Properties and examples,
    with Walid Hachem, Jamal Najim and David Renfrew.
    In preparation.
  • 10.
    Large deviations of subgraph counts for sparse Erdős–Rényi graphs, arXiv
    slides
    with Amir Dembo.
    Preprint.
  • 9.
    Maximum for the characteristic polynomial of a random permutation matrix, arXiv
    slides
    with Ofer Zeitouni.
    Commun. Pure Appl. Math, to appear.
  • 8.
    Circular law for the sum of random permutation matrices, arXiv
    slides
    with Anirban Basak and Ofer Zeitouni.
    Electronic Journal of Probability, 23:paper no. 33, 51 pp., 2018. doi:10.1214/18-EJP162
  • 7.
    The circular law for random regular digraphs. arXiv
    slides
    Ann. Inst. H. Poincaré, Probab. Statist., 55(4):2111–2167, 2019. doi:10.1214/18-AIHP943
  • 6.
    Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and ESDs, arXiv
    slides
    with Walid Hachem, Jamal Najim and David Renfrew.
    Electronic Journal of Probability, 23:paper no. 110, 61 pp., 2018. doi:10.1214/18-EJP230
  • Spectral properties of non-Hermitian random matrices. url
    PhD Thesis, UCLA.
  • 5.
    Lower bounds for the smallest singular value of structured random matrices. arXiv
    Annals of Probability, 46(6):3442–3500, 2018. doi:10.1214/17-AOP1251
  • 4.
    Size biased couplings and the spectral gap for random regular graphs, arXiv
    with Larry Goldstein and Toby Johnson.
    Annals of Probability, 46(1):72–125, 2018. doi:10.1214/17-AOP1180
  • 3.
    The circular law for random regular digraphs with random edge weights. arXiv
    Random Matrices: Theory Appl., 06, 1750012, 2017. doi:10.1142/S2010326317500125.
  • 2.
    Discrepancy properties for random regular digraphs. arXiv
    Random Struct. Algor., 50:23–58, 2016. doi:10.1002/rsa.20643.
  • 1.
    On the singularity of adjacency matrices for random regular digraphs. arXiv
    Probab. Theory Relat. Fields, 167(1):143--200, Feb 2017. doi:10.1007/s00440-015-0679-8.

Slides:

  • Pseudospectra of structured random matrices. pdf
    Random matrices workshop, Oberwolfach, Dec 2019
  • Large deviations for sparse random graphs. pdf
    Random matrices and random graphs workshop, CIRM, Luminy, Apr 2019
  • Large deviations of subgraph counts for sparse random graphs. pdf
    Amir Dembo’s 60th birthday conference, Stanford, Dec 2018
  • Pseudospectrum and spectral anti-concentration: Beyond mean field models. pdf
    IPAM QLA Culminating Workshop, Lake Arrowhead, CA, June 2018
  • The maximum of the characteristic polynomial for a random permutation matrix. pdf
    Random matrices and free probability workshop, IPAM, UCLA, May 2018
  • Circular laws for random regular digraphs. pdf
    AMS Fall Western Sectional Meeting, UC Riverside, Nov 2017
  • Inhomogeneous circular laws for random matrices with non-identically distributed entries. pdf
    Probability and Mathematical Physics Seminar, UC Davis, April 2017
  • Random regular digraphs: singularity and spectrum. pdf
    Probability Seminar, Stanford, Nov 2015