SIMIS Seminar series on Random matrix theory, Integrable systems and Applications: 2D Coulomb Gases and Their Statistical Behaviors

Meng Yang (Great Bay University)

Abstract: We consider 2D Coulomb gases with the external potential Q(z) = |z|2 −2clog|z −a|, where c > 0 and a ∈ C. Equivalently, this model can be realised as N eigenvalues of the complex Ginibre matrix of size (c + 1)N ×(c+1)N conditioned to have deterministic eigenvalue a with multiplicity cN. Depending on the values of c and a, the droplet reveals a phase transition: it is doubly connected in the post-critical regime and simply connected in the pre-critical regime. In both regimes, we derive precise large-N expansions of the free energy up to the O(1) term, providing a non-radially symmetric example that confirms the Zabrodin-Wiegmann conjecture made for general planar Coulomb gas ensembles. As a consequence, our results provide asymptotic behaviours of moments of the characteristic polynomial of the complex Ginibre matrix, where the powers are of order O(N). Furthermore, by combining with a duality formula, we obtain precise large deviation probabilities of the smallest eigenvalue of the Laguerre unitary ensemble. This talk is based on the joint work with Sung-Soo Byun and Seong-Mi Seo.

Time: 18th April 2025, 11:00h-12:00h

Location: Room 1310, SIMIS

Send comments or questions to: Miguel Tierz to tierz at simis.cn

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Biography of the speaker: I am Meng Yang, an assistant professor at the Great Bay University. I completed my Ph.D at the University of South Florida. Before joining the GBU, I worked at UCLouvain and the University of Copenhagen. My primary research focus includes mathematical physics, random matrix theory and related areas, such as Riemann–Hilbert problems, orthogonal polynomials, multiple orthogonal polynomials, Coulomb gases, etc..