Determinantal Coulomb Gas Ensembles with a Class of Discrete Rotational Symmetric Potentials
Sung‐Soo Byun, Meng Yang
Abstract
.We consider determinantal Coulomb gas ensembles with a class of discrete rotational symmetric potentials whose droplets consist of several disconnected components. Under the insertion of a point charge at the origin, we derive the asymptotic behavior of the correlation kernels both at the macro- and microscopic scales. At the macroscopic scale, we use these results to show that there are strong correlations among the particles on the boundary of the droplets. At the microscopic scale, this gives rise to the universal edge scaling limit established by Hedenmalm and Wennman for a general class of potentials with connected droplets. For the proofs, we use the nonlinear steepest descent method on the matrix Riemann–Hilbert problem to derive the asymptotic behaviors of the associated planar orthogonal polynomials and their norms up to the first subleading terms.Keywordsdeterminantal Coulomb gas ensemblesedge universalitylemniscate archipelagogeneralized Christoffel–Darboux formulaorthogonal polynomialparabolic cylinder functionMSC codes60B2042C0534M5082B21