Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model

Abstract

We derive bulk asymptotics of skew-orthogonal polynomials πm(β), β=1, 4, defined with respect to the weight exp(−2NV(x)), V(x)=gx4/4+tx2/2, g>0 and t<0. We assume that as m,N→∞, there exists an ϵ>0 such that ϵ≤(m/N)≤λcr−ϵ, where λcr is the critical value that separates skew-orthogonal polynomials with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between skew-orthogonal polynomials and orthogonal polynomials and using asymptotic results of orthogonal polynomials derived by Bleher and Its [“Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problems, and universality in the matrix model,” Ann. Math. 150, 185 (1999)]. Finally, we apply these asymptotic results of skew-orthogonal polynomials and their recursion coefficients in the generalized Christoffel–Darboux formula [Ghosh, S., “Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory,” J. Phys. A 39, 8775 (2006)] to obtain level densities and sine kernels in the bulk of the spectrum for orthogonal and symplectic ensembles of random matrices.