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 (sop) $π^{\bt}_{m}$, $\beta =1$, 4, defined w.r.t. the weight $\exp (-2NV(x))$, $V(x)=g{x}^4/4+t{x}^2/2$, $g>0$ and $t<0$. We assume that as $m,N\to \infty$ there exists an $\epsilon >0$, such that $\epsilon \le (m/N)\le {\lambda }_{\mathrm{cr}}-\epsilon$, where ${\lambda }_{\mathrm{cr}}$ is the critical value which separates sop 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 sop and orthogonal polynomials (op) and using asymptotic results of op derived in \cite{bleher}. Finally, we apply these asymptotic results of sop and their recursion coefficients in the generalized Christoffel-Darboux formula (GCD) \cite{ghosh3} to obtain level densities and sine-kernels in the bulk of the spectrum for orthogonal and symplectic ensembles of random matrices.

6 pages

References in corpus (2)

• Generalized Christoffel-Darboux formula for skew-orthogonal polynomials and random matrix theory
• Skew-orthogonal polynomials, differential systems and random matrix theory