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

We derive bulk asymptotics of skew-orthogonal polynomials (sop) ${\pi }_m^{\text{bt}}$, $\beta =1$, 4, defined w.r.t. the weight $\exp (-2NV(x))$, $V (x)=gx^4/4+tx^2/2$,$g>0$and$t<0$\mathrm{.}Weassumethatas$m,N \to\infty$ there exists an $ϵ>0$, such that $\epsilon\leq (m/N)\leq \lambda_{\rm cr}-\epsilon$,where$\lambda_{\rm cr}$ 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.