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We derive bulk asymptotics of skew-orthogonal polynomials (sop)$\pi^{\bt}_{m}$, $\beta=1$, 4, defined w.r.t. the weight $\exp(-2NV(x))$, $V(x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ thereexists an $\epsilon > 0$, such that $\epsilon\leq (m/N)\leq \lambda_{\rmcr}-\epsilon$, where $\lambda_{\rm cr}$ is the critical value which separatessop with two cuts from those with one cut. Simultaneously we derive asymptoticsfor the recursive coefficients of skew-orthogonal polynomials. The proof isbased on obtaining a finite term recursion relation between sop and orthogonalpolynomials (op) and using asymptotic results of op derived in \cite{bleher}.Finally, we apply these asymptotic results of sop and their recursioncoefficients in the generalized Christoffel-Darboux formula (GCD) \cite{ghosh3}to obtain level densities and sine-kernels in the bulk of the spectrum fororthogonal and symplectic ensembles of random matrices.

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