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Abstract: We derive bulk asymptotics of skew-orthogonal polynomials sop$\pi^{\bt} {m}$, $\beta=1$, 4, defined w.r.t. the weight $\exp-2NVx$, $Vx=gx^4-4+tx^2-2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ thereexists an $\epsilon > 0$, such that $\epsilon\leq m-N\leq \lambda { mcr}-\epsilon$, where $\lambda { m cr}$ 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.



Author: Saugata Ghosh

Source: https://arxiv.org/



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