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arXiv · 0808.1256

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

Saugata Ghosh
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Abstract

We derive bulk asymptotics of skew-orthogonal polynomials (sop) $π^{\bt}_{m}$, $β=1$, 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 $m,N \to\infty$ there exists an $ε> 0$, such that $ε\leq (m/N)\leq λ_{\rm cr}-ε$, where $λ_{\rm cr}$ 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.

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Saugata Ghosh. 2008-08-08. Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model. https://doi.org/10.1063/1.3093266

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