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 (β) m , β = 1, 4, defined w.r.t. the weight exp(−2N V (x)), V (x) = gx 4 /4 + tx 2 /2, g > 0 and t < 0. We assume that as m, N → ∞ there exists an ǫ > 0, such that ǫ ≤ (m/N ) ≤ λcr − ǫ, where λcr 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 [1]. Finally, we apply these asymptotic results of sop and their recursion coefficients in the generalized Christoffel-Darboux formula (GCD) [8] to obtain level densities and sine-kernels in the bulk of the spectrum for orthogonal and symplectic ensembles of random matrices.

Generalized Christoffel–Darboux formula for classical skew-orthogonal polynomials

“…We would like to emphasize that the key step in deriving the asymptotic results of SOP is to obtain and solve finite term recursion relations between SOP and OP. Recently, this technique has been used to obtain bulk asymptotics of SOP corresponding to quartic double well potential [7]. Till date, this seems the easier method to study the asymptotic behavior of SOP rather than solving the 2d × 2d Riemann Hilbert problem [6,45].…”

“…The second and perhaps a more enriching method is to evaluate the kernel functions directly in terms of skew-orthogonal polynomials/skew-orthogonal functions (SOP) and use asymptotic properties of these functions [27]. For this, we need to develop the theory of SOP [3][4][5][6][7][8][28][29][30][31][32] so that we can have further insight into orthogonal and symplectic ensembles of random matrices.…”