Partition function for multi-cut matrix models

Abstract

We consider the partition function Z N of a random matrix model with polynomial potential V(ξ) = t 1 ξ + t 2 ξ 2 + ··· + t 2d ξ 2d . It is known that the second logarithmic derivative of Z N with respect to the times t k can be expressed in terms of the recurrence coefficients of the related orthogonal polynomials. An explicit formula for the recurrence coefficients of the orthogonal polynomials in the limit N → oo for multi-cut regular V (ξ) has been derived in [ 10] through the Riemann-Hilbert approach. The expression for Z N in the limit N → ∞ has been derived in [7] through a mean-field approach. We show that the above asymptotic formulae satisfy the same relations that hold for finite N.