Universality in the two matrix model with a monomial quartic and a general even polynomial potential

In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with the probability measure Z−1 n exp (−n (tr(V (M1) +W (M2)− τM1M2)) dM1dM2, in the case where W = y 4 4 and V is a general even polynomial. We studied the correlation kernel for the eigenvalues of the matrix M1 in the limit as n → ∞. We extended the results of Duits and Kuijlaars in [14] to the case when the limiting eigenvalue density for M1 is supported on multiple intervals. The results are achieved by constructing the parametrix to a Riemann-Hilbert problem obtained in [14] with theta functions and then showing that this parametrix is well-defined for all n by studying the theta divisor.