Universality in the two‐matrix model: a Riemann‐Hilbert steepest‐descent analysis

The eigenvalue statistics of a pair ( M 1 , M 2 ) of n × n Hermitian matrices taken randomly with respect to the measure can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest‐descent analysis of a 4 × 4 matrix‐valued Riemann‐Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case W ( y ) = y 4 /4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M 1 (when averaged over M 2 ) in the global and local regime as n → ∞ in the one‐cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.