Universality in the two-matrix model: A Riemann-Hilbert Steepest-Descent Analysis
2009 (English)In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 62, no 8, 1076-1153 p.Article in journal (Refereed) Published
The eigenvalue statistics of a pair (M(1), M(2)) of n x n Hermitian matrices taken randomly with respect to the measure 1/Z(n) exp (-n Tr(V(M(1)) + W(M(2)) - tau M(1)M(2)))dM(1) dM(2) can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 x 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 -> infinity 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.
Place, publisher, year, edition, pages
2009. Vol. 62, no 8, 1076-1153 p.
IdentifiersURN: urn:nbn:se:kth:diva-93328DOI: 10.1002/cpa.20269ISI: 000267381600004OAI: oai:DiVA.org:kth-93328DiVA: diva2:515688
QC 201206262012-04-142012-04-142012-06-26Bibliographically approved