Double scaling limit in random matrix models and a nonlinear hierarchy of differential equations

Authors: Bleher P.1; Eynard B.2

Source: Journal of Physics A: Mathematical and General, Volume 36, Number 12, 2003 , pp. 3085-3105(21)

Publisher: Institute of Physics Publishing

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Abstract:

We study the critical behaviour of a random Hermitian one-matrix model with nonsymmetric interaction at a critical point, in which the eigenvalue density function has a zero of degree 2m, m 1, inside a cut. We prove that in the generic case, m 1, the model exhibits a third-order phase transition in temperature. We formulate an ansatz for the double scaling limit of recurrence coefficients, which is consistent with the quasiperiodic asymptotics of recurrence coefficients in the low temperature region, and from this ansatz we derive the Painlevé II hierarchy of ordinary differential equations for the recurrence coefficients. In addition, we derive an integral kernel which governs the double scaling limit of correlation functions.

Language: English

Document Type: Miscellaneous

Affiliations: 1: Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N Blackford Street, Indianapolis, IN 46202, USA 2: Service de Physique Théorique, Saclay, F-91191 Gif-sur-Yvette Cedex, France

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