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Double scaling limit in the random matrix model: The Riemann‐Hilbert approach
Author(s) -
Bleher Pavel M.,
Its Alexander R.
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10065
Subject(s) - scaling limit , mathematics , semiclassical physics , unitary matrix , scaling , integrable system , quartic function , limit (mathematics) , random matrix , riemann hypothesis , mathematical analysis , matrix (chemical analysis) , kernel (algebra) , mathematical physics , pure mathematics , unitary state , quantum mechanics , physics , eigenvalues and eigenvectors , geometry , materials science , political science , law , composite material , quantum
We prove the existence of the double scaling limit in the unitary matrix model with quartic interaction, and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the ψ function for the Hastings‐McLeod solution to the Painlevé II equation. The proof is based on the Riemann‐Hilbert approach, and the central point of the proof is an analysis of analytic semiclassical asymptotics for the ψ function at the critical point in the presence of four coalescing turning points. © 2003 Wiley Periodicals, Inc.