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Title: Universality of the double scaling limit in random matrix models
Authors: Claeys, Tom ×
Kuijlaars, Arno #
Issue Date: 2006
Publisher: John wiley & sons inc
Series Title: Communications on pure and applied mathematics vol:59 issue:11 pages:1573-1603
Abstract: We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential.


We study unitary random matrix ensembles in the critical case
where the limiting mean eigenvalue density vanishes quadratically
at an interior point of the support. We establish universality of
the limits of the eigenvalue correlation kernel at such a critical
point in a double scaling limit. The limiting kernels are constructed
out of functions associated with the second Painlev\'e equation.
This extends a result of Bleher and Its for the
special case of a critical quartic potential.

The two main tools we use are equilibrium measures and
Riemann-Hilbert problems. In our treatment of equilibrium measures
we allow a negative density near the critical point, which
enables us to treat all cases simultaneously. The asymptotic
analysis of the Riemann-Hilbert problem is done with the
Deift/Zhou steepest descent analysis. For the construction of
a local parametrix at the critical point we introduce a
modification of the approach of Baik, Deift, and Johansson
so that we are able to satisfy the required jump properties
exactly.
URI: 
ISSN: 0010-3640
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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