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We study unitary random matrix ensembles of the form \[Z_{n,N}^{-1} |\det M|^{2\alpha} e^{-N \mathrm{Tr} V(M)}dM,\] where $ \alpha>-1/2$ and $V$ is such that the limiting mean eigenvalue density for $n,N\to\infty$ and $n/N\to 1$ vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight $|x|^{2\alpha}e^{-NV(x)}$. Here the main focus is on the construction of a local parametrix near the origin with $\psi$-functions associated with a special solution $q_\alpha$ of the Painlevé II equation $q”=sq+2q^3-\alpha$. We show that $q_\alpha$ has no real poles for $\alpha > -1/2$, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of $q_\alpha$ in the double scaling limit.
Tom Claeys 1 ; Arno B. J. Kuijlaars 1 ; Maarten Vanlessen 1
@article{10_4007_annals_2008_168_601,
author = {Tom Claeys and Arno B. J. Kuijlaars and Maarten Vanlessen},
title = {Multi-critical unitary random matrix ensembles and the general {Painlev\'e} {II} equation},
journal = {Annals of mathematics},
pages = {601--641},
publisher = {mathdoc},
volume = {168},
number = {2},
year = {2008},
doi = {10.4007/annals.2008.168.601},
mrnumber = {2434886
},
zbl = {1179.15037},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.601/}
}
TY - JOUR AU - Tom Claeys AU - Arno B. J. Kuijlaars AU - Maarten Vanlessen TI - Multi-critical unitary random matrix ensembles and the general Painlevé II equation JO - Annals of mathematics PY - 2008 SP - 601 EP - 641 VL - 168 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.601/ DO - 10.4007/annals.2008.168.601 LA - en ID - 10_4007_annals_2008_168_601 ER -
%0 Journal Article %A Tom Claeys %A Arno B. J. Kuijlaars %A Maarten Vanlessen %T Multi-critical unitary random matrix ensembles and the general Painlevé II equation %J Annals of mathematics %D 2008 %P 601-641 %V 168 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.601/ %R 10.4007/annals.2008.168.601 %G en %F 10_4007_annals_2008_168_601
Tom Claeys; Arno B. J. Kuijlaars; Maarten Vanlessen. Multi-critical unitary random matrix ensembles and the general Painlevé II equation. Annals of mathematics, Tome 168 (2008) no. 2, pp. 601-641. doi: 10.4007/annals.2008.168.601
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