1 June 2013 A critical phenomenon in the two-matrix model in the quartic/quadratic case
Maurice Duits, Dries Geudens
Duke Math. J. 162(8): 1383-1462 (1 June 2013). DOI: 10.1215/00127094-2208757


We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a 4×4 Riemann–Hilbert problem related to the Hastings–McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appeared in the literature before. First, we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit. Second, we briefly discuss the relation with the tacnode singularity in noncolliding Brownian motions that was recently analyzed. Although the limiting density in that model also vanishes like a square root at a certain interior point, the process at the local scale is different from the process that we obtain in the two-matrix model.


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Maurice Duits. Dries Geudens. "A critical phenomenon in the two-matrix model in the quartic/quadratic case." Duke Math. J. 162 (8) 1383 - 1462, 1 June 2013. https://doi.org/10.1215/00127094-2208757


Published: 1 June 2013
First available in Project Euclid: 28 May 2013

zbMATH: 1286.60006
MathSciNet: MR3079252
Digital Object Identifier: 10.1215/00127094-2208757

Primary: 60B20 , 82B27
Secondary: 15B52 , 30E25 , 31A05 , 42C05

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 8 • 1 June 2013
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