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A critical phenomenon in the two-matrix model in the quartic/quadratic case
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-7598-4521
2013 (English)In: Duke mathematical journal, ISSN 0012-7094, Vol. 162, no 8, 1383-1462 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2013. Vol. 162, no 8, 1383-1462 p.
Keyword [en]
Double Scaling Limit, Riemann-Hilbert Problem, Random-Matrix Theory, Biorthogonal Polynomials, Eigenvalue Correlations, Exponential Weights, Brownian Motions, External Source, Pearcey Process, Universality
National Category
URN: urn:nbn:se:kth:diva-134290DOI: 10.1215/00127094-2208757ISI: 000319794500001ScopusID: 2-s2.0-84880293602OAI: diva2:665713
Knut and Alice Wallenberg Foundation, KAW 2010.0063

QC 20131120

Available from: 2013-11-20 Created: 2013-11-20 Last updated: 2013-12-09Bibliographically approved

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Duits, Maurice
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