Non-Intersecting Squared Bessel Paths: Critical Time and Double Scaling Limit

  • A. B. J. KuijlaarsEmail author
  • A. Martínez-Finkelshtein
  • F. Wielonsky


We consider the double scaling limit for a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n → ∞ that intersects the hard edge at x = 0 at a critical time t = t *. In a previous paper, the scaling limits for the positions of the paths at time t ≠ t * were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as n → ∞ of the correlation kernel at critical time t * and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued Riemann-Hilbert problem by the Deift-Zhou steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions of a particular third-order linear differential equation, and its matching with a global parametrix.


Riemann Surface Critical Time Random Matrix Theory Bessel Process Correlation Kernel 
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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • A. B. J. Kuijlaars
    • 1
    Email author
  • A. Martínez-Finkelshtein
    • 2
    • 3
  • F. Wielonsky
    • 4
  1. 1.Department of MathematicsKatholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Department of Statistics and Applied MathematicsUniversity of AlmeríaAlmeríaSpain
  3. 3.Instituto Carlos I de Física Teórica y ComputacionalGranada UniversityGranadaSpain
  4. 4.Laboratoire d’Analyse, Topologie et ProbabilitésUniversité de ProvenceMarseille Cedex 20France

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