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    Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights

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    Kuijlaars-Non intersecting.pdf (626.4Kb)
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    URI: http://hdl.handle.net/10835/1629
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    Author/s
    Kuijlaars, A. B. J.; Martínez-Finkelshtein, Andrei; Wielonsky, F.
    Date
    2009
    Abstract
    We study 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 $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using t...
    Palabra/s clave
    Procesos de Bessel
    Polinomios ortogonales
    Pesos de Bessel
    Squared Bessel processes
    Orthogonal polynomials
    Modified Bessel weights
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    • Artículos de revista Dpto. Matemáticas [119]

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