Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights
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 fil...
| Main Authors: | , , |
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| Format: | info:eu-repo/semantics/article |
| Language: | English |
| Published: |
2012
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| Subjects: | |
| Online Access: | http://hdl.handle.net/10835/1629 |
| _version_ | 1789406321300209664 |
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| author | Kuijlaars, A. B. J. Martínez-Finkelshtein, Andrei Wielonsky, F. |
| author_facet | Kuijlaars, A. B. J. Martínez-Finkelshtein, Andrei Wielonsky, F. |
| author_sort | Kuijlaars, A. B. J. |
| collection | DSpace |
| description | 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 the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-1629 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-16292023-04-12T19:36:35Z Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights Kuijlaars, A. B. J. Martínez-Finkelshtein, Andrei Wielonsky, F. Procesos de Bessel Polinomios ortogonales Pesos de Bessel Squared Bessel processes Orthogonal polynomials Modified Bessel weights 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 the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest. 2012-08-01T08:31:56Z 2012-08-01T08:31:56Z 2009 info:eu-repo/semantics/article http://hdl.handle.net/10835/1629 en info:eu-repo/semantics/openAccess Communications in Mathematical Physics Volume 286, Issue 1 |
| spellingShingle | Procesos de Bessel Polinomios ortogonales Pesos de Bessel Squared Bessel processes Orthogonal polynomials Modified Bessel weights Kuijlaars, A. B. J. Martínez-Finkelshtein, Andrei Wielonsky, F. Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights |
| title | Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights |
| title_full | Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights |
| title_fullStr | Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights |
| title_full_unstemmed | Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights |
| title_short | Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights |
| title_sort | non-intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights |
| topic | Procesos de Bessel Polinomios ortogonales Pesos de Bessel Squared Bessel processes Orthogonal polynomials Modified Bessel weights |
| url | http://hdl.handle.net/10835/1629 |
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