Non-intersecting squared Bessel paths: critical time and double scaling limit
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 reg...
| 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/10835/1590 |
| _version_ | 1789406495510626304 |
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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 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. |
| format | info:eu-repo/semantics/article |
| id | oai:repositorio.ual.es:10835-1590 |
| institution | Universidad de Cuenca |
| language | English |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oai:repositorio.ual.es:10835-15902023-04-12T19:37:56Z Non-intersecting squared Bessel paths: critical time and double scaling limit Kuijlaars, A. B. J. Martínez-Finkelshtein, Andrei Wielonsky, F. Double scaling Model of n Squared Bessel processes Doble escala Modelo de n Procesos cuadrados de Bessel 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. 2012-07-12T11:52:40Z 2012-07-12T11:52:40Z 2011 info:eu-repo/semantics/article http://hdl.handle.net/10835/1590 en info:eu-repo/semantics/openAccess Communications in Mathematical Physics Vol. 308, Number 1 (2011) |
| spellingShingle | Double scaling Model of n Squared Bessel processes Doble escala Modelo de n Procesos cuadrados de Bessel Kuijlaars, A. B. J. Martínez-Finkelshtein, Andrei Wielonsky, F. Non-intersecting squared Bessel paths: critical time and double scaling limit |
| title | Non-intersecting squared Bessel paths: critical time and double scaling limit |
| title_full | Non-intersecting squared Bessel paths: critical time and double scaling limit |
| title_fullStr | Non-intersecting squared Bessel paths: critical time and double scaling limit |
| title_full_unstemmed | Non-intersecting squared Bessel paths: critical time and double scaling limit |
| title_short | Non-intersecting squared Bessel paths: critical time and double scaling limit |
| title_sort | non-intersecting squared bessel paths: critical time and double scaling limit |
| topic | Double scaling Model of n Squared Bessel processes Doble escala Modelo de n Procesos cuadrados de Bessel |
| url | http://hdl.handle.net/10835/1590 |
| work_keys_str_mv | AT kuijlaarsabj nonintersectingsquaredbesselpathscriticaltimeanddoublescalinglimit AT martinezfinkelshteinandrei nonintersectingsquaredbesselpathscriticaltimeanddoublescalinglimit AT wielonskyf nonintersectingsquaredbesselpathscriticaltimeanddoublescalinglimit |