On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble
We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the n × n Laguerre unitary ensemble. The probability that this ratio is greater than r > 1 is expressed in terms of an n × n Hankel determinant with a perturbed Laguerre weight. The limit...
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| Format: | Journal article |
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IOP Publishing
2018
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| _version_ | 1826287711938412544 |
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| author | Atkin, MR Charlier, C Zohren, S |
| author_facet | Atkin, MR Charlier, C Zohren, S |
| author_sort | Atkin, MR |
| collection | OXFORD |
| description | We study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the n × n Laguerre unitary ensemble. The probability that this ratio is greater than r > 1 is expressed in terms of an n × n Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as n → ∞ is found as an integral over (0, ∞) containing two functions q1(x) and q2(x). These functions satisfy a system of two coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. We compute asymptotic behaviours of these functions as rx → 0+ and (r − 1)x → ∞, as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x. |
| first_indexed | 2024-03-07T02:02:45Z |
| format | Journal article |
| id | oxford-uuid:9deda560-8319-4dde-9866-d4fb32f39cf9 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:02:45Z |
| publishDate | 2018 |
| publisher | IOP Publishing |
| record_format | dspace |
| spelling | oxford-uuid:9deda560-8319-4dde-9866-d4fb32f39cf92022-03-27T00:46:37ZOn the ratio probability of the smallest eigenvalues in the Laguerre unitary ensembleJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:9deda560-8319-4dde-9866-d4fb32f39cf9Symplectic Elements at OxfordIOP Publishing2018Atkin, MRCharlier, CZohren, SWe study the probability distribution of the ratio between the second smallest and smallest eigenvalue in the n × n Laguerre unitary ensemble. The probability that this ratio is greater than r > 1 is expressed in terms of an n × n Hankel determinant with a perturbed Laguerre weight. The limiting probability distribution for the ratio as n → ∞ is found as an integral over (0, ∞) containing two functions q1(x) and q2(x). These functions satisfy a system of two coupled Painlevé V equations, which are derived from a Lax pair of a Riemann–Hilbert problem. We compute asymptotic behaviours of these functions as rx → 0+ and (r − 1)x → ∞, as well as large n asymptotics for the associated Hankel determinants in several regimes of r and x. |
| spellingShingle | Atkin, MR Charlier, C Zohren, S On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble |
| title | On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble |
| title_full | On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble |
| title_fullStr | On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble |
| title_full_unstemmed | On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble |
| title_short | On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble |
| title_sort | on the ratio probability of the smallest eigenvalues in the laguerre unitary ensemble |
| work_keys_str_mv | AT atkinmr ontheratioprobabilityofthesmallesteigenvaluesinthelaguerreunitaryensemble AT charlierc ontheratioprobabilityofthesmallesteigenvaluesinthelaguerreunitaryensemble AT zohrens ontheratioprobabilityofthesmallesteigenvaluesinthelaguerreunitaryensemble |