تقرير
On a distinguished family of random variables and Painlev\'e equations
| العنوان: | On a distinguished family of random variables and Painlev\'e equations |
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| المؤلفون: | Assiotis, Theodoros, Bedert, Benjamin, Gunes, Mustafa Alper, Soor, Arun |
| المصدر: | Prob. Math. Phys. 2 (2021) 613-642 |
| سنة النشر: | 2020 |
| المجموعة: | Mathematics Mathematical Physics |
| مصطلحات موضوعية: | Mathematics - Probability, Mathematical Physics |
| الوصف: | A family of random variables $\mathbf{X}(s)$, depending on a real parameter $s>-\frac{1}{2}$, appears in the asymptotics of the joint moments of characteristic polynomials of random unitary matrices and their derivatives, in the ergodic decomposition of the Hua-Pickrell measures and conjecturally in the asymptotics of the joint moments of Hardy's function and its derivative. Our first main result establishes a connection between the characteristic function of $\mathbf{X}(s)$ and the $\sigma$-Painlev\'e III' equation in the full range of parameter values $s>-\frac{1}{2}$. Our second main result gives the first explicit expression for the density and all the complex moments of the absolute value of $\mathbf{X}(s)$ for integer values of $s$. Finally, we establish an analogous connection to another special case of the $\sigma$-Painlev\'e III' equation for the Laplace transform of the sum of the inverse points of the Bessel point process. Comment: Improvements in exposition and a number of references added. To appear PMP |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.2140/pmp.2021.2.613 |
| URL الوصول: | http://arxiv.org/abs/2009.04760 |
| رقم الأكسشن: | edsarx.2009.04760 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.2140/pmp.2021.2.613 |
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