25 pages, 1 figure. Title changed, size reduced, added numerics of gap probabilities and its small s asymptotics for general parameters, to appear in Physica D: Nonlinear Phenomena
本文考虑由$$θx^α\int_0^1J_{\frac{α+1}θ,\frac{1}θ}(ux)J_{α+1,θ}((uy)^θ)u^α\,\mathrm{d} u, \qquad α>-1, \qquad θ>0,$$定义的Wright广义Bessel核$K^{(α,θ)}(x,y)$，其中$$J_{a,b}(x)=\sum_{j=0}^\infty\frac{(-x)^j}{j!Γ(a+bj)},\qquad a\in\mathbb{C},\qquad b>-1,$$是Wright对Bessel函数的推广.这种非对称核是随机矩阵理论中经典Bessel核（对应于$θ=1$）的推广，是某些Muttalib-Borodin系综相关核的硬边尺度极限。我们证明，如果$θ$是理性的，即，$θ=\frac{m}{n}$与$m,n\in\mathbb{N}$、$gcd(m,n)=1$和$α> m-1-\frac{m}{n}$，Wright广义贝塞尔核在Its-Izergin-Korepin-Slavnov意义下是可积的。然后我们得到这个核在几个标度区间的并集上的Fredholm行列式，它也可以解释为这些区间上的间隙概率（没有发现粒子的概率）。可积的结构，使我们能够获得一个系统的耦合偏微分方程与相应的Fredholm行列式以及哈密顿解释。因此，我们能够表示在一个单一的间隔$(0,s)$的间隙概率在一个非线性常微分方程组的解决方案。
英文摘要：
In this paper, we consider the Wright's generalized Bessel kernel $K^{(α,θ)}(x,y)$ defined by $$θx^α\int_0^1J_{\frac{α+1}θ,\frac{1}θ}(ux)J_{α+1,θ}((uy)^θ)u^α\,\mathrm{d} u, \qquad α>-1, \qquad θ>0,$$ where $$J_{a,b}(x)=\sum_{j=0}^\infty\frac{(-x)^j}{j!Γ(a+bj)},\qquad a\in\mathbb{C},\qquad b>-1,$$ is Wright's generalization of the Bessel function. This non-symmetric kernel, which generalizes the classical Bessel kernel (corresponding to $θ=1$) in random matrix theory, is the hard edge scaling limit of the correlation kernel for certain Muttalib-Borodin ensembles. We show that, if $θ$ is rational, i.e., $θ=\frac{m}{n}$ with $m,n\in\mathbb{N}$, $gcd(m,n)=1$, and $α> m-1-\frac{m}{n}$, the Wright's generalized Bessel kernel is integrable in the sense of Its-Izergin-Korepin-Slavnov. We then come to the Fredholm determinant of this kernel over the union of several scaled intervals, which can also be interpreted as the gap probability (the probability of finding no particles) on these intervals. The integrable structure allows us to obtain a system of coupled partial differential equations associated with the corresponding Fredholm determinant as well as a Hamiltonian interpretation. As a consequence, we are able to represent the gap probability over a single interval $(0,s)$ in terms of a solution of a system of nonlinear ordinary differential equations.
Pure $SU(2)$ gauge theory partition function and generalized Bessel kernel
Center for Advanced Studies, Skolkovo Institute of Science and Technology（斯科尔科沃科学技术学院高级研究中心）；National Research University Higher School of Economics（国立高等经济大学）；Bogolyubov Institute for Theoretical Physics（博戈留博夫理论物理研究所）；Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350, Université de Tours（图尔大学CNRS/UMR 7350理论与数学物理实验室）
