On Wright’s generalized Bessel kernel

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• 作者：Lun Zhang p>lunzhang@fudan.edu.cnp>
• 关键词：Wright&rsquo ; s generalized Bessel kernel ; Integrable representation ; Muttalib-Borodin ensembles ; Fredholm determinant ; Gap probability ; Coupled nonlinear differential equations
• 刊名：Physica D: Nonlinear Phenomena
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：340
• 期：Complete
• 页码：27-39
• 全文大小：541 K
• 卷排序：340

文摘

In this paper, we consider the Wright’s generalized Bessel kernel K(α,θ)(x,y)K(α,θ)(x,y) defined by θxα∫01Jα+1θ,1θ(ux)Jα+1,θ((uy)θ)uαdu,α>−1,θ>0, where Ja,b(x)=∑j=0∞(−x)jj!Γ(a+bj),a∈C,b>−1, is Wright’s generalization of the Bessel function. This non-symmetric kernel, which generalizes the classical Bessel kernel (corresponding to θ=1θ=1) in random matrix theory, is the hard edge scaling limit of the correlation kernel for certain Muttalib–Borodin ensembles. We show that, if θθ is rational, i.e., θ=mn with m,n∈Nm,n∈N, gcd(m,n)=1gcd(m,n)=1, and α>m−1−mn, 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)(0,s) in terms of a solution of a system of nonlinear ordinary differential equations.