2021/09/02 by Shinsuke M. Nishigaki, Nishigaki, Shinsuke M. · 1 citation
Mathematics · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Lattice (hep-lat) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2109.00790
openalex publication_date 2021/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The J 'anossy density for a determinantal point process is the probability\ndensity that an interval I contains exactly p points except for those at\nk designated loci. The J 'anossy density associated with an integrable\nkernel \K doteq (\φ(x)\ψ(y)-\ψ(x)\φ(y))/(x-y) is shown\nto be expressed as a Fredholm determinant\n\Det( mathbbI-\\K|I) of a transformed kernel\n\\K doteq\n(\\φ(x)\\ψ(y)-\\ψ(x)\\φ(y))/(x-y).\nWe observe that \\K satisfies Tracy and Widom's criteria if\n\K does, because of the structure that the map (\φ,\n\ψ)\↦ (\\φ, \\ψ) is a meromorphic\n\SL(2,\ℝ) gauge transformation between covariantly constant\nsections. This observation enables application of the Tracy--Widom method to\nJ 'anossy densities, expressed in terms of a solution to a system of\ndifferential equations in the endpoints of the interval. Our approach does not\nexplicitly refer to isomonodromic systems associated with Painlev 'e\nequations employed in the preceding works. As illustrative examples we compute\nJ 'anossy densities with k=1, p=0 for Airy and Bessel kernels, related to\nthe joint distributions of the two largest eigenvalues of random Hermitian\nmatrices and of the two smallest singular values of random complex matrices.\n