2013/12/31 by Yuta Takahashi, Y. Takahashi, Makoto Katori
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Determinantal point process #Eigenvalues and eigenvectors #Kernel (algebra) #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Orthogonal polynomials #Partition function (quantum field theory) #Physics #Point process #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Unitary group #Unitary matrix #Unitary state #cond-mat.stat-mech #hep-th #math-ph #math.MP #math.PR #nlin.SI
paper · pdf · doi:10.1063/1.4894235
published as J. Math. Phys. 55 (2014) 093302/1-24 · v2: 29 pages, 5 figures, minor corrections made for publication in J. Math. Phys
arxiv created 2014/08/16 · openalex publication_date 2014/09/01 · arxiv updated 2014/09/09 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The partition function of the Chern-Simons theory on the three-sphere with the unitary group U(N) provides a one-matrix model. The corresponding N-particle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the Stieltjes-Wigert orthogonal polynomials. The matrix model and the point process are regarded as q-extensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the N-particle system to an infinite-dimensional determinantal point process in N → ∞, in which the correlation kernel is expressed by Jacobi's theta functions. We show that the matrix model obtained by this limit realizes the oscillatory matrix model in Chern-Simons theory discussed by de Haro and Tierz.