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The partition function of the Chern-Simons theory on the three-sphere withthe unitary group $U(N)$ provides a one-matrix model. The corresponding$N$-particle system can be mapped to the determinantal point process whosecorrelation kernel is expressed by using the Stieltjes-Wigert orthogonalpolynomials. The matrix model and the point process are regarded as$q$-extensions of the random matrix model in the Gaussian unitary ensemble andits eigenvalue point process, respectively. We prove the convergence of the$N$-particle system to an infinite-dimensional determinantal point process in$N \to \infty$, in which the correlation kernel is expressed by Jacobi's thetafunctions. We show that the matrix model obtained by this limit realizes theoscillatory matrix model in Chern-Simons theory discussed by de Haro and Tierz.

Oscillatory matrix model in Chern-Simons theory and Jacobi-theta determinantal point process