  Applying the machinery of random matrix theory and Toeplitz determinants we
study the level $k$, $U(N)$ Chern-Simons theory coupled with fundamental matter
on $S^2\times S^1$ at finite temperature $T$. This theory admits a discrete
matrix integral representation, i.e. a unitary discrete matrix model of
two-dimensional Yang-Mills theory. In this study, the effective partition
function and phase structure of the Chern-Simons matter theory, in a special
case with an effective potential namely the Gross-Witten-Wadia potential, are
investigated. We obtain an exact expression for the partition function of the
Chern-Simons matter theory as a function of $k,N,T,$ for finite values and in
the asymptotic regime. In the Gross-Witten-Wadia case, we show that ratio of
the Chern-Simons matter partition function and the continuous two-dimensional
Yang-Mills partition function, in the asymptotic regime, is the Tracy-Widom
distribution. Consequently, using the explicit results for free energy of the
theory, new second-order and third-order phase transitions are observed.
Depending on the phase, in the asymptotic regime, Chern-Simons matter theory is
represented either by a continuous or discrete two-dimensional Yang-Mills
theory, separated by a third-order domain wall.
