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Phase Space Distribution for Two-Gap Solution in Unitary Matrix Model

We analyze the dynamics of weakly coupled finite temperature $U(N)$ gauge theories on $S^3$ by studying a class of effective unitary matrix model. Solving Dyson-Schwinger equation at large $N$, we find that different phases of gauge theories are characterized by gaps in eigenvalue distribution over a unit circle. In particular, we obtain no-gap, one-gap and two-gap solutions at large $N$ for a class of matrix model we are considering. The same effective matrix model can equivalently be written as a sum over representations (or Young diagrams) of unitary group. We show that at large $N$, Young diagrams corresponding to different phases can be classified in terms of discontinuities in number of boxes in two consecutive rows. More precisely, the representation, where there is no discontinuity, corresponds to no-gap and one-gap solution, where as, a diagram with one discontinuity corresponds to two-gap phase, mentioned above. This observation allows us to write a one to one relation between eigenvalue distribution function and Young tableaux distribution function for each saddle point, in particular for two-gap solution. We find that all the saddle points can be described in terms of free fermions with a phase space distribution for no-gap, one-gap and two-gap phases.