The Hagedorn transition as a Plancherel wall: Tracy-Widom statistics, Polyakov loops and partial deconfinement

Abstract

We study the Hagedorn transition of the single-winding Dutta-Gopakumar unitary matrix model at finite rank N. Its Schur expansion is a sum of Plancherel probabilities restricted by the finite-rank wall on the number of rows, and the Vershik-Kerov-Logan-Shepp limit shape reaches this wall when the number of boxes equals N^2/4. Two scaling windows around the Hagedorn point resolve the wall at different scales. In the outer window the Baik-Deift-Johansson theorem gives a Tracy-Widom crossover, and summing it identifies Liu's critical free-energy coefficient with the first moment of the GUE Tracy-Widom distribution. In the inner window the same sum determines the canonical probability law of the Polyakov loop at the Hagedorn point: the normalized loop tr U /N is uniformly distributed on a disk of radius one half, so the order parameter does not self-average at large N. This law is identical to the limiting distribution of the number of letters divided by N^2: at the Hagedorn point all letter numbers below the finite-rank wall enter with equal canonical weight, with the wall providing the sole cutoff. Under the partial-deconfinement dictionary the uniform disk becomes a linear probability density for the deconfined color fraction, and its fully deconfined endpoint is the Young-diagram endpoint of Berenstein and Yan. The Tracy-Widom mean also fixes the first finite-N correction to the Polyakov-loop Laplace transform and to all of its fixed radial moments. Bessel-Toeplitz evaluations up to N = 100 confirm the free-energy coefficient, the uniform law, and the predicted correction.