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 N2/4. 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. The intensity |tr U|2/N2 has the same limiting law as the Schur degree n/N2: at the Hagedorn point the canonical degree weights become asymptotically equal below the macroscopic finite-rank wall. 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.
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BibTeX
@misc{koch2026hagedorn,
title = {{The Hagedorn transition as a Plancherel wall: Tracy-Widom statistics, Polyakov loops and partial deconfinement}},
author = {Robert de Mello Koch and Minkyoo Kim and Hyunwoo Oh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12588},
eprint = {2609.12588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12588},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - The Hagedorn transition as a Plancherel wall: Tracy-Widom statistics, Polyakov loops and partial deconfinement
AU - Robert de Mello Koch
AU - Minkyoo Kim
AU - Hyunwoo Oh
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.12588
UR - https://arxiv.org/abs/2609.12588
AB - 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. The intensity |tr U|^2/N^2 has the same limiting law as the Schur degree n/N^2: at the Hagedorn point the canonical degree weights become asymptotically equal below the macroscopic finite-rank wall. 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.
ER -