Hagedorn 相变作为 Plancherel 墙：Tracy-Widom 统计、Polyakov 环与部分退禁闭
The Hagedorn transition as a Plancherel wall: Tracy-Widom statistics, Polyakov loops and partial deconfinement
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中文总结 AI 辅助
本研究通过 Plancherel 墙揭示 Dutta-Gopakumar 矩阵模型的 Hagedorn 相变，得到 Tracy-Widom 统计与 Polyakov 环均匀分布，并建立部分退禁闭对应。
中文摘要 AI 辅助
我们研究了有限秩 N 下单缠绕 Dutta-Gopakumar 酉矩阵模型的 Hagedorn 相变。其 Schur 展开是受行数有限秩墙限制的 Plancherel 概率之和，当盒子数等于 N^2/4 时，Vershik-Kerov-Logan-Shepp 极限形状达到此墙。Hagedorn 点周围的两个标度窗口在不同标度下解析该墙。在外窗口中，Baik-Deift-Johansson 定理给出 Tracy-Widom 交叉，对其求和可将 Liu 的临界自由能系数识别为 GUE Tracy-Widom 分布的一阶矩。在内窗口中，同样的求和决定了 Hagedorn 点处 Polyakov 环的典范概率律：归一化环 tr U /N 均匀分布在半径为二分之一的圆盘上，因此序参量在大 N 下不自平均。该律与字母数除以 N^2 的极限分布相同：在 Hagedorn 点，所有低于有限秩墙的字母数以相等的典范权重进入，墙提供唯一截断。在部分退禁闭词典下，均匀圆盘变为退禁闭颜色分数的线性概率密度，其完全退禁闭端点是 Berenstein 和 Yan 的杨图端点。Tracy-Widom 均值还确定了 Polyakov 环 Laplace 变换及其所有固定径向矩的第一个有限 N 修正。Bessel-Toeplitz 计算直至 N=100 证实了自由能系数、均匀律和预测的修正。
英文摘要
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 $\mathrm{tr}\,U/N$ is uniformly distributed on a disk of radius one half, so the order parameter does not self-average at large $N$. The intensity $|\mathrm{tr}\,U|^2/N^2$ 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.
发表机构
- School of Science, Huzhou Normal University（湖州师范学院理学院）
- Mandelstam Institute for Theoretical Physics, School of Physics, University of the Witwatersrand（威特沃特斯兰德大学物理学院曼德尔斯坦理论物理研究所）
- Research Institute of Basic Sciences, Seoul National University（首尔大学基础科学研究院）
- Department of Physics and Astronomy & Center for Theoretical Physics, Seoul National University（首尔大学物理与天文学系及理论物理中心）
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