论文标题
Lee - Curie的Yang零 - Weiss Ferromagnet，单一的Hermite多项式和向后的热流动
Lee--Yang zeroes of the Curie--Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow
论文作者
论文摘要
实际线上的向后热流从初始条件$ z^n $始于经典的$ n $ th hermite多项式，其零是根据wigner semicircle law分配的。同样，具有周期性初始条件$（\ sin \fracθ2）^n $的向后热流导致Hermite多项式的三角类似物。这些多项式与Weiss模型的分区函数密切相关，并出现在Mirabelli对有限自由概率的工作中。我们将$ n $ th统一的Hermite多项式与预期的特征多项式的多项式联系起来，这是通过在单一$ u（n）$上运行布朗尼运动获得的单一随机矩阵。我们确定单位赫尔米特多项式零的全球分布是自由的单一正态分布。我们还计算了这些多项式的渐近学，或者等效地，在复杂的外场中curie-Weiss模型的自由能。我们确定了该模型的Lee-yang零的全局分布。最后，我们表明，向后热流应用于高度实地的多项式（分别是三角多项式），这会诱导其根部的渐近分布水平，这是一种自由的布朗运动（分别是自由的布朗尼运动）。
The backward heat flow on the real line started from the initial condition $z^n$ results in the classical $n$-th Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large $n$ limit. Similarly, the backward heat flow with the periodic initial condition $(\sin \frac θ2)^n$ leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie--Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the $n$-th unitary Hermite polynomial to the expected characteristic polynomial of a unitary random matrix obtained by running a Brownian motion on the unitary group $U(n)$. We identify the global distribution of zeroes of the unitary Hermite polynomials as the free unitary normal distribution. We also compute the asymptotics of these polynomials or, equivalently, the free energy of the Curie--Weiss model in a complex external field. We identify the global distribution of the Lee--Yang zeroes of this model. Finally, we show that the backward heat flow applied to a high-degree real-rooted polynomial (respectively, trigonometric polynomial) induces, on the level of the asymptotic distribution of its roots, a free Brownian motion (respectively, free unitary Brownian motion).
