Lee-Yang zeroes of the Curie-Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow
Lee-Yang zeroes of the Curie-Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow
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发表时间：
2022-03
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通讯作者：
Z. Kabluchko
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作者：
Z. Kabluchko
。实线上从初始条件zn开始的后向热flow得到经典的n次Hermite多项式，其零点在大的n极限上按维格纳半圆定律分布。类似地，具有周期初始条件(Sinfl2)n的反向热θow导致Hermite多项式的三角或么正类似。这些多项式与居里-魏斯模型的配分函数密切相关，并出现在Mirabelli关于finite自由几率的工作中。我们将n次酉Hermite多项式与酉群U(N)上的布朗运动所得到的酉随机矩阵的期望特征多项式联系起来。证明了酉Hermite多项式的零点的整体分布为自由酉正态分布。我们还计算了这些多项式的渐近性，或者等价地计算了复外fi场中居里-魏斯模型的自由能。我们证明了该模型的Lee-Yang零点的全局分布。最后，我们证明了高次实根多项式(分别为三角多项式)的反向热flow在其根的渐近分布水平上诱导出自由布朗运动(分别为自由酉布朗运动)。
. 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 θ 2 ) 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).