Dyson gas on a curved contour
Dyson gas on a curved contour
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DOI：
10.1088/1751-8121/ac5a8f
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发表时间：
2021-11
期刊：
影响因子：
--
通讯作者：
P. Wiegmann;A. Zabrodin
中科院分区：
文献类型：
--
作者：
P. Wiegmann;A. Zabrodin
引入并研究了平面上任意光滑封闭轮廓上温度为β的对数气体模型。该模型在单位圆上推广了戴森气体（β系综）。我们通过迭代“循环方程”来计算自由能（N是粒子的数量）的大N扩展的非消失项，该循环方程是关于轮廓的再参数化和扩张的Ward恒等式。我们证明了自由能的主要贡献是通过与轮廓相关的诺伊曼跳跃算子的谱行列式表示的，或者等价地通过诺伊曼-庞加莱（双层）算子的Fredholm行列式表示的。这一结果将戴森气体的统计力学与支撑轮廓的内部和外部域的光谱几何联系起来。
We introduce and study a model of a logarithmic gas with inverse temperature β on an arbitrary smooth closed contour in the plane. This model generalizes Dyson’s gas (the β-ensemble) on the unit circle. We compute the non-vanishing terms of the large N expansion of the free energy (N is the number of particles) by iterating the ‘loop equation’ that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann–Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour.