There is a close connection between the ground state of non-interacting fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: (i) we consider the full family of admissible quantum boundary conditions (i.e., self-adjoint extensions) for the Laplacian on a bounded interval, and the corresponding projection correlation kernels; (ii) we construct the grand canonical extensions at finite temperature of the projection kernels, interpolating from Poisson to random matrix eigenvalue statistics. The scaling limits in the bulk and at the edges are studied in a unified framework, and the question of universality is addressed. Whether the finite temperature determinantal processes correspond to the eigenvalue statistics of some matrix models is, a priori, not obvious. We complete the picture by constructing a finite temperature extension of the Haar measure on the classical compact groups. The eigenvalue statistics of the resulting grand canonical matrix models (of random size) corresponds exactly to the grand canonical measure of free fermions with classical boundary conditions.

中文翻译：

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自由费米子和经典紧群

具有经典（吸收、反射和周期）边界条件的盒子中非相互作用费米子的基态与经典紧群的特征值统计之间存在密切联系。相关的行列式点过程可以在两个自然方向上扩展：（i）我们考虑有界区间上拉普拉斯算子的可容许量子边界条件（即自伴随扩展）的全族，以及相应的投影相关核；(ii) 我们在投影内核的有限温度下构建盛大规范扩展，从泊松插值到随机矩阵特征值统计。在一个统一的框架中研究了体积和边缘的缩放限制，并解决了普遍性问题。有限温度行列过程是否与某些矩阵模型的特征值统计相对应，先验地并不明显。我们通过在经典紧群上构建 Haar 测度的有限温度扩展来完成这幅图。生成的正则矩阵模型（随机大小）的特征值统计与具有经典边界条件的自由费米子的正则度量完全对应。