某些β系综的大偏差原理
Large deviations principle for some beta-ensembles
摘要：
设L是射影复流形X上的正线丛。考虑L的p阶张量幂幂的全纯截面空间。这个空间的基的行列式，连同X中的加权紧集上的某个给定的概率测度，自然地导出β系综，即，紧集上的随机点过程。在物理上，这个一般的设定对应于X中的自由费米子气体，并且可能允许一些随机矩阵模型。当p趋于无穷大时，与这种β-集合相关联的经验测度几乎必然收敛于一个平衡测度。我们建立了一个大偏差原则（LDP）与这些经验措施的有效收敛速度。我们的研究涵盖了实球面或实欧几里得空间的紧致子集上的某些β-系综的情形。
英文摘要：
Let L be a positive line bundle over a projective complex manifold X. Consider the space of holomorphic sections of the tensor power of order p of L. The determinant of a basis of this space, together with some given probability measure on a weighted compact set in X, induces naturally a beta-ensemble, i.e., a random point process on the compact set. Physically, this general setting corresponds to a gas of free fermions in X and may admit some random matrix models. The empirical measures, associated with such beta-ensembles, converge almost surely to an equilibrium measure when p goes to infinity. We establish a large deviations principle (LDP) with an effective speed of convergence for these empirical measures. Our study covers the case of some beta-ensembles on a compact subset of a real sphere or of a real Euclidean space.
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