大N极限下黎曼对称空间上的高斯分布
Gaussian distributions on Riemannian symmetric spaces in the large N limit
摘要：
我们考虑某些黎曼对称空间上的高斯分布。与欧几里德的情况相反，计算这种分布的归一化因子是具有挑战性的，我们称之为分拆函数。在某些情况下，例如埃尔米特正定矩阵空间或双曲空间，可以使用随机矩阵理论的技术精确计算它们。然而，在大多数情况下，这是重要的应用，如空间的对称正定（SPD）矩阵或西格尔域，这是唯一可能的数字。此外，当我们考虑例如高维SPD矩阵时，用于计算分拆函数的已知算法可能变得非常慢。受理论物理概念的启发，我们将讨论如何在大$N$极限下近似分拆函数：随着基础对称空间的维数（更准确地说，是其秩）变大，这种近似变得越来越好。我们将在SPD矩阵和相关空间的情况下给出首阶项的公式。此外，我们将通过作为鞍点方程产生的奇异积分方程来表征Siegel域的大$N$极限。
英文摘要：
We consider Gaussian distributions on certain Riemannian symmetric spaces. In contrast to the Euclidean case, it is challenging to compute the normalization factors of such distributions, which we refer to as partition functions. In some cases, such as the space of Hermitian positive definite matrices or hyperbolic space, it is possible to compute them exactly using techniques from random matrix theory. However, in most cases which are important to applications, such as the space of symmetric positive definite (SPD) matrices or the Siegel domain, this is only possible numerically. Moreover, when we consider, for instance, high-dimensional SPD matrices, the known algorithms for computing partition functions can become exceedingly slow. Motivated by notions from theoretical physics, we will discuss how to approximate the partition functions in the large $N$ limit: an approximation that gets increasingly better as the dimension of the underlying symmetric space (more precisely, its rank) gets larger. We will give formulas for leading order terms in the case of SPD matrices and related spaces. Furthermore, we will characterize the large $N$ limit of the Siegel domain through a singular integral equation arising as a saddle-point equation.
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