论文标题
$ p $ - adic随机矩阵产品的限制和波动
Limits and fluctuations of $p$-adic random matrix products
论文作者
论文摘要
我们表明，在$ \ mathbb {q} _p $上的随机矩阵的产品和角落的单数数字（也称为史密斯正常形式）由Hall-littlewood多项式的构造，与复杂随机矩阵和Heckman-Opdam超模型函数的单数关系之间的经典关系的结构相同。这意味着$ \ text {gl} _n（\ mathbb {z} _p）$的haar分配元素的产物的奇异数字形成了一个离散的时间马尔可夫链，该分散时间分布在Hall-littlewood过程中，并且矩阵的数量在产品中发挥了作用。我们为Hall-Littlewood过程提供了一种精确的采样算法，该算法通过将它们与类似于Pushtasep的相互作用的粒子系统联系起来而产生。通过分析该粒子系统的渐近行为，我们表明，此类产品的奇异数量遵守了大量定律，它们的波动动态地融合到独立的布朗尼运动。在大型矩阵大小的极限下，我们还表明，在此类$ \ text {gl} _n（\ mathbb {z} _p）$ Corners的$ \ text {gl} _n（\ text {gl} _n（\ text {gl} _n）中，lyapunov指数的类似物具有通用限制。
We show that singular numbers (also known as invariant factors or Smith normal forms) of products and corners of random matrices over $\mathbb{Q}_p$ are governed by the Hall-Littlewood polynomials, in a structurally identical manner to the classical relations between singular values of complex random matrices and Heckman-Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of $\text{GL}_N(\mathbb{Z}_p)$ form a discrete-time Markov chain distributed as a Hall-Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall-Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of $\text{GL}_N(\mathbb{Z}_p)$ corners.
