Products of Random Matrices, Non-Commutative Branching Random Walks and Multitype Branching Random Walks in Random Environment

随机矩阵的乘积，即具有随机实值条目(和固定维度)的独立、同分布矩阵的乘积，在各种模型中作为基本对象出现，并且其本身具有重要意义，因为它可以被视为非交换(半)群上乘法随机游动的典型模型。这个项目的目的是双重的，基本概念是基础研究和应用的相互丰富。首先，我们希望利用我们在随机矩阵乘积方面的经验来研究应用概率中的模型；特别是具有分支机制的模型，其中极值粒子的研究在过去几年中引起了极大的关注。其次，研究这些模型以及进一步的模型，如多元金融时间序列或深度学习中的随机梯度下降，将在随机矩阵乘积理论中产生具有挑战性的新问题，这是我们想要解决的。

A product of random matrices, i. e., a product of independent, identically distributed matrices with random real-valued entries (and a fixed dimension), arises as a fundamental object in various models and is of importance in its own right, for it can be seen as the archetypical model of a multiplicative random walk on a non-commutative (semi-)group. The aim of this project is twofold, the fundamental concept being mutual enrichment of basic research and applications. Firstly, we want to use our experience with products of random matrices to study models from applied probability; in particular models with a branching mechanism, where the study of extremal particles has attracted a great deal of attention in the last few years. Secondly, studying these and further models like multivariate financial time series or stochastic gradient descent in deep learning will give rise to challenging new problems in the theory of products of random matrices, which we want to solve.