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

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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.