From interacting particle systems to topological data analysis

Abstract

We first study duality in stochastic interacting particle systems. In contrast to previous work the main emphasis will be on particle systems with non-conserved internal degrees of freedom. Starting from the known symmetries of the intensity matrix, probabilistic, algebraic and combinatorial techniques will be employed to construct invariant measures, duality functions and shock measures, with a view also on a generalization of the second-class particle technique to mark the microscopic position of shocks. For the bricklayers' process we devise a reversed route starting from shock measures in order to uncover duality functions and the underlying symmetry. Then we go on to use expertise from non-reversible interacting particle systems to tackle convergence problems in persistent homology as used in the framework of topological data analysis. The idea is to use the super paramagnetic clustering to devise a non-reversible Markov chain that allows for the numerical computation of persistent homology and that exhibits fast convergence to a given measure on ensembles of simplicial complexes that are relevant for topological data analysis. The efficiency of this algorithm will be tested numerically on data sets with known topological properties. Moreover, it will be investigated whether there universal asymptotic properties of random null models that can serve for benchmarking purposes in the distinction of relevant information from noise. (AU)