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Recent developments have found unexpected connections between non-commutative probability theory and algebraic topology. In particular, Boolean cumulants functionals seem to be important for describing morphisms of homotopy operadic…
A parallel splitting method is proposed for solving systems of coupled monotone inclusions in Hilbert spaces. Convergence is established for a wide class of coupling schemes. Unlike classical alternating algorithms, which are limited to two…
For a Coupled Map Lattice with a specific strong coupling emulating Stavskaya's probabilistic cellular automata, we prove the existence of a phase transition using a Peierls argument, and exponential convergence to the invariant measures…
Systems of interacting random replicators are studied using generating functional techniques. While replica analyses of such models are limited to systems with symmetric couplings, dynamical approaches as presented here allow specifically…
Nonreciprocal coupling between photonic modes enables a range of advanced functionalities, though the available approaches for its practical implementation remain limited. Here, we introduce a novel strategy for achieving nonreciprocal…
This review paper provides an introduction of Markov chains and their convergence rates which is an important and interesting mathematical topic which also has important applications for very widely used Markov chain Monte Carlo (MCMC)…
We consider the distribution of cycle counts in a random regular graph, which is closely linked to the graph's spectral properties. We broaden the asymptotic regime in which the cycle counts are known to be approximately Poisson, and we…
We introduce the concept of directional coupling, i.e., the selective transfer of a state between adjacent quantum wires, in the context of quantum computing and short-distance communication. Our analysis rests upon a mathematical analogy…
State-switching models such as hidden Markov models or Markov-switching regression models are routinely applied to analyse sequences of observations that are driven by underlying non-observable states. Coupled state-switching models extend…
Nonlinear time series models with exogenous regressors are essential in econometrics, queuing theory, and machine learning, though their statistical analysis remains incomplete. Key results, such as the law of large numbers and the…
In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional…
Most popular dimension reduction (DR) methods like t-SNE and UMAP are based on minimizing a cost between input and latent pairwise similarities. Though widely used, these approaches lack clear probabilistic foundations to enable a full…
We review the main tools which allow for the statistical characterization of weighted networks. We then present two case studies, the airline connection network and the scientific collaboration network, which are representative of critical…
Problems in the field of open quantum systems often involve an environment that strongly influences the dynamics of excited states. Here we present a numerical method to model optical spectra of non-Markovian open quantum systems. The…
Monte Carlo algorithms often aim to draw from a distribution $\pi$ by simulating a Markov chain with transition kernel $P$ such that $\pi$ is invariant under $P$. However, there are many situations for which it is impractical or impossible…
Spectral Method is a commonly used scheme to cluster data points lying close to Union of Subspaces by first constructing a Random Geometry Graph, called Subspace Clustering. This paper establishes a theory to analyze this method. Based on…
We have developed a steady state theory of complex transport networks used to model the flow of commodity, information, viruses, opinions, or traffic. Our approach is based on the use of the Markov chains defined on the graph…
Chirality in integrated quantum photonics has emerged as a promising route towards achieving scalable quantum technologies with quantum nonlinearity effects. Topological photonic waveguides, which utilize helical optical modes, have been…
We introduce a new framework that yields spectral bounds on norms of functions of transition maps for finite, homogeneous Markov chains. The techniques employed work for bounded semigroups, in particular for classical as well as for quantum…
In this paper, we study consistent and partially exchangeable sequences of Markov chains on a finite state space. We provide a characterisation of the admissible transition rates via a decomposition into individual and coordinated motion of…