## Max Xu : Random multiplicative functions and applications

- Probability ( 262 Views )Random multiplicative functions are probabilistic models for multiplicative arithmetic functions, such as Dirichlet characters or the Liouville function. In this talk, I will first quickly give an overview of the area, and then focus on some of the recent works on proving central limit theorems, connections to additive combinatorics, as well as some other deterministic applications. Part of the talk is based on joint work with Soundararajan, with Harper and Soundararajan (in progress) and with Angelo and Soundararajan (in progress).

## Erin Beckman : Shape Theorem for the Brownian Frog Model

- Probability ( 131 Views )The frog model is a type of branching random walk model. Active "frogs" move according to random walks, and if they encounter a sleeping frog on their walk, the sleeping frog becomes active and begins an independent random walk. Over the past 20 years, recurrence properties and asymptotic behavior of this system (and many generalizations) have been studied extensively. One way to generalize this system is to consider the continuous version: Brownian motion frogs moving in R^d. In this talk, we will describe a continuous variant of the problem and show a limiting shape theorem analogous to prior discrete results.

## Jessica Zuniga : On the spectral analysis of second-order Markov chains.

- Probability ( 107 Views )In this talk we consider second-order finite Markov chains that are > trajectorially reversible, a property that is a generalization of the > notion of reversibility for usual Markov chains. Specifically, we > study spectral properties of second-order Markov chains that have a > tendency to not return to their previous state. We confirm that > resorting to second-order chains can be an option to improve the speed > of convergence to equilibrium. This is joint work with Persi Diaconis > and Laurent Miclo.

## Galen Reeves : Non-asymptotic bounds for approximate message passing via Gaussian coupling

- Probability ( 0 Views )Approximate message passing (AMP) has emerged as a powerful framework for the design and analysis of iterative algorithms for high dimensional inference problems involving regression and low-rank matrix factorization. The basic form of an AMP algorithm consists of a recursion defined on a random matrix. Under suitable conditions, the distribution of this recursion can be well approximated by a Gaussian process whose mean and covariance are defined via a recursive process called state evolution. This talk will briefly summarize some of the key ideas in AMP (no background is assumed). I will then describe a new approach for analyzing these algorithms that constructs an explicit coupling between the AMP iterates and a Gaussian process, Under mild regularity conditions, this coupling argument provides simple and interpretable guarantees on the non-asymptotic behavior of AMP algorithms. Related work can be found in the arXiv papers: https://arxiv.org/abs/2405.08225 and https://arxiv.org/abs/2306.15580