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- Wednesday, March 27, 2019, 11:00am, Fuqua McKinley Seminar Room, Probability Seminar
*Tales of Random Projections of High-dimensional Measures*

Kavita Ramanan (Brown)- Properties of random projections of high-dimensional probability measures are of interest in a variety of fields, including asymptotic convex geometry, and high-dimensional statistics and data analysis. A particular question of interest is to identify what properties of the high-dimensional measure are captured by its lower-dimensional projections. While fluctuations of these projections have been well studied over the past decade, we describe more recent work on large deviations principles and associated conditional limit theorems for (possibly multidimensional) projections. This talk is based on joint works with Nina Gantert, Steven Kim and Yin-Ting Liao.

- Thursday, March 28, 2019, 3:15pm, 119 Physics, Probability Seminar
*Beyond Mean-Field Limits: Local Dynamics on Sparse Graphs*

Kavita Ramanan (Brown)- Many applications can be modeled as a large system of homogeneous interacting particles on a graph in which the infinitesimal evolution of each particle depends on its own state and the empirical distribution of the states of neighboring particles. When the graph is a clique, it is well known that the dynamics of a typical particle converges in the limit, as the number of vertices goes to infinity, to a nonlinear Markov process, often referred to as the McKean-Vlasov or mean-field limit. In this talk, we focus on the complementary case of scaling limits of dynamics on certain sequences of sparse graphs, including regular trees and sparse Erdos-Renyi graphs, and obtain a novel characterization of the dynamics of the neighborhood of a typical particle.

- Thursday, April 11, 2019, 3:15pm, 119 Physics, Probability Seminar
*title*

Sayan Banerjee (UNC-Chapel Hill)- abstract

- Thursday, April 25, 2019, 3:15pm, 119 Physics, Probability Seminar
*The Stabilisation of Equilibria in Evolutionary Game Dynamics through Mutation*

Johan Brauer (City University of London)- The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross-learning, as well as of systems of co-evolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We report on our investigation of a new notion of equilibria of RD, called mutation limits, which is based on the inclusion of a naturally arising, simple form of mutation, but is invariant under the specific choice of mutation parameters. We prove the existence of such mutation limits for a large range of games, and consider an interesting subclass, that of attracting mutation limits. Attracting mutation limits are approximated by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence, offer an approximate dynamic solution of the underlying game, especially if the original dynamic has no asymptotically stable equilibria. Therefore, the presence of mutation will indeed stabilise the system in certain cases and make attracting mutation limits near-attainable. Furthermore, the relevance of attracting mutation limits as a game theoretic equilibrium concept is emphasised by the relation of (mutation-)perturbed RD to the Q-learning algorithm in the context of multi-agent reinforcement learning. However, in contrast to the guaranteed existence of mutation limits, attracting mutation limits do not exist in all games, raising the question of their characterization.

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