Georgia Institute of Technology College of Sciences

Transport maps serve as a powerful tool to transfer information from source to target measures. However, this transfer of information is possible only if the transport map is sufficiently regular, which is often difficult to show. I will explain how taking the source measure to be an infinite-dimensional measure, and building transport maps based on stochastic processes, solves some of these challenges both in the continuous and discrete settings. 

A wide array of network growth models have been proposed across various domains as test beds to understand questions such as the effect of network change point (when a shock to the network changes the probabilistic rules of its evolution) or the role of attributes in driving the emergence of network structure and subsequent centrality measures in real world systems. 

The goal of this talk will be to describe three specific settings where continuous time branching processes give mathematical insight into asymptotic properties of such models. In the first setting, a natural network change point model can be directly embedded into continuous time thus leading to an understanding of long range dependence of the initial network system on subsequent properties imply the difficulty in understanding and estimating network change point. In the second application, we will describe a notion of resolvability where convergence of a simple macroscopic functional in a model of networks with vertex attributes, coupled with stochastic approximation techniques implies local weak convergence of a standard model of nodal attribute driven network evolution to a limit infinite random structure driven by a multitype continuous time branching process. In the second setting, continuous time branching processes only emerge in the limit. In the final setting we will describe network evolution models with delay where once again such processes arise only in the limit. 

The emergence of large connected structures in networks has been a central topic in random graph theory since its inception, forming a foundation for understanding fundamental processes such as the spread of influence or epidemics, and the robustness of networked systems. The field witnessed significant growth from the early 2000s, fueled by a surge in experimental work from statistical physics that introduced fascinating concepts such as universality. Broadly speaking, universality suggests that the formation of a giant component in random graphs often depends primarily on macroscopic statistical properties like the degree distribution. In the theoretical literature, two universality classes have emerged, both closely related to Aldous’ seminal work on critical random graphs and the theory of multiplicative coalescents. In this talk, I will present a third universality class that emerges in the setting of percolation on random graphs with infinite-variance degree distributions. The new universality class exhibits fundamentally different behavior compared to multiplicative coalescents and reveals surprising phenomena concerning the width of the critical window—phenomena that were unforeseen in the substantial physics literature on this topic. Based on joint work with Shankar Bhamidi and Remco van der Hofstad.

Statistical mechanics explains that systems in thermal equilibrium spend a greater fraction of their time in states with apparent order because these states have lower energy. This explanation is remarkable, and powerful, because energy is a "local" property of states. While non-equilibrium steady states can similarly exhibit order, there can be no local property analogous to energy that explains why, as Landauer argued 50 years ago. However, recent experiments suggest that a local property called “rattling” predicts which states are favored, at least for a broad class of non-equilibrium systems.

I will present a Markov chain theory that explains when and why rattling predicts non-equilibrium order. In brief, it "works" when the correlation between a Markov chain's effective potential and the logarithm of its exit rates is high. It is therefore important to estimate this correlation in different classes of Markov chains. As an example, I will discuss estimates of the correlation exhibited by reaction kinetics on disordered energy landscapes, including dynamics of the random energy model and the Sherrington–Kirkpatrick spin glass. (Joint work with Dana Randall.)