Georgia Institute of Technology College of Sciences

First-passage percolation on the square lattice is a random growth model in which each edge of Z^2 is assigned an i.i.d. nonnegative weight.  The passage time between two points is the smallest total weight of a nearest-neighbor path connecting them, and a path achieving this minimum is called a geodesic.  Typically, the number of edges in a geodesic is comparable to the Euclidean distance between its endpoints.  However, when the edge-weights take the value 0 with probability exactly 1/2, a strikingly different behavior occurs: geodesics travel primarily on critical clusters of zero-weight edges, whose internal graph distance scales superlinearly with Euclidean distance.  Determining the precise degree of this superlinear scaling is a challenging and ongoing endeavor.  I will discuss recent progress on this front (joint with David Harper, Xiao Shen, and Evan Sorensen), along with complementary results on a dual problem, where we restrict path lengths and analyze passage times (joint with Jack Hanson and Daniel Slonim).

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.