Georgia Institute of Technology College of Sciences

In first-passage percolation (FPP), we let \tau_v be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of \tau_v, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of F^{-1}(1/2+1/2^k) diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of k^{7/8} F^{-1}(1/2+1/2^k) converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam.

This talk will be given on Bluejeans at the link https://bluejeans.com/547955982/2367

We will motivate this talk by exhibiting recent progress on (either general or symmetric anisotropic) bootstrap percolation models in $d$-dimensions. Then, we will discuss our intention to start a deeper study of non-symmetric models for $d\ge 3$. It looks like some proportion of them could be related to first passage percolation models (in lower dimensions).

This talk will be online at https://bluejeans.com/216376580/6460

Random graphs with latent geometric structure, where the edges are generated depending on some hidden random vectors, find broad applications in the real world, including social networks, wireless communications, and biological networks. As a first step to understand these models, the question of when they are different from random graphs with independent edges, i.e., Erd\H{o}s--R\'enyi graphs, has been studied recently. It was shown that geometry in these graphs is lost when the dimension of the latent space becomes large. In this talk, we focus on the case when there exist different notions of noise in the geometric graphs, and we show that there is a trade-off between dimensionality and noise in detecting geometry in the random graphs.

Statistical inference of stochastic processes based on high-frequency observations has been an active research area for more than a decade. The most studied problem is the estimation of the quadratic variation of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of truncated moments of Lévy processes, a new efficient estimator is developed for a class of Lévy processes of unbounded variation. The proposed method is based on an iterative debiasing procedure of truncated realized quadratic variations. This is joint work with Cooper Bonience and Yuchen Han.