Purdue Probability Seminar, Fall 2024

This talk will discuss some recent progress on understanding the structure of semi-infinite geodesics and their associated Busemann functions in the inhomogeneous exactly solvable exponential last-passage percolation model. In contrast to the homogeneous model, this generalization admits linear segments of the limit shape and an associated richer structure of semi-infinite geodesic behaviors. Depending on certain choices of the inhomogeneity parameters, we show that the model exhibits new behaviors of semi-infinite geodesics, which include wandering semi-infinite geodesics with no asymptotic direction, isolated asymptotic directions of semi-infinite geodesics, and non-trivial intervals of directions with no semi-infinite geodesics. Based on joint work-in-progress with Elnur Emrah (Bristol) and Timo Seppäläinen (Madison)

There has been enormous progress in the last few decades concerning random geometric objects in two dimensions which interact nicely with conformal maps. Such objects include Schramm-Loewner evolution (SLE), Liouville quantum gravity (LQG), and discrete analogs thereof. However, much less is known about analogs of these objects in dimension $d\geq 3$. I will give an overview of a few known results and many open problems concerning random geometry in dimension $d\geq 3$. Some of the known results come from recent joint works with Jian Ding and Zijie Zhuang, with Ahmed Bou-Rabee, and with Federico Bertacco. I will not assume any background knowledge about random geometry for $d=2$.

The "true" self-avoiding random walk (TSAW) is a model for random motion that is very strongly self-repelling. Over 25 years ago, Toth showed that the one-dimensional version of this walk is strongly super-diffusive in the sense that the position of the walk after $n$ steps should converge in distribution when scaled by $n^{2/3}$. Toth was able to characterize what the limiting distribution of the walk would be if it exists, but he was not able to prove that the limiting distribution does in fact exist. Later, motivated by Toth's earlier results, Toth and Werner constructed a continuous time limit called the true self-repelling motion which they argued should be the functional scaling limit for the TSAW, but once again they did not prove this functional limiting distribution. In this talk I will show how one can use joint Ray-Knight theorems to prove that the TSAW does indeed converge in distribution to the true self-repelling motion. This is based on joint work (in progress) with Elena Kosygina.

In this talk, I will present a method for obtaining one-point lower tail large deviation principles (LDPs) for integrable models, including the stochastic six-vertex model, q-PNG, and ASEP. A crucial/novel element of this method involves establishing log-concavity estimates on tail probabilities. Based on two joint works and one ongoing collaboration with Yuchen Liao and Matteo Mucciconi.

I will describe the model of two-player nonzero-sum stopping games in discrete time, define the concepts of epsilon-equilibrium and random stopping times, and show how a stochastic variation of Ramsey's Theorem from Graph Theory can be used to prove the existence of epsilon-equilibrium in random stopping times in two-player nonzero-sum stopping games in discrete time. Joint work with Eran Shmaya (Stony Brook University)

Consider the following randomly forced reaction-diffusion equation: \(\partial_t u = \tfrac12\partial^2_x u + b(u) + \sigma(u)\dot{W},\)subject to "nice initial data" where (the random forcing) denotes a "space-time white noise." The function (the interaction term) is assumed to be positive, bounded, globally Lipschitz continuous, and bounded uniformly away from the origin, and the function (the sink/source term) is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the classical Osgood condition \(\int_1^\infty {\rm d}y/b(y)<\infty\) implies that, with probability one, any reasonable solution theory leads us to instantaneous and complete blowup. In light of the celebrated work of Fujita (1963) and many others on the blowup problem for nonrandom PDEs, our result warns how the introduction of an iota of random forcing can fundamentally change the qualitative behavior of the solution to a near-critical reaction-diffusion equation. The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities, and the study of the "spatial ergodicity of stochastic convolutions" using Malliavin's calculus and Poincaré inequalities that were recently developed by Le Chen, David Nualart, Fei Pu, and the speaker (2021, 2022). This is based on joint work with Mohammud Foondun and Eulalia Nualart.

Many classical results for the Brownian motion rely on its special properties such as independent increments and Markov property. When moving away from the Brownian motion, strong local nondeterminism (SLND) turned out to be a useful tool for studying Gaussian random fields including solutions to SPDEs. In this talk, I will give an overview of SLND and some recent works on SLND of SPDEs. I will also present a uniform dimension result for systems of nonlinear stochastic heat equations (ongoing joint work with Davar Khoshnevisan, Fei Pu and Yimin Xiao).

In this talk, we consider the stochastic wave and heat equations on the real line and driven by a Gaussian noise which behaves, in the space variable, like a fractional Brownian motion with Hurst index $H\in (0, 1)$. First, we explain how such a noise may be mathematically modeled, and then we define the notion of mild solution to our equations. Secondly, we focus on our main objective: prove that the solution of each of the above equations is continuous in terms of the parameter $H$, with respect to the convergence in law in the space of continuous functions. This talk is based on joint work with Luca M. Giordano (Milano) and Maria Jolis (Barcelona).

We develop an extension of the Stein-Malliavin calculus which allows to measure the Wasserstein distance between the probability distributions of $ (X, Y)$ and $(Z,Y)$, where $X,Y$ are arbitrary random vectors and $ Z\sim N(0, \sigma ^{2})$ is independent of $Y$. We apply this method to study the asymptotic independence for sequences of random vectors A particular focus will be on the spatial average of solutions to SPDEs and on certain parameter estimators.

Activated random walk (ARW) is a toy particle system that conjecturally exhibits 'self-organized criticality,' a phenomenon which was first studied by physicists in the late 1980s in the context of complex real-world systems like forest fires and earthquakes. I will discuss recent results for ARW and the stochastic sandpile (a related model) on Z and other graphs, plus many open questions.

In this talk, I will describe a model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In this stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, the model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, I will discuss the large-window asymptotics of graph-theoretic and topological characteristics such as higher-order degree distribution, Betti numbers of the associated Dowker complex as well as simplex counts. In particular, for the latter quantity there are regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. I will conclude by a simulation study and an application to the collaboration network extracted from the arXiv dataset. If time permits, I will also comment on results in the dynamic case, when the networks changes over time. This talk is based on joint work with M. Brun, P. Juhász, and M. Otto