Matthew Junge and Shirshendu Chatterjee

## Seminar Schedule:

**The seminar meets on Tuesdays from 4:30 to 5:30 pm EDT.**

**Videoconference Link (via Zoom**):

**Time: August 31, 4:30 – 5:30 pm EDT****Speaker: Michael Damron, Georgia Institute of Technology**

**Seminar Link:**

**Zoom**

**Title:**Dynamical First-Passage Percolation

**Abstract:**In first-passage percolation (FPP), we place i.i.d. nonnegative weights on the edges of the cubic lattice Z^d and study the induced weighted graph metric T = T(x,y). Letting F be the common distribution function of the weights, it is known that if F(0) is less than the threshold p_c for Bernoulli percolation, then T(x,y) grows like a linear function of the distance |x-y|. In 2015, Ahlberg introduced a dynamical model of first-passage percolation, in which the weights are resampled according to Poisson clocks, and considered the growth of T(x,y) as time varies. He showed that when F(0) < p_c, the model has no “exceptional times” at which the order of the growth is anomalously large or small. I will discuss recent work with J. Hanson, D. Harper, and W.-K. Lam, in which we study this question in two dimensions in the critical regime, where F(0) = p_c, and T(x,y) typically grows sublinearly. We find that the existence of exceptional times depends on the behavior of F(x) for small positive x, and we characterize the dimension of the exceptional sets for all but a small class of such F.

**Time: September 14, 4:30 – 5:30 pm EDT****Speaker: Matthew Junge, Baruch College, CUNY**

**Seminar Link:**

**Zoom**

**Title:**Pandemic REUs

**Abstract:**The pandemic changed many things, REU Programs included. I will discuss challenges and advantages of mentoring undergraduates in math research from afar. Some results about interacting particle systems—namely, the frog model and ballistic annihilation—from this summer will also be presented.

**Time: October 05, 4:30 – 5:30 pm EDT****Speaker: David Aldous,**Professor, Emeritus and Professor of the Graduate School, UC Berkeley

**Seminar Link:**

**Zoom**

**Title:**Two processes on compact spaces

**Abstract: **It can be found here.

**Time**: October 12, 4:30 – 5:30 pm EDT

**Speaker**: Lily Reeves, Cornell University

**Seminar Link:**

**Zoom**

**Title:**Chemical distance for 2d critical percolation and random cluster model

**Abstract:**In 2-dimensional critical percolation, with positive probability, there is a path that connects the left and right side of a square box. The chemical distance is the expected length of the shortest such path conditional on its existence. In this talk, I will introduce the best known estimates for chemical distance. I will then discuss analogous estimates for the radial chemical distance (the expected length of the shortest path from the origin to distance n), as well as recent extensions of these estimates to the random cluster model. A portion of this talk is based on joint work with Philippe Sosoe.

**Time: October 19, 3:30 – 4:30 pm EDT**

**Seminar Link:**

**Zoom**

**Title:**Non-equilibrium multi-scale analysis and coexistence in competing

first-passage percolation

**Abstract:**We consider a natural random growth process with competition on Z^d called first-passage percolation in a hostile environment, that consists of two first-passage percolation processes FPP_1 and FPP_\lambda that compete for the occupancy of sites. Initially FPP_1 occupies the origin and spreads through the edges of Z^d at rate 1, while FPP_\lambda is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter p. A seed remains dormant until FPP_1 or FPP_\lambda attempts to occupy it after which it spreads through the edges of Z^d at rate \lambda. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on Z^d. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of FPP_\lambda could favor FPP_1. A central contribution of our work is the development of a novel multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity.

Based on a joint work with Tom Finn (Univ. of Bath).

**Time: October 26, 3:30 – 4:30 pm EDT**

**Speaker: Lisa Hartung, Johannes Gutenberg University**

**Seminar Link:**

**Zoom**

**Title:**Branching Brownian motion with self repulsion

**Abstract:**We consider a model of branching Brownian motion with self repulsion. Self-repulsion is introduced via change of measure that penalises particles spending time in an \epsilon-neighbourhood of each other. We derive a simplified version of the model where only branching events are penalised. This model is almost exactly solvable and we derive a precise description of the particle numbers and branching times. In the limit of weak penalty, an interesting universal time-inhomogeneous branching process emerges. The position of the maximum is governed by a F-KPP type reaction-diffusion equation with a time dependent reaction term. This is joint work with Anton Bovier.

**Time: November 02, 4:30 – 5:30 pm EDT**

**Speaker:**Souvik Dhara, MIT

**Seminar Link:**

**Zoom**

**Title:**An unexpected phase-transition for percolation on networks

**Abstract: **The talk concerns the critical behavior for percolation on inhomogeneous random networks on n vertices, where the weights of the vertices follow a power-law distribution with exponent τ∈(2,3). Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero, as n→ ∞. We show that the critical window for percolation phase transition is given by πc(λ)=λn^{−}^{(3}^{−}^{τ)/2}, for λ∈(0,λc), where λc>0 is an explicit constant. Rather surprisingly, it turns out that a giant component of size √n emerges for λ > λc. Thus, the critical window turns out to be of finite length, which is in sharp contrast with the previously studied critical behaviors for τ∈(3,4) and τ >4 regimes. The rescaled vector of maximum component sizes are shown to converge in distribution to an infinite vector of non-degenerate random variables that can be described in terms of components of a one-dimensional inhomogeneous percolation model on Z_{+} studied in a seminal work by Durrett and Kesten (1990). Based on joint work with Shankar Bhamidi, Remco van der Hofstad.

**Time: November 09, 4:30 – 5:30 pm EDT**

**Speaker:**Erin Beckman, Concordia University

**Seminar Link:**

**Zoom**

**Title:**Cooperative motion random walks

**Abstract:**Cooperative motion random walks form a family of random walks where each step is dependent upon the distribution of the walk itself. Movement is promoted at locations of high probability and dampened in locations of low probability. These processes are a generalization of the hipster random walk introduced by Addario-Berry et. al. in 2020. We study the process through a recursive equation satisfied by its CDF, allowing the evolution of the walk to be related to a finite difference scheme. I will discuss this relationship and how PDEs can be used to describe the distributional convergence of asymmetric and symmetric cooperative motion. This talk is based on joint work with Louigi Addario-Berry and Jessica Lin.

**Time: November 23, 4:30 – 5:30 pm EDT**

**Speaker:**Evan Sorensen, University of Wisconsin—Madison

**Seminar Link:**

**Zoom**

**Title:**Busemann functions and semi-infinite geodesics in a semi-discrete space

**Abstract:**In the last 10-15 years, Busemann functions have been a key tool for studying semi-infinite geodesics in planar first and last-passage percolation. We study Busemann functions in the semi-discrete Brownian last-passage percolation (BLPP) model and use these to derive geometric properties of the full collection of semi-infinite geodesics in BLPP. This includes a characterization of uniqueness and coalescence of semi-infinite geodesics across all asymptotic directions. To deal with the uncountable set of points in BLPP, we develop new methods of proof and uncover new phenomena, compared to discrete models. For example, for each asymptotic direction, there exists a random countable set of initial points out of which there exist two semi-infinite geodesics in that direction. Further, there exists a random set of points, of Hausdorff dimension ½, out of which, for some random direction, there are two semi-infinite geodesics that split from the initial point and never come back together. We derive these results by studying variational problems for Brownian motion with drift.

**Time: November 30, 4:30 – 5:30 pm EST**

**Speaker: Lionel Levine,**Cornell University

**Seminar Link:**

**Zoom**

**Title:**Exact sampling and fast mixing of Activated Random Walk

**Abstract:**Activated Random Walk (ARW) is an interacting particle system on the d-dimensional lattice Z^d: Random walkers fall asleep at a fixed rate and wake up any sleeping particles they encounter. On a finite subset V of Z^d it defines a Markov chain on {0,1}^V. We prove that when V is a Euclidean ball intersected with Z^d, the mixing time of the ARW Markov chain is at most 1+o(1) times the volume of the ball. The proof uses an exact sampling algorithm for the stationary state, a coupling with internal DLA, and an upper bound on the time when internal DLA fills the entire ball. We conjecture cutoff at time z times the volume of the ball, where z<1 is the limiting density of the stationary state. Joint work with Feng Liang.

**Time: December 07, 4:30 – 5:30 pm EST**

**Speaker: Evita Nestoridi,**Princeton University

**Seminar Link:**

**Zoom**

**Title:**Limit Profiles of Reversible Markov Chains

**Abstract:**The limit profile captures the exact shape of the distance of a Markov chain from stationarity. Lately, there have been new techniques developed with the aim of studying limit profiles. These techniques have been particularly helpful in studying the limit profile of certain interchange processes, such as the random-transpositions and star transpositions. In this talk, I will give an overview of these new results and discuss a few open questions.