Erin Beckman (Concordia)
Title: Cooperative Motion Random Walk
Abstract: We look at a family of random walks which are defined by a type of cooperative motion. These processes are a generalization of the totally asymmetric hipster random walk studied by Addario-Berry et. al. [PTRF, ’20]. We study the processes through a recursive distributional equation, which allows the evolution of the walk to be related to a finite difference scheme. We discuss distributional convergence of the process and its relation to the solution of a Hamilton-Jacobi equation.
The talk is based on joint work with Louigi Addario-Berry and Jessica Lin.
Duncan Dauvergne (Princeton)
Title: Learning from the directed landscape
Abstract: The directed landscape is a random `directed metric’ on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. It is expected to be the universal scaling limit for all models in the KPZ universality class for random growth. In this talk, I will describe its construction in terms of the Airy line ensemble, give an extension of this construction for optimal length disjoint paths in the directed landscape, and show how these constructions reveal surprising Brownian structures in the directed landscape.
This talk is based on joint work with J. Ortmann, B. Virag, and L. Zhang.
Michael Damron (GA Tech)
Title: Translation-invariant nearest neighbor graphs
Abstract: Given edge-lengths te assigned to the edges of 𝑍𝑑 for 𝑑 ≥ 2 , each vertex draws a directed edge to its closest neighbor to form the “nearest neighbor” graph. Nanda-Newman studied these graphs when the te‘s are i.i.d. and continuously distributed and showed that the undirected version has only finite components, with size distribution whose tail decays like 1/n!. I will discuss recent work with B. Bock and J. Hanson in which the 𝑡! ‘s are only assumed to be translation-invariant and distinct. Here, we can show that there are no doubly—infinite paths and completely characterize the set of possible graphs. In particular, for d=2, the number of infinite components is either 0,1, or 2, and for 𝑑 ≥ 3, it can be any nonnegative integer. I will also mention relations to both geodesic graphs from first-passage percolation and the coalescing walk model of Chaika-Krishnan.
Jasmine Foo (U. Minnesota)
Title:Evolutionary dynamics of cancer initiation
Abstract: The process of cancer initiation from healthy epithelial tissue can be modeled using stochastic spatial processes. In particular, cancer is often caused by genetic mutations which confer a fitness advantage to a cell, leading to a clonal expansion of its progeny through the tissue. In this talk I will discuss some models of this evolutionary process, and explore how tissue architecture may impact cancer initiation.
Reza Gheissari (UC Berkeley)
Title: Exact thresholds for random-cluster dynamics via coarse-graining
Abstract: The random-cluster model at parameters (p,q) is a model of dependent percolation generalizing independent bond percolation, and at integer q corresponding in a precise way to the Ising (q=2) and Potts (q≥3) models. In joint work with A. Sinclair, we study the mixing time of Glauber dynamics for the random-cluster model on Zd for general d≥3. Using a dynamical coarse-graining scheme that handles high and low temperatures simultaneously, we prove optimal mixing time bounds at all high-temperatures, as well as at all low-temperatures under a weak mixing condition conjectured to hold up to the critical point. In particular, this gives an optimal MCMC sampler from the Ising model on Zd at all off-critical temperatures, as well as for the Potts model on Zd at all high-temperatures.
Chris Hoffman (U. Washington)
Title: Bi-infinite geodesics in First and Last Passage Percolation in Z2
Abstract: First and last passage percolation on Z2 are two classes of random spatial stochastic processes that are believed to be in the KPZ universality class. One of the most important questions in first passage percolation is whether there exists bi-infinite geodesics a.s. A bi-infinite geodesic is a path 𝛾 such that for any two points 𝑥, 𝑦 ∈ Z2 on the path the shortest between 𝑥, 𝑦 ∈ Z2 lies along 𝛾. I will discuss recent results about the existence of two sided geodesics in first passage percolation and for the analogous problem in last passage percolation
This talk will cover joint work with Alan Sly, Riddhi Basu and Daniel Ahlberg
Leo Petrov (UVA)
Title: Reversing nonequilibrium systems
Abstract: A typical stochastic particle model for nonequilibrium thermodynamics starts from a densely packed initial configuration, and evolves by emanating particles into the “rarefaction fan”. Imagine having air and vacuum in two halves of a room, and removing the separating barrier. I will explain how for very special (integrable) stochastic particle systems one can explicitly “undo” the rarefaction, and construct another Markov chain which “puts the air back into its half of the room”. I will also discuss the corresponding stationary processes preserving each time-t nonequilibrium measure.
Bálint Virág (U. Toronto)
Title: Introduction to random plane geometry.
Abstract: If lengths 1 and 2 are assigned randomly to each edge in Z2, what are the fluctuations of distances between far away points?
This problem is open, yet we know, in great detail, what to expect. The directed landscape, a universal random plane geometry, provides the answer to such questions.
What is the directed landscape? What does it teach us about longest increasing subsequences in random permutations, about random polymers, about models for spread of infection, about tetris, about random Schrodinger operators, and about cell biology?