Stochastics Seminar
joint with the University of Arizona
Spring 2021 Wednesday 3:004:00 Utah Time
Zoom information: Meeting ID: 998 1181 2123 Passcode: Email the organizers
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Date  Speaker  Title (click for abstract, if available) 

January 20 
Chris Janjigian
Purdue University 
Firstpassage percolation defines a random pseudometric on Z^d by attaching to each nearestneighbor edge of the lattice a nonnegative weight. Geodesics are paths which realize the distance between sites. This project considers the question of what the environment looks like on a geodesic through the lens of the empirical distribution on that geodesic when the weights are i.i.d.. We obtain upper and lower tail bounds for the upper and lower tails which quantify and limit the intuitive statement that the typical weight on a geodesic should be small compared to the marginal distribution of an edge weight.

January 27 
Tom Alberts
University of Utah 
Recently Peltola and Wang introduced the multiple SLE(0) process as the deterministic limit of the random multiple SLE(kappa) curves as kappa goes to zero. They prove this result by means of a ``small kappa’’ large deviations principle, but the limiting curves also turn out to have important geometric characterizations that are independent of their relation to SLE(kappa). In particular, they show that the SLE(0) curves can be generated by a deterministic Loewner evolution driven by multiple points, and the vector field describing the evolution of these points must satisfy a particular system of algebraic equations. We show how to generate solutions to these algebraic equations in two ways: first in terms of the poles and critical points of an associated real rational function, and second via the wellknown CalogerMoser integrable system with particular initial velocities. Although our results are purely deterministic they are again motivated by taking limits of probabilistic constructions, which I will explain. 
February 3 
Yevgeniy Kovchegov
Oregon State University 
We introduce a oneparameter family of critical GaltonWatson tree measures invariant under the operation of Horton pruning (cutting tree leaves followed by series reduction). Under a regularity condition, this family of measures are the attractors of critical GaltonWatson trees under consecutive Horton pruning. The invariant GaltonWatson (IGW) measures with i.i.d. exponential edge lengths are the only GaltonWatson measures invariant with respect to all admissible types of generalized dynamical pruning (an operation of erasing a tree from leaves down to the root). 
February 10 
Sam Gralla
University of Arizona 
General relativity implies that images of black holes will contain narrow rings of light with a precisely predicted, nearly circular shape. Motivated by the prospect of comparing (future) radiointerferometric observations with fundamental theory, we have studied the geometry of plane curves in terms of their interferometric observable, the "projected position function". This has led to some fun connections with classical results and curves (Cauchy surface area theorem, Reuleaux triangle, Cartesian oval). 
February 17 
Pierre Yves Gaudreau Lamarre
University of Chicago 
In this talk, I will discuss recent progress in the understanding of the structure in the spectrum of random Schrödinger operators. More specifically, I will introduce the concept of number rigidity in point processes and discuss recent efforts to understand its occurrence in the spectrum of random Schrödinger operators.

February 24 
No talk this week


March 3 
Grigorios Pavliotis
Imperial College London 
In this talk I will present some recent results on mean field limits for interacting diffusions. We study problems for which the mean field limit exhibits phase transitions, in the sense that the limiting McKeanVlasov PDE can have more than one stationary states, at a sufficiently strong interaction strength/low temperature. We provide a general characterization of first and second order phase transitions for mean field dynamics on the torus and we study fluctuations around the mean field limit. As a case study, we consider the combined mean field/homogenization limit for noisy Kuramoto oscillators. In addition, we study the breakdown of linear response theory for the mean field dynamics at the phase transition point. Applications of this type of dynamics to models for opinion formation and to sampling and optimization algorithms are also discussed. 
March 10 
Iddo BenAri
University of Connecticut 
In this talk I will discuss the discretetime voter model for opinion dynamics and its quasistationary distribution (QSD). The focus will be on the sequence of QSDs corresponding to the model on complete bipartite graphs with a "large" partition whose size tends to infinity and a "small" partition of constant size. In this case, the QSDs converge to a nontrivial limit featuring a consensus, except for a random number of dissenting vertices in the large partition which follows the heavytailed Sibuya distribution. The results rely on duality between the voter model and coalescing random walks through timereversal. Time permitting, I'll expand the discussion on the duality and its application to a broader class of processes. The research presented in this talk was carried out during the 2019 UConn Markov Chains REU and is joint work with Hugo Panzo and student participants Philip Speegle and R. Oliver VandenBerg. arXiv:2004.10187. 
March 17 
Robert Sims
University of Arizona 
We prove that uniformly small shortrange perturbations do not close the bulk gap above the ground state of frustrationfree quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finitesystem Hamiltonians uniform in the system size. To obtain this result, we adapt the BravyiHastingsMichalakis strategy to the GNS representation of the infinitesystem ground state. This is joint work with Bruno Nachtergaele and Amanda Young. 
March 24 Special Time: 3PM  4PM 
Makiko Sasada
University of Tokyo 
In this talk, I will introduce infinite versions of four wellstudied discrete integrable models, namely the ultradiscrete KdV equation, the discrete KdV equation, the ultradiscrete Toda equation, and the discrete Toda equation. These systems are understood as "deterministic vertex model”, which are discretely indexed in space and time, and their deterministic dynamics is defined locally via lattice equations. They have another formulation via the generalized Pitman’s transform, which is a new crucial observation. We show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Also, a detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. This talk is based on a joint work with David Croydon and Satoshi Tsujimoto. 
March 31 
No talk this week


April 7 
Jason Schweinsberg
UC San Diego 
Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to onedimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the Biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave. This is joint work with Matt Roberts. 
April 14 
Mohammad Latifi
University of Arizona 
A correspondence between S^1 trace of the Gaussian free field on the unit disk and a distribution of Verblunsky coefficients leads to an intriguing identity which we call the supertelescoping formula. Using this formula we construct an exactly solvable nonhomogeneous 1D Ising model. We further proceed with a natural construction of a statistical field model, with explicit hamiltonian, where the partition function is given by the Riemann zeta function. We finish with a discussion of the LeeYang theorem in relation to this lattice model and explore its connections to the Riemann hypothesis. 
April 21 
Nick Ercolani/Jonathan RamalheiraTsu
University of Arizona 
Classical constructions from soliton theory are making a reappearance in many novel contexts within mathematical physics and representation theory. One of the most notable recent examples of this concerns cellular automata known as boxball systems (BBS). We present results related to the phase shift phenomena of interacting solitons in this BBS setting and, time permitting, will indicate some of its potential applications. This is joint work with Jonathan RamalheiraTsu that is an outgrowth of work in his recent PhD thesis. 
April 28 
Anton Izosimov
University of Arizona 
The pentagram map, introduced by Richard Schwartz in 1992, is a discrete dynamical system on planar polygons. By definition, the image of a polygon P under the pentagram map is the polygon whose vertices are intersections of shortest diagonals of P (i.e. diagonals connecting second nearest vertices). The pentagram map is a completely integrable system which can be thought of as a lattice version of the Boussinesq model in hydrodynamics.

May 5 
Arjun Krishnan
University of Rochester 
Busemann functions are objects of interest in first and
lastpassage percolation. Determining the correlations of Busemann
function increments is important because of their relationship to the
second KPZ relationship that relates the two fluctuation exponents in
the model. We show that the correlations of adjacent Busemann
increments in lastpassage percolation with general weights are, in
fact, directly related to the timeconstant of lastpassage percolation
with exponential weights (a wellknown integrable model). Using this
relationship, we give an easily checkable condition that determines when
adjacent Busemann increments are negatively correlated.

Stochastics Seminar for Spring 2021 is organized at the University of Utah by Tom Alberts, Davar Khoshnevisan, Firas RassoulAgha.
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This web page is maintained by Tom Alberts.
Past Seminars:
 Fall 2020 
 Fall 2019  Spring 2020
 Fall 2018  Spring 2019
 Fall 2017  Spring 2018
 Fall 2016  Spring 2017
 Fall 2015  Spring 2016
 Fall 2014  Spring 2015
 Fall 2013  Spring 2014
 Fall 2012  Spring 2013
 Fall 2011  Spring 2012
 Fall 2010  Spring 2011
 Fall 2009  Spring 2010
 Fall 2008  Spring 2009
 Fall 2007  Spring 2008
 Fall 2006  Spring 2007
 Fall 2005  Spring 2006
 Fall 2004  Spring 2005
 Fall 2003  Spring 2003
 Fall 2002  Spring 2002
 Fall 2001
 Winter 2000
 Fall 1999
 Spring 1998