First-passage percolation defines a random pseudo-metric on Z^d by attaching to each nearest-neighbor edge of the lattice a non-negative 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.

Based on joint work-in-progress with Michael Damron, Wai-Kit Lam, and Xiao Shen which was started at the AMS MRC on Spatial Stochastic Models in 2019.

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 well-known Caloger-Moser 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.

Joint work with Nam-Gyu Kang (KIAS) and Nikolai Makarov (Caltech).

We introduce a one-parameter family of critical Galton-Watson 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 Galton-Watson trees under consecutive Horton pruning. The invariant Galton-Watson (IGW) measures with i.i.d. exponential edge lengths are the only Galton-Watson measures invariant with respect to all admissible types of generalized dynamical pruning (an operation of erasing a tree from leaves down to the root).

This is a joint work with Ilya Zaliapin (University of Nevada Reno) and Guochen Xu (Oregon State University).

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) radio-interferometric 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).

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.

Based on joint work with Promit Ghosal (MIT) and Yuchen Liao (U Michigan).

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 McKean-Vlasov 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.

In this talk I will discuss the discrete-time 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 heavy-tailed Sibuya distribution. The results rely on duality between the voter model and coalescing random walks through time-reversal. 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.

We prove that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free 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 finite-system Hamiltonians uniform in the system size. To obtain this result, we adapt the Bravyi-Hastings-Michalakis strategy to the GNS representation of the infinite-system ground state. This is joint work with Bruno Nachtergaele and Amanda Young.

In this talk, I will introduce infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete 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.

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional 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.

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 super-telescoping formula. Using this formula we construct an exactly solvable non-homogeneous 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 Lee-Yang theorem in relation to this lattice model and explore its connections to the Riemann hypothesis.

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 box-ball 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 Ramalheira-Tsu that is an outgrowth of work in his recent PhD thesis.

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.
In this talk, we will discuss polygons which are projectively equivalent to their image under the pentagram map. One can think of such polygons as pentagram map "solitons". The result is that a real convex polygon P is projectively equivalent to its image under the pentagram map if and only if there is a conic inscribed in P and a conic circumscribed about P. I will explain the idea of the proof, which is based on the theory of commuting difference operators, elliptic curves, and theta functions.

Busemann functions are objects of interest in first- and last-passage 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 last-passage percolation with general weights are, in fact, directly related to the time-constant of last-passage percolation with exponential weights (a well-known integrable model). Using this relationship, we give an easily checkable condition that determines when adjacent Busemann increments are negatively correlated.

Joint work with I. Alevy.