The 41st Midwest Probability Colloquium

Conference Abstracts

Harry Crane (Rutgers University) (Two talks)

Probabilistic Symmetry and Network Models, I & II

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Abstract: Talk 1: Basic symmetries and network sampling. The first lecture focuses
on exchangeability and sampling considerations in network modeling. After introducing
some well known random graph models, I compare structural results for graphon models
(Aldous, 1979; Hoover, 1980; Kallenberg, 2006) and edge exchangeable models (Crane-
Dempsey, 2018). I discuss qualitative similarities and differences between these
cases and show how the main results extend naturally to higher-order structures and
more general invariance principles (Crane-Dempsey, 2019; Crane-Towsner, 2018).
Talk 2: Dynamic network models. The second lecture focuses on network dynamics.
In this setting, I discuss graph-valued Markov process models that satisfy a certain
Markov projectivity condition. These processes evolve on the state space of graphs
with countable vertex set in such a way that the process induced by projecting to
any subset of vertices preserves the Markov property. I discuss a general structural
result for processes of this type (Crane, 2017; Crane-Towsner, 2019+) and then
present a special subclass of combinatorial Levy processes, for which additional
results are known (Crane, 2018).
Many questions about the material in both lectures remain open. I will highlight some
such problems throughout the lectures.
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Greg Lawler (University of Chicago) (Two talks)

Talk 1: Complex Gaussian Fields and (Brownian Motion and Random Walk) Loop Soups

Abstract: There has been a lot of work relating Gaussian fields and their squares with random walks and Brownian motion. Among the tools are the Brownian and random walk loop soups. I will give an introduction from the perspective of discrete time loop measures and describe an improvement of a result of Trujillo Ferreras and myself on the convergence of random walk loops to Brownian loops. This is related to the convergence of the square of a discrete Gaussian field to the square of the continuous field. This is joint work with Peter Panov. Talk 2: Brownian Loops, Multiple Radial Schramm-Lowner Evolution and Dyson Brownian Motion Abstract: There are many models from statistical physics that give measures to configurations of interacting or non-self-intersecting curves. The limit objects for the curves are often Schramm-Loewner evolution (SLE) paths and it is not obvious on the continuous level how to tilt individual paths to get interacting paths. I will discuss how the Brownian loop measure helps describe this. I will then discuss a recent result with Vivian Healey showing how multiple radial SLE paths lead naturally to a driving function that is Dyson Brownian motion.

Louis-Pierre Arguin (City University of New York)

Large Values of the Riemann Zeta Function in Short IntervalsAbstract:cIn a seminal paper in 2012, Fyodorov & Keating proposed a series of conjectures describing the statistics of large values of zeta in short intervals of the critical line. In particular, they relate these statistics to the ones of log-correlated Gaussian fields. In this lecture, I will present recent results that answer many aspects of these conjectures. Connections to problems in number theory will also be discussed.

Jasmine Foo (University of Minnesota)

Statistial Models of Cancer Evolution

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.

Janko Gravner (University of California, Davis)

Long-range Bootstrap Percolation

iteratively enlarges a set of occupied sites by adjoining points with at least

a threshold number of occupied neighbors. The initially occupied set is random,

given by a uniform product measure with a low density. Bootstrap percolation on

integer lattices is a classic model. This talk will focus on recent results for this

model for graphs with long-range connectivity, specifically Hamming graphs and their

Cartesian products with lattices, which are joint work with David Sivakoff and

Erik Slivken.