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Theses Doctoral

Some Exactly Solvable Models And Their Asymptotics

Rychnovsky, Mark

In this thesis, we present three projects studying exactly solvable models in the KPZ universality class and one project studying a generalization of the SIR model from epidemiology. The first chapter gives an overview of the results and how they fit into the study of KPZ universality when applicable. Each of the following 4 chapters corresponds to a published or submitted article.

In the first project, we study an oriented first passage percolation model for the evolution of a river delta. We show that at any fixed positive time, the width of a river delta of length L approaches a constant times L²/³ with Tracy-Widom GUE fluctuations of order L⁴/⁹. This result can be rephrased in terms of a particle system generalizing pushTASEP. We introduce an exactly solvable particle system on the integer half line and show that after running the system for only finite time the particle positions have Tracy-Widom fluctuations.

In the second project, we study n-point sticky Brownian motions: a family of n diffusions that evolve as independent Brownian motions when they are apart, and interact locally so that the set of coincidence times has positive Lebesgue measure with positive probability. These diffusions can also be seen as n random motions in a random environment whose distribution is given by so-called stochastic flows of kernels. For a specific type of sticky interaction, we prove exact formulas characterizing the stochastic flow and show that in the large deviations regime, the random fluctuations of these stochastic flows are Tracy-Widom GUE distributed. An equivalent formulation of this result states that the extremal particle among n sticky Brownian motions has Tracy-Widom distributed fluctuations in the large n and large time limit. These results are proved by viewing sticky Brownian motions as a diffusive limit of the exactly solvable beta random walk in random environment.

In the third project, we study a class of probability distributions on the six-vertex model, which originates from the higher spin vertex model. For these random six-vertex models we show that the behavior near their base is asymptotically described by the GUE-corners process.
In the fourth project, we study a model for the spread of an epidemic. This model generalizes the classical SIR model to account for inhomogeneity in the infectiousness and susceptibility of individuals in the population. A first statement of this model is given in terms of infinitely many coupled differential equations. We show that solving these equations can be reduced to solving a one dimensional first order ODE, which is easy to solve numerically. We use the explicit form of this ODE to characterize the total number of people who are ever infected before the epidemic dies out. This model is not related to the KPZ universality class.

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More About This Work

Academic Units
Mathematics
Degree
Ph.D., Columbia University
Published Here
April 21, 2021