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Some Exactly Solvable Models and Their Asymptotics.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	Some Exactly Solvable Models and Their Asymptotics./
Author:	Rychnovsky, Mark.
Description:	1 online resource (293 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 82-10, Section: B.
Contained By:	Dissertations Abstracts International82-10B.
Subject:	Statistics. -
Online resource:	click for full text (PQDT)
ISBN:	9798597099170

Some Exactly Solvable Models and Their Asymptotics.
Rychnovsky, Mark.

Some Exactly Solvable Models and Their Asymptotics. - 1 online resource (293 pages)

Source: Dissertations Abstracts International, Volume: 82-10, Section: B.

Thesis (Ph.D.)--Columbia University, 2021.

Includes bibliographical references

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.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798597099170Subjects--Topical Terms:

556824
Statistics.
Subjects--Index Terms:

Exactly solvable modelsIndex Terms--Genre/Form:

554714
Electronic books.

Some Exactly Solvable Models and Their Asymptotics.
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502 $a Thesis (Ph.D.)--Columbia University, 2021.
504 $a Includes bibliographical references
520 $a 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.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
650 4 $a Statistics. $3 556824
650 4 $a Epidemiology. $3 635923
653 $a Exactly solvable models
653 $a Interacting particle systems
653 $a KPZ universality
653 $a Kardar-Parisi-Zhang equation
655 7 $a Electronic books. $2 local $3 554714
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690 $a 0463
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710 2 $a Columbia University. $b Mathematics. $3 1179054
773 0 $t Dissertations Abstracts International $g 82-10B.
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