FitzGerald, William Robert (2019) Exactly solvable interacting particle systems. PhD thesis, University of Warwick.

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Abstract

The results of Chapter 2 are related to point-to-line last passage percolation and directed polymers in the KPZ universality class. We prove an equality in law between the invariant measure of a reflected system of Brownian motions and a vector of point-to-line last passage percolation times in a discrete random environment. A consequence describes the distribution of the all-time supremum of Dyson Brownian motion with drift. A finite temperature version relates the point-to-line partition functions of two directed polymers, with an inverse-gamma and a Brownian environment, and generalises Dufresne’s identity. Our proof introduces an interacting system of Brownian motions with an invariant measure given by a field of point-to-line log partition functions for the log-gamma polymer.

The results of Chapter 3 are related to the probability of a gap in a Pfaffian point process with applications to random matrices, random polynomials and systems of coalescing and annihilating Brownian motions. We find the leading order exponent and constant term in the asymptotics of the probability of a large gap in a Pfaffian point process for a large class of kernels including both translation invariant kernels and non translation invariant kernels. We analyse the asymptotics of Fredholm determinants and Fredholm Pfaffians by findind probability expressions, inspired by the work of Kac [72], which can be analysed by classical results for random walks. This has applications to the largest real eigenvalue and the bulk scaling limit of the real eigenvalues of the real Ginibre ensemble; the fixed time particle positions and persistence probabilities for coalescing and annihilating Brownian motions; and persistence probabilities for random polynomials. We also find a new example of a Pfaffian point process given by an absorbed point process at the origin for coalescing and annihilating Brownian motions which is related to the real zeroes of a Gaussian power series.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Brownian motion processes, Percolation, Pfaffian systems, Kernel functions
Official Date:	September 2019
Dates:	Date Event September 2019 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Warren, Jon (Reader in statistics) ; Tribe, Roger ; Zaboronski, Oleg V.
Sponsors:	Engineering and Physical Sciences Research Council
Format of File:	pdf
Extent:	207 leaves
Language:	eng

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