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

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Official URL: http://webcat.warwick.ac.uk/record=b3491704~S15
Abstract
The results of Chapter 2 are related to pointtoline 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 pointtoline last passage percolation times in a discrete random environment. A consequence describes the distribution of the alltime supremum of Dyson Brownian motion with drift. A finite temperature version relates the pointtoline partition functions of two directed polymers, with an inversegamma 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 pointtoline log partition functions for the loggamma 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 (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: 


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:  
Extent:  207 leaves  
Language:  eng 
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