Ortmann, Janosch (2011) Random matrices, large deviations and reflected Brownian motion. PhD thesis, University of Warwick.

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Abstract

In this thesis we present results in large deviations theory, free probability and the theory of reflected Brownian motion. We study the large deviations behaviour of the block structure of a non-crossing partition chosen uniformly at random. This allows us to apply the free momentcumulant formula of Speicher to express the spectral radius of a non-commutative random variable in terms of its free cumulants. Next the distributions of three quadratic functionals of the free Brownian bridge are studied: the square norm, the signature and the Lévy area of the free Brownian bridge. We introduce two representation of the free Brownian bridge as series involving free semicircular variables, analogous to classical results due to Lévy and Kac. The latter representation extends to all semicircular processes. For each of the three quadratic functionals we give the R-transform, from which we extract information about the distribution, including free infinite divisibility and smoothness of the density. We also apply our result about the spectral radius to compute the maximum of the support for Lévy area and square norm. In both cases we obtain implicit equations. The final chapter of the thesis is devoted to the study of a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. This is motivated by recent developments in the theory of directed polymer and percolation models, in which existence of an invariant measure in product form plays a role. Informally, RBM is defined by running a standard Brownian motion in the polyhedral domain and giving it a singular drift whenever it hits one of the boundaries, kicking the process back into the interior. Our process is obtained by replacing this singular drift by a continuous one, involving a continuous potential. RBM has an invariant measure in product form if and only if a certain skew-symmetry condition holds. We show that this result extends to our generalisation. Applications include examples motivated by queueing theory, Brownian motion with rank-dependent drift and a process with close connections to the δ-Bose gas.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Brownian motion processes, Large deviations, Free probability theory
Date:	January 2011
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	O'Connell, Neil, 1968-
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC)
Extent:	ix, 136 leaves : ill.
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/50020

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