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Random matrices, large deviations and reflected Brownian motion
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Ortmann, Janosch (2011) Random matrices, large deviations and reflected Brownian motion. PhD thesis, University of Warwick.
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WRAP_THESIS_Ortmann_2011.pdf - Submitted Version Download (1288Kb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2582723~S1
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 (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Brownian motion processes, Large deviations, Free probability theory | ||||
Official Date: | January 2011 | ||||
Dates: |
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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 |
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