Ben Fleming (2008-2011)


Particle system realisations of determinantal processes


Special classes of non-intersecting or interlacing particle systems, inspired by various statistical models, and their description in terms of determinantal correlation functions are the main themes of this thesis. Another unifying aspect of our subject matter is that the particle systems permit scaling limits in which their join law becomes identical to the joint law of the eigenvalues for certain ensembles of random matrices.

Particle systems resulting from a queueing model, as well as particle systems relating to tilings of hexagons, are reviewed from the viewpoint of the joint PDF and their scaling to random matric forms. An interlacing particle system relating to a limit of an \( a \times b \times c \) hexagon with \( a \) large is introduced, and forms the majority of the subject matter for this thesis. The particle system is analyzed by the computation of single line PDFs and correlation functions, as well as density profiles and scaled correlation functions in certain scaled limits.

This in turn is possible due to their being an underlying determinantal structure to the joint PDFs which carries to the correlations themselves. The functional forms obtained involve classical orthogonal polynomials, and their asymptotic properties allow scaled limits to be calculated. The scaled functional forms exhibit a universality property, being common (mostly) to a class of models which exhibit fluctuations known from random matrix theory.

Finally, a particle system relating to tilings of the Aztec diamond is studied from the viewpoint of the joint PDF, allowing seemingly original methods to be used to show known results such as the total number of possible tilings of an Aztec diamond, and the so-called arctic circle boundary.

Published papers:

A finitization of the bead process (with P.J. Forrester and E. Nordenstam), Probability Theory and Related Fields DOI: 10.1007/s00440-010-0324-5

Interlaced particle systems and tilings of the Aztec diamond (with P.J. Forrester), Journal of Statistical Physics 141, 441-459 (2011)