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# The Pfaffian Schur process

## Citation

Vuletic, Mirjana (2009) The Pfaffian Schur process. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-05292009-161729

## Abstract

This thesis consists of an introduction and three independent chapters.

In Chapter 2, we define the shifted Schur process as a measure on sequences of strict partitions. This process is a generalization of the shifted Schur measure introduced by Tracy-Widom and Matsumoto, and is a shifted version of the Schur process introduced by Okounkov-Reshetikhin. We prove that the shifted Schur process defines a Pfaffian point process. Furthermore, we apply this fact to compute the bulk scaling limit of the correlation functions for a measure on strict plane partitions which is an analog of the uniform measure on ordinary plane partitions. This allows us to obtain the limit shape of large strict plane partitions distributed according to this measure. The limit shape is given in terms of the Ronkin function of the polynomial P(z,w)=-1+z+w+zw and is parameterized on the domain representing half of the amoeba of this polynomial. As a byproduct, we obtain a shifted analog of famous MacMahon's formula.

In Chapter 3, we generalize the generating formula for plane partitions known as MacMahon's formula, as well as its analog for strict plane partitions. We give a 2-parameter generalization of these formulas related to Macdonald's symmetric functions. Our formula is especially simple in the Hall--Littlewood case. We also give a bijective proof of the analog of MacMahon's formula for strict plane partitions.

In Chapter 4, generating functions of plane overpartitions are obtained using various methods: nonintersecting paths, RSK type algorithms and symmetric functions. We give t-generating formulas for cylindric partitions. We also show that overpartitions correspond to domino tilings and give some basic properties of this correspondence. This is a joint work with Sylvie Corteel and Cyrille Savelief.

Item Type: Thesis (Dissertation (Ph.D.)) correlation functions; generating formulas; point processes; Schur process; symmetric functions California Institute of Technology Physics, Mathematics and Astronomy Mathematics Public (worldwide access) Borodin, Alexei Borodin, Alexei (chair)Wales, David B.Rains, Eric M.Duits, Maurice 18 May 2009 CaltechETD:etd-05292009-161729 http://resolver.caltech.edu/CaltechETD:etd-05292009-161729 No commercial reproduction, distribution, display or performance rights in this work are provided. 2280 CaltechTHESIS Imported from ETD-db 02 Jun 2009 26 Dec 2012 02:49

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