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Elliptic Combinatorics and Markov Processes


Betea, Dan Dumitru (2012) Elliptic Combinatorics and Markov Processes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/MMZG-5G61.


We present combinatorial and probabilistic interpretations of recent results in the theory of elliptic special functions (due to, among many others, Frenkel, Turaev, Spiridonov, and Zhedanov in the case of univariate functions, and Rains in the multivariate case). We focus on elliptically distributed random lozenge tilings of the hexagon which we analyze from several perspectives. We compute the N-point function for the associated process, and show the process as a whole is determinantal with correlation kernel given by elliptic biorthogonal functions. We furthermore compute transition probabilities for the Markov processes involved and show they come from the multivariate elliptic difference operators of Rains. Properties of difference operators yield an efficient sampling algorithm for such random lozenge tilings. Simulations of said algorithm lead to new arctic circle behavior. Finally we introduce elliptic Schur processes on bounded partitions analogous to the Schur process of Reshetikhin and Okounkov ( and to the Macdonald processes of Vuletic, Borodin, and Corwin). These give a somewhat different (and faster) sampling algorithm from these elliptic distributions, but in principle should encompass more than just tilings of a hexagon.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Combinatorics, Special Functions, Probability
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Rains, Eric M.
Thesis Committee:
  • Rains, Eric M. (chair)
  • Owhadi, Houman
  • Wales, David B.
  • Pak, Igor
Defense Date:22 May 2012
Record Number:CaltechTHESIS:05312012-201348939
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:7115
Deposited By: Dan Betea
Deposited On:04 Jun 2012 16:57
Last Modified:03 Oct 2019 23:56

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