Non UBC
DSpace
Moll, Alexander
2017-02-07T09:23:21
2016-04-13T10:29
Jack measures on partitions occur naturally in the study of continuum circular log-gases in generic background potentials V at arbitrary values \beta of Dyson’s inverse temperature. Our main result is a law of large numbers (LLN) and central limit theorem (CLT) for Jack measures in the macroscopic scaling limit, which corresponds to the large N limit in the log-gas. Precisely, the emergent limit shape and macroscopic fluctuations of profiles of these random Young diagrams are the push-forwards along V of the uniform measure on the circle (LLN) and of the restriction to the circle of a Gaussian free field on the upper half-plane (CLT), respectively. At \beta=2, this recovers Okounkov’s LLN for Schur measures (2003) and coincides with Breuer-Duits’ CLT for biorthogonal ensembles (2013).
Our limit theorems follow from an all-order expansion (AOE) of joint cumulants of linear statistics, which has the same form as the all-order 1/N refined topological expansion for the log-gas on the line due to Chekhov-Eynard (2006) and Borot-Guionnet (2012). To prove our AOE, we rely on the Lax operator for the quantum Benjamin-Ono hierarchy with periodic profile V exhibited in collective field variables by Nazarov-Sklyanin (2013). The characterization of the limit laws as push-forwards follows from factorization formulas for resolvents of Toeplitz operators with symbol V due to Krein and Calderón-Spitzer-Widom (1958).
https://circle.library.ubc.ca/rest/handle/2429/59436?expand=metadata
63 minutes
video/mp4
Author affiliation: MIT
Banff (Alta.)
10.14288/1.0319126
eng
Unreviewed
Vancouver : University of British Columbia Library
Banff International Research Station for Mathematical Innovation and Discovery
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Graduate
BIRS Workshop Lecture Videos (Banff, Alta)
Mathematics
Probability theory and stochastic processes
Statistical mechanics, structure of matter
Random partitions and the quantum Benjamin-Ono hierarchy
Moving Image
http://hdl.handle.net/2429/59436