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GOE fluctuations for the maximum of the top path in alternating sign matrices

Ayyer, Arvind; Chhita, Sunil; Johansson, Kurt

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Authors

Arvind Ayyer

Kurt Johansson



Abstract

The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter ∆. When ∆ = 0, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all ∆, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at ∆ = 1/2, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy–Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger’s proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free fermion case.

Citation

Ayyer, A., Chhita, S., & Johansson, K. (2023). GOE fluctuations for the maximum of the top path in alternating sign matrices. Duke Mathematical Journal, 172(10), 1961-2104. https://doi.org/10.1215/00127094-2022-0075

Journal Article Type Article
Acceptance Date Jul 6, 2022
Online Publication Date Aug 15, 2023
Publication Date Jul 15, 2023
Deposit Date Jul 7, 2022
Publicly Available Date Jul 15, 2023
Journal Duke Mathematical Journal
Print ISSN 0012-7094
Electronic ISSN 1547-7398
Publisher Duke University Press
Peer Reviewed Peer Reviewed
Volume 172
Issue 10
Pages 1961-2104
DOI https://doi.org/10.1215/00127094-2022-0075
Public URL https://durham-repository.worktribe.com/output/1201674
Publisher URL https://www.dukeupress.edu/duke-mathematical-journal
Related Public URLs https://arxiv.org/abs/2109.02422

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