The Annals of Probability

Airy point process at the liquid-gas boundary

Vincent Beffara, Sunil Chhita, and Kurt Johansson

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Abstract

Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function, we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.

Article information

Source
Ann. Probab., Volume 46, Number 5 (2018), 2973-3013.

Dates
Received: February 2017
Revised: November 2017
First available in Project Euclid: 24 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1535097644

Digital Object Identifier
doi:10.1214/17-AOP1244

Mathematical Reviews number (MathSciNet)
MR3846843

Subjects
Primary: 60G55: Point processes 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Domino tilings Airy kernel point process two-periodic Aztec diamond

Citation

Beffara, Vincent; Chhita, Sunil; Johansson, Kurt. Airy point process at the liquid-gas boundary. Ann. Probab. 46 (2018), no. 5, 2973--3013. doi:10.1214/17-AOP1244. https://projecteuclid.org/euclid.aop/1535097644


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