The Annals of Applied Probability

Tracy–Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder

Arjun Krishnan and Jeremy Quastel

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The free-energy fluctuations of the discrete directed polymer in $1+1$ dimensions is conjecturally in the Tracy–Widom universality class at all finite temperatures and in the intermediate disorder regime. Seppäläinen’s log-gamma polymer was proven to have GUE Tracy–Widom fluctuations in a restricted temperature range by Borodin, Corwin and Remenik [Comm. Math. Phys. 324 (2013) 215–232]. We remove this restriction, and extend this result into the intermediate disorder regime. This result also identifies the scale of fluctuations of the log-gamma polymer in the intermediate disorder regime, and thus verifies a conjecture of Alberts, Khanin and Quastel [Ann. Probab. 42 (2014) 1212–1256]. Using a perturbation argument, we show that any polymer that matches a certain number of moments with the log-gamma polymer also has Tracy–Widom fluctuations in intermediate disorder.

Article information

Ann. Appl. Probab., Volume 28, Number 6 (2018), 3736-3764.

Received: October 2016
Revised: April 2018
First available in Project Euclid: 8 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments

Tracy–Widom distribution universality log-gamma directed polymer


Krishnan, Arjun; Quastel, Jeremy. Tracy–Widom fluctuations for perturbations of the log-gamma polymer in intermediate disorder. Ann. Appl. Probab. 28 (2018), no. 6, 3736--3764. doi:10.1214/18-AAP1404.

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