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

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.