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

Abstract

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 et al. [11]. 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 et al. [3]. 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.

Tracy-Widom Asymptotics for a River Delta Model

“…Most of them can be realized as more or less direct degenerations of the higher-spin stochastic six-vertex model. This includes particle systems such as exclusion processes (q-TASEP [22,10,33,43] and other models [12,6,36,54]), directed polymers ( [17,21,18,32,40,42]), and the stochastic six-vertex model [3,1,11,19,24].…”

“…Note that in these bounds we can make δ 1 , δ 2 , δ 3 as small as desired by choosing ε small. Equations (39) and (40) follow from the fact that f 1 , and f 2 are holomorphic in the compact set B ε (λ). And equation (41) follows from Lemma 2.7.…”

“…And equation (41) follows from Lemma 2.7. Note that along D 0 , z is purely imaginary, so (39), (40), and (41) show that the full exponential in the integrand in (11) is bounded above by e 2δ 3 e −(f 1 (λ)−δ 1 )ω 3 −(f 2 (λ)−δ 2 )ω .…”

Stochastic growth in time-dependent environments

“…Mathematical proofs would require performing a more careful analysis of the function H along the tails of the contours, and proving a number of estimates to justify the convergence. Such justifications can be found for instance in [23,41,42] in a similar context. Making all these justifications in the present context (inhomogeneous weights) constitutes a mathematical challenge, but from the physical point of view it does not seem necessary, and thus we will proceed via saddle point analysis without attempting to prove rigorously a theorem.…”

“…provided the distribution of weights w has a sufficient number of finite moments [61]. This estimate is based on the universality of convergence of directed polymer free energy at high temperature to the KPZ equation [3,24] (see also [42]). In the case of the log-gamma polymer, one can check that indeed as γ goes to infinity,…”