Fluctuation exponents for stationary exactly solvable lattice polymer models via a Mellin transform framework

Abstract

We develop a Mellin transform framework which allows us to simultaneously analyze the four known exactly solvable 1`1 dimensional lattice polymer models: the log-gamma, strict-weak, beta, and inversebeta models. Using this framework we prove the conjectured fluctuation exponents of the free energy and the polymer path for the stationary point-to-point versions of these four models. The fluctuation exponent for the polymer path was previously unproved for the strict-weak, beta, and inverse-beta models.

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique

“…One promising probabilistic approach capable of identifying the KPZ scaling exponents is the coupling method introduced by E. Cator and P. Groeneboom in the context of Hammersley's process [25]. This is a particularly versatile scheme that has since been further developed and fruitfully adapted to a variety of KPZ class models [8,9,10,11,12,14,27,29,30,67,73,74,83]. In broad strokes, the method compares a model under study with its stationary versions through suitable couplings, and likely produces results as long as the latter models are sufficiently tractable.…”

Tracy-Widom Asymptotics for a River Delta Model

“…In [30] it was shown that the heat kernel for the time reversed Beta RWRE converges to the stochastic heat equation with multiplicative noise. In [9] it was shown using a stationary version of the model that a Beta RWRE conditioned to have atypical velocity has wandering exponent 2/3 (see also [26]), as expected in general for directed polymers in 1 + 1 dimensions. The stationary structure of Bernoulli-exponential FPP was computed in [48] (In [48] Bernoulli-exponential FPP is referred to as the Bernoulli-exponential polymer).…”

“…In what follows, we will use (25) or (26) when we say that an infinite sum is δ-equivalent to its first term.…”

Large deviations and wandering exponent for random walk in a dynamic beta environment

“…Theorem 5.1 is the same as Proposition 1.1 in [5], via the connections (4.1) and (4.2) between the RWRE and the polymer. Theorem 5.1 is also proved in Section 4.1 of the first preprint version [3] of this paper.…”

“…By looking at the random walk paths under the Doob-transformed RWRE in reverse direction, we can view this model as a stationary directed polymer model, called the beta polymer. We establish this connection in the present section, and then use it in the next two sections to rely on recently published estimates in [5] for the technical work behind our Theorems 2.4 and 2.5. The polymer model described here is case (1.4) on p. 4 of [5], with their parameter triple pµ, β, θq corresponding to our pα, β, λq.…”