On some special directed last-passage percolation models

Abstract

Abstract. We investigate extended processes given by last-passage times in directed models defined using exponential variables with decaying mean. In certain cases we find the universal Airy process, but other cases lead to nonuniversal and trivial extended processes.

Stochastic growth in time-dependent environments

“…where F GUE is the CDF of the TW GUE distribution. When the parameter θ goes to 0, one recovers the zerotemperature model studied by Johannsson [12] (in the sense that F n goes to the zero-temperature free energy defined in [12]). Using the asymptotics ψ (x)…”

“…hence the transition at a = 1/3 at zero-temperature. Considering the limit θ → 0 of (20) an asymptotic analysis recovers the one-point distribution results from [12]. We may also let θ go to zero simultaneously as n goes to infinity and study the crossover between zero and finite temperature.…”

“…where E i,j are independent exponential random variables with parameter a i + b j . This is the model studied in [12], which considers in particular the case a i = b j = i a . Let us scale u in (124) as u = e −sθ −1 .…”

“…The asymptotic behaviour of F n will depend on whether σ n stays bounded or diverges as n → ∞. In the zero temperature limit θ → 0, the threshold found in [12] was for a = 1/3. Interestingly, the result is different for positive θ.…”

“…In a seminal paper, Johansson [11] obtained an exact and concise formula for the optimal energy for a directed polymer at zero temperature on the square lattice (of coordinate i, j) with random exponential on site energies. He then extended in [12] the solution to an inhomogeneous disorder, where the amplitude of the disorder was chosen 1/(i a + j a ), with a 0. He discovered a sharp transition at a = 1/3: δh ∝ t 1/3−a with universal TW fluctuations if a > 1/3 and δh < ∞ with non universal fluctuations if a < 1/3.…”