Sublinear Variance for Directed Last-Passage Percolation

Abstract

A range of first-passage percolation type models are believed to demonstrate the related properties of sublinear variance and superdiffusivity. We show that directed last-passage percolation with Gaussian vertex weights has a sublinear variance property. We also consider other vertex weight distributions.Corresponding results are obtained for the ground state of the 'directed polymers in a random environment' model.

Gaussian fluctuations of replica overlap in directed polymers

“…As mentioned previously, the proof of Theorem 1.1 relies on proving the superconcentration of log Z T . Similar results have been obtained in [1,24,12] for different models. Our approach follows [13], and a crucial input is an estimate on the spatial variations of the solution to the KPZ equation, see Proposition 3.2 below.…”

Asymptotic behavior for a version of directed percolation on the honeycomb lattice

“…Furthermore, it is known that the boundary of the Domany-Kinzel model has the same distribution as the one-dimensional last passage percolation model [10]. A threedimensional version of Domany-Kinzel model with occupation probability 1 along two spatial directions was considered in Ref.…”

Subgaussian concentration and rates of convergence in directed polymers

“…The assumption of gaussian disorder is also strongly used there. In [14] estimates of the variance of directed last passage percolation are obtained via a coupling method, which appears difficult to extend to the case of polymers. In [7] exponential concentration estimates on the scale (|v|/ log |v|) 1/2 were obtained for first passage percolation, for a large class of disorders.…”