Cube root fluctuations for the corner growth model associated to the exclusion process: Balazs, Cator and Seppalainen

Cube root fluctuations for the corner growth model associated to the exclusion process
Balázs Márton, Eric Cator, Timo Seppäläinen
ELECTRONIC JOURNAL OF PROBABILITY 11: pp. 1094-1132. (2006).

Abstract: We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order t^{2/3}. With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order t^{1/3}, and also that the transversal fluctuations of the maximal path have order t^{2/3}. We adapt and then build on a recent study of Hammersley's process by Cator and Groeneboom, and also utilize the competition interface introduced by Ferrari, Martin and Pimentel. The arguments are entirely probabilistic, and no use is made of the combinatorics of Young tableaux or methods of asymptotic analysis.

Reflections: This shows the rarefaction fan fluctuation orders of Prahofer and Spohn but not the scaling functions. I'm not yet completely sure about what they really prove as far as the distribution of edge occupancy and also the transversal fluctuations (as opposed to, say, Johansson). Perhaps their large deviation estimates are what is needed to put through the Johansson argument. The competition interface idea is pretty interesting. I need to still reflect on what this tells about the fluctuations of the location of the second class particle.