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.

Asymmetric simple exclusion process and modified random matrix ensembles: Taro Nagao and Tomohiro Sasamoto

Asymmetric simple exclusion process and modified random matrix ensembles
Taro Nagao and Tomohiro Sasamoto
Nuclear Physics B
Volume 699, Issue 3, 8 November 2004, Pages 487-502.
arXiv: http://arxiv.org/abs/cond-mat/0405321

Abstract: We study the fluctuation properties of the asymmetric simple exclusion process (ASEP) on an infinite one-dimensional lattice. When $N$ particles are initially situated in the negative region with a uniform density $\rho_-=1$, Johansson showed the equivalence of the current fluctuation of ASEP and the largest eigenvalue distribution of random matrices. We extend Johansson's formula and derive modified ensembles of random matrices, corresponding to general ASEP initial conditions. Taking the scaling limit, we find that a phase change of the asymptotic current fluctuation occurs at a critical position.

Reflections: This paper arrives at the Johansson type results for (essentially) one-sided LPP without appealing to Young Tableaux and RSK, but rather by just using Greens functions. The Greens functions correspond to the Tracy and Widom ASEP formulas, for p=1, q=0. They derive the one-sided LPP Prahofer and Spohn conjecture from these methods. The results here are similar to the BBP results on eigenvalues of perturbed Wishart ensembles.

Purpose

The purpose of this blog is my attempt at keeping track of mathematical papers which I've read or looked at. I will try to write a LATEX bbl entry for every article and then summarize the topic, the results, what it uses, and where it is subsequently used. I'm not sure if this is the best way to organize this information, but for the time being it will need to do. Feel free to comment on your thoughts about these articles and also about my annotation of them.