6 research outputs found
Occupation times of long-range exclusion and connections to KPZ class exponents
With respect to a class of long-range exclusion processes on \ZZ^d, with single particle transition rates of order , starting under Bernoulli invariant measure with density , we consider the fluctuation behavior of occupation times at a vertex and more general additive functionals. Part of our motivation is to investigate the dependence on , and with respect to the variance of these functionals and associated scaling limits.
In the case the rates are symmetric, among other results, we find the scaling limits exhaust a range of fractional Brownian motions with Hurst parameter .
However, in the asymmetric case, we study the asymptotics of the variances, which when and points to a curious dichotomy between long-range strength parameters . In the former case, the order of the occupation time variance is the same as under the process with symmetrized transition rates, which are calculated exactly. In the latter situation, we provide consistent lower and upper bounds and other motivations that this variance order is the same as under the asymmetric short-range model, which is connected to KPZ class scalings of the space-time bulk mass density fluctuations.The research of CB was supported in part by the French Ministry of Education through the grant ANR JCJC EDNHS. PG thanks FCT (Portugal) for support through the research project PTDC/MAT/109844/2009 and CNPq (Brazil) for support through the research project 480431/2013-2. PG thanks CMAT for support by "FEDER" through the "Programa Operacional Factores de Competitividade COMPETE" and by FCT through the project PEst-C/MAT/UI0013/2011. SS was supported in part by ARO grant W911NF-14-1-0179
A pedestrian's view on interacting particle systems, KPZ universality, and random matrices
These notes are based on lectures delivered by the authors at a Langeoog
seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a
mixed audience of mathematicians and theoretical physicists. After a brief
outline of the basic physical concepts of equilibrium and nonequilibrium
states, the one-dimensional simple exclusion process is introduced as a
paradigmatic nonequilibrium interacting particle system. The stationary measure
on the ring is derived and the idea of the hydrodynamic limit is sketched. We
then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and
explain the associated universality conjecture for surface fluctuations in
growth models. This is followed by a detailed exposition of a seminal paper of
Johansson that relates the current fluctuations of the totally asymmetric
simple exclusion process (TASEP) to the Tracy-Widom distribution of random
matrix theory. The implications of this result are discussed within the
framework of the KPZ conjecture.Comment: 52 pages, 4 figures; to appear in J. Phys. A: Math. Theo