Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks

Fluctuations from a hydrodynamic limit of a one-dimensional asymmetric system come at two levels. On the central limit scale n^{1/2} one sees initial fluctuations transported along characteristics and no dynamical noise. The second order of fluctuations comes from the particle current across the characteristic. For a system made up of independent random walks we show that the second-order fluctuations appear at scale n^{1/4} and converge to a certain self-similar Gaussian process. If the system is in equilibrium, this limiting process specializes to fractional Brownian motion with Hurst parameter 1/4. This contrasts with asymmetric exclusion and Hammersley's process whose second-order fluctuations appear at scale n^{1/3}, as has been discovered through related combinatorial growth models.Comment: Published at http://dx.doi.org/10.1214/009117904000000946 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Strong law of large numbers for the interface in ballistic deposition

We prove a hydrodynamic limit for ballistic deposition on a multidimensional lattice. In this growth model particles rain down at random and stick to the growing cluster at the first point of contact. The theorem is that if the initial random interface converges to a deterministic macroscopic function, then at later times the height of the scaled interface converges to the viscosity solution of a Hamilton-Jacobi equation. The proof idea is to decompose the interface into the shapes that grow from individual seeds of the initial interface. This decomposition converges to a variational formula that defines viscosity solutions of the macrosopic equation. The technical side of the proof involves subadditive methods and large deviation bounds for related first-passage percolation processes

Fluctuation bounds for the asymmetric simple exclusion process

We give a partly new proof of the fluctuation bounds for the second class particle and current in the stationary asymmetric simple exclusion process. One novelty is a coupling that preserves the ordering of second class particles in two systems that are themselves ordered coordinatewise.Comment: Minor improvements made to text. 24 page

A convexity property of expectations under exponential weights

Take a random variable X with some finite exponential moments. Define an exponentially weighted expectation by E^t(f) = E(e^{tX}f)/E(e^{tX}) for admissible values of the parameter t. Denote the weighted expectation of X itself by r(t) = E^t(X), with inverse function t(r). We prove that for a convex function f the expectation E^{t(r)}(f) is a convex function of the parameter r. Along the way we develop correlation inequalities for convex functions. Motivation for this result comes from equilibrium investigations of some stochastic interacting systems with stationary product distributions. In particular, convexity of the hydrodynamic flux function follows in some cases.Comment: After completion of this manuscript we learned that our main results can be obtained as a special case of some propositions in Karlin: Total Positivity, Vol.