We discuss necessary and sufficient conditions for the convergence of disordered asymmetric zero-range process to the critical invariant measures.
We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for asymmetric nearest neighbour zero range processes with non homogeneous jump rates. The class of environments considered is close to that consider
ed by Andjel, Ferrari, Guiol and Landim, while our class of processes is broader. We also give a simpler proof of a result of Ferrari and Sisko with weaker assumptions.
The dynamical discrete web (DDW), introduced in recent work of Howitt and Warren, is a system of coalescing simple symmetric one-dimensional random walks which evolve in an extra continuous dynamical parameter s. The evolution is by independent updat
ing of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed s. In this paper, we study the existence of exceptional (random) values of s where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of such exceptional s. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by Haggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in DDW is rather different from the situation for dynamical random walks of Benjamini, Haggstrom, Peres and Steif. In particular, we prove that there are exceptional values of s for which the walk from the origin S^s(n) has limsup S^s(n)/sqrt n leq K with a nontrivial dependence of the Hausdorff dimension on K. We also discuss how these and other results extend to the dynamical Brownian web, a natural scaling limit of DDW. The scaling limit is the focus of a paper in preparation; it was studied by Howitt and Warren and is related to the Brownian net of Sun and Swart.
