We introduce trap models on a finite volume $k$-level tree as a class of Markov jump processes with state space the leaves of that tree. They serve to describe the GREM-like trap model of Sasaki and Nemoto. Under suitable conditions on the parameters
of the trap model, we establish its infinite volume limit, given by what we call a $K$-process in an infinite $k$-level tree. From this we deduce that the $K$-process also is the scaling limit of the GREM-like trap model on extreme time scales under a fine tuning assumption on the volumes.
We obtain scaling limit results for asymmetric trap models and their infinite volume counterparts, namely asymmetric K processes. Aging results for the latter processes are derived therefrom.
We consider symmetric trap models in the d-dimensional hypercube whose ordered mean waiting times, seen as weights of a measure in the natural numbers, converge to a finite measure as d diverges, and show that the models suitably represented converge
to a K process as d diverges. We then apply this result to get K processes as the scaling limits of the REM-like trap model and the Random Hopping Times dynamics for the Random Energy Model in the hypercube in time scales corresponding to the ergodic regime for these dynamics.
We consider the two dimensional version of a drainage network model introduced by Gangopadhyay, Roy and Sarkar, and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the conv
ergence criteria proposed by Fontes, Isopi, Newman and Ravishankar.
The dynamical discrete web (DyDW),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 time parameter tau. The evolution is by independen
t updating of the underlying Bernoulli variables indexed by discrete space-time that define the discrete web at any fixed tau. In this paper, we study the existence of exceptional (random) values of tau where the paths of the web do not behave like usual random walks and the Hausdorff dimension of the set of exceptional such tau. Our results are motivated by those about exceptional times for dynamical percolation in high dimension by H{a}ggstrom, Peres and Steif, and in dimension two by Schramm and Steif. The exceptional behavior of the walks in the DyDW is rather different from the situation for the dynamical random walks of Benjamini, H{a}ggstrom, Peres and Steif. For example, we prove that the walk from the origin S^tau_0 violates the law of the iterated logarithm (LIL) on a set of tau of Hausdorff dimension one. We also discuss how these and other results extend to the dynamical Brownian web, the natural scaling limit of the DyDW.
We study K-processes, which are Markov processes in a denumerable state space, all of whose elements are stable, with the exception of a single state, starting from which the process enters finite sets of stable states with uniform distribution. We s
how how these processes arise, in a particular instance, as scaling limits of the trap model in the complete graph, and subsequently derive aging results for those models in this context.
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.
