We introduce a general model of trapping for random walks on graphs. We give the possible scaling limits of these Randomly Trapped Random Walks on Z. These scaling limits include the well-known fractional kinetics process, the Fontes-Isopi-Newman singular diffusion as well as a new broad class we call spatially subordinated Brownian motions. We give sufficient conditions for convergence and illustrate these on two important examples.1. Introduction. We present here a general class of trapping mechanisms for random walks. This class includes the usual "effective" models of trapping, from the Continuous Time Random Walks (CTRW) (see [27]), to the Bouchaud Trap Models (BTM) (see [11][12][13][14] and [6]). It is in fact much wider. This higher level of generality is needed for the study of random walks on classical random structures, where the trapping is not introduced ab initio as in the CTRW or the BTM, but is created by the complexity of the underlying geometry. We introduce the class of models for general graphs, but restrict the study in this paper to the case of the line Z. We obtain a rather complete understanding of the asymptotic behavior of these trapped walks on Z. We give first a description of all possible scaling limits, and then proceed to give wide sufficient conditions for convergence to each of the possible scaling limits. We illustrate this by two simple examples, one effective and the other geometric, where we exhibit a rich transition picture between those different asymptotic regimes and scaling limits.The behavior of these models in higher dimension or other graphs is open. It seems clear that, when the underlying graph is transient, the asymptotic
In this paper we present rigorous results on the critical behavior of the Activated Random Walk model. We conjecture that on a general class of graphs, including Z d , and under general initial conditions, the system at the critical point does not reach an absorbing state. We prove this for the case where the sleep rate λ is infinite. Moreover, for the one-dimensional asymmetric system, we identify the scaling limit of the flow through the origin at criticality. The case λ < +∞ remains largely open, with the exception of the one-dimensional totally-asymmetric case, for which it is known that there is no fixation at criticality.
We study an infinite system of moving particles, where each particle is of type A or B. Particles perform independent random walks at rates D A > 0 and D B 0, and the interaction is given by mutual annihilation A + B → ∅. The initial condition is i.i.d. with finite first moment. We show that this system is site-recurrent, that is, each site is visited infinitely many times. We also generalize a lower bound on the density decay of Bramson and Lebowitz by considering a construction that handles different jump rates.
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