Luiz Renato Fontes

On the dynamics of trap models in Z^d

We consider trap models on Z^d, namely continuous time Markov jump process on Z^d with embedded chain given by a generic discrete time random walk, and whose mean waiting time at x is given by tau_x, with tau = (tau_x, x in Z^d) a family of positive iid random variables in the basin of attraction of an alpha-stable law, 0<alpha<1. We may think of x as a trap, and tau_x as the depth of the trap at x. We are interested in the trap process, namely the process that associates to time t the depth of the currently visited trap. Our first result is the convergence of the law of that process under suitable scaling. The limit process is given by the jumps of a certain alpha-stable subordinator at the inverse of another alpha-stable subordinator, correlated with the first subordinator. For that result, the requirements for the embedded random walk are a) the validity of a law of large numbers for its range, and b) the slow variation at infinity of the tail of the distribution of its time of return to the origin: they include all transient random walks as well as all planar random walks, and also many one dimensional random walks. We then derive aging results for the process, namely scaling limits for some two-time correlation functions thereof, a strong form of which requires an assumption of transience, stronger than a, b. The above mentioned scaling limit result is an averaged result with respect to tau. Under an additional condition on the size of the intersection of the ranges of two independent copies of the embeddded random walk, roughly saying that it is small compared with the size of the range, we derive a stronger scaling limit result, roughly stating that it holds in probability with respect to tau. With that additional condition, we also strengthen the aging results, from the averaged version mentioned above, to convergence in probability with respect to tau.

Convergence of Symmetric Trap Models in the Hypercube

We consider symmetric trap models in the d-dimensional hypercube whose ordered mean waiting times, seen as weights of a measure in ℕ*, converge to a finite measure as d→∞, and show that the models suitably represented converge to a K process as d→∞. 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.

The K-process on a tree as a scaling limit of the GREM-like trap model

We introduce trap models on a finite volume

Phase Transitions in Layered Systems

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Scaling limits and aging for asymmetric trap models on the complete graph and K processes

-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

On an epidemic model on finite graphs

K

Scaling limit of the radial Poissonian web

-process in an infinite

GREM-like K processes on trees with infinite depth

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K-processes, scaling limit and aging for the REM-like trap model

-level tree. From this we deduce that the

Asymptotic behavior and aging of a low temperature cascading 2-GREM dynamics at extreme time scales

K