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Sharp phase transition for Cox percolation
We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence, satisfies a local boundedness condition and can be constructed from a discrete iid random field, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction that allows us to compare different notions of influence.
Phase transitions for the Boolean model of continuum percolation for Cox point processes
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.
Aging in the GREM-like trap model
The GREM-like trap model is a continuous time Markov jump process on the leaves of a finite volume L-level tree whose transition rates depend on a trapping landscape built on the vertices of the whole tree. We prove that the natural two-time correlation function of the dynamics ages in the infinite volume limit and identify the limiting function. Moreover, we take the limit L→ ∞ of the two-time correlation function of the infinite volume L-level tree. The aging behavior of the dynamics is characterized by a collection of clock processes, one for each level of the tree. We show that for any L, the joint law of the clock processes converges. Furthermore, any such limit can be expressed through Neveu's continuous state branching process. Hence, the latter contains all the information needed to describe aging in the GREM-like trap model both for finite and infinite levels.
Sharp phase transition for Cox percolation
We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence and satisfies a local boundedness condition, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction.
Extremal aging for trap models
In the seminal work [5], Ben Arous and Cerný give a general characterization of aging for trap models in terms of α-stable subordinators with α ∈ (0,1). Some of the important examples that fall into this universality class are Random Hopping Time (RHT) dynamics of Random Energy Model (REM) and p-spin models observed on exponential time scales. In this paper, we explain a different aging mechanism in terms of extremal processes that can be seen as the extension of α-stable aging to the case α=0. We apply this mechanism to the RHT dynamics of the REM for a wide range of temperature and time scales. The other examples that exhibit extremal aging include the Sherrington Kirkpatrick (SK) model and p-spin models [6, 9], and biased random walk on critical Galton-Watson trees conditioned to survive [11].