Alexander Drewitz

Survival Probability of a Random Walk Among a Poisson System of Moving Traps

We review some old and prove some new results on the survival probability of a random walk among a Poisson system of moving traps on \({\mathbb{Z}}^{d}\), which can also be interpreted as the solution of a parabolic Anderson model with a random time-dependent potential. We show that the annealed survival probability decays asymptotically as e\({}^{-{\lambda }_{1}\sqrt{t}}\) for d = 1, as e\({}^{-{\lambda }_{2}t/\log t}\) for d = 2, and as e\({}^{-{\lambda }_{d}t}\) for d ≥ 3, where λ1 and λ2 can be identified explicitly. In addition, we show that the quenched survival probability decays asymptotically as e\({}^{-\tilde{{\lambda }}_{d}t}\), with \(\tilde{{\lambda }}_{d} > 0\) for all d ≥ 1. A key ingredient in bounding the annealed survival probability is what is known in the physics literature as the Pascal principle, which asserts that the annealed survival probability is maximized if the random walk stays at a fixed position. A corollary of independent interest is that the expected cardinality of the range of a continuous time symmetric random walk increases under perturbation by a deterministic path.

On chemical distances and shape theorems in percolation models with long-range correlations

In this paper, we provide general conditions on a one parameter family of random infinite subsets of Zd to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of Cerný and Popov [“On the internal distance in the interlacement set,” Electron. J. Probab. 17(29), 1–25 (2012)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of Antal and Pisztora [“On the chemical distance for supercritical Bernoulli percolation...

An Introduction to Random Interlacements

Random Walk, Green Function, Equilibrium Measure.- Random Interlacements: First Definition and Basic Properties.- Random Walk on the Torus and Random Interlacements.- Poisson Point Processes.- Random Interlacements Point Process.- Percolation of the Vacant Set.- Source of Correlations and Decorrelation via Coupling.- Decoupling Inequalities.- Phase Transition of Vu.- Coupling of Point Measures of Excursions.

Local percolative properties of the vacant set of random interlacements with small intensity

Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown in arXiv:0704.2560 and arXiv:0808.3344, and the infinite connected component, when it exists, is almost surely unique, see arXiv:0805.4106. In this paper we study local percolative properties of the vacant set of random interlacements at level u for all dimensions d>=3 and small intensity parameter u>0. We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level u. Our results imply that finite connected components of the vacant set at level u are unlikely to be large. These results were proved in arXiv:1002.4995 for d>=5. Our approach is different from that of arXiv:1002.4995 and works for all d>=3. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

Selected Topics in Random Walks in Random Environment

Random walk in random environment (RWRE) is a fundamental model of statistical mechanics, describing the movement of a particle in a highly disordered and inhomogeneous medium as a random walk with random jump probabilities. It has been introduced in a series of papers as a model of DNA chain replication and crystal growth (see Chernov [10] and Temkin [51, 52]), and also as a model of turbulent behavior in fluids through a Lorentz gas description (Sinai 1982 [42]). It is a simple but powerful model for a variety of complex large-scale disordered phenomena arising from fields such as physics, biology, and engineering. While the one-dimensional model is well-understood in the multidimensional setting, fundamental questions about the RWRE model have resisted repeated and persistent attempts to answer them. Two major complications in this context stem from the loss of the Markov property under the averaged measure as well as the fact that in dimensions larger than one, the RWRE is not reversible anymore. In these notes we present a general overview of the model, with an emphasis on the multidimensional setting and a more detailed description of recent progress around ballisticity questions.

Asymptotic direction in random walks in random environment revisited

Recently Simenhaus in [Sim07] proved that for any elliptic random walk in random environment, transience in the neighborhood of a given direction is equivalent to the a.s. existence of a deterministic asymptotic direction and to transience in any direction in the open half space defined by this asymptotic direction. Here we prove an improved version of this result and review some open problems.

Transience of the vacant set for near-critical random interlacements in high dimensions

The model of random interlacements is a one-parameter family

Poisson Point Processes

\mathcal I^u,

Random Interlacements: First Definition and Basic Properties

u \ge 0,