Frank den Hollander

Bio: Frank den Hollander is an academic researcher from Leiden University. The author has contributed to research in topics: Random walk & Random field. The author has an hindex of 14, co-authored 54 publications receiving 546 citations.

Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

Abstract: In this paper, we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let β denote the inverse temperature and let Λ β C ℤ 2 be a square box with periodic boundary conditions such that lim β→∞ |Λ β | = ∞. We run the dynamics on Λ β , starting from a random initial configuration where all of the droplets (clusters of plus-spins and clusters of particles, respectively) are small. For large β and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single critical droplet somewhere in Λ β . Using potential-theoretic methods, we compute the average nucleation time (the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to 1 as β → ∞. It turns out that this time grows as Ke Γβ /|Λ β | for Glauber dynamics and as Kβe Γβ /|Λ β | for Kawasaki dynamics, where Γ is the local canonical (resp. grand-canonical) energy, to create a critical droplet and K is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on |Λ β |). The fact that the average nucleation time is inversely proportional to |Λ β | is referred to as homogeneous nucleation because it says that the critical droplet for the transition appears essentially independently in small boxes that partition Λ β .

Law of Large Numbers for a Class of Random Walks in Dynamic Random Environments

Abstract: In this paper we consider a class of one-dimensional interacting particle systems in equilibrium, constituting a dynamic random environment, together with a nearest-neighbor random walk that on occupied/vacant sites has a local drift to the right/left. We adapt a regeneration-time argument originally developed by Comets and Zeitouni for static random environments to prove that, under a space-time mixing property for the dynamic random environment called cone-mixing, the random walk has an a.s. constant global speed. In addition, we show that if the dynamic random environment is exponentially mixing in space-time and the local drifts are small, then the global speed can be written as a power series in the size of the local drifts. From the first term in this series the sign of the global speed can be read off. The results can be easily extended to higher dimensions.

Construction of the incipient infinite cluster for spread-out oriented percolation above 4+1 dimensions

Abstract: We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ d × ℤ+, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ n (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ d × ℤ+, summing this probability over x  ℤ d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n −1, we prove existence of a limiting measure ℚ∞, with ℚ∞ = ℙ∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1.

A new inductive approach to the lace expansion for self-avoiding walks

Abstract: We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤ d where loops of length m are penalised by a factor e −β/m p (0 4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0.

Invasion percolation on regular trees

Abstract: We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its r-point function for any r=2 and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

A Definitive Book on Seed Ecology@@@Seeds: Ecology, Biogeography, and Evolution of Dormancy and Germination

Abstract: Introduction. Ecologically Meaningful Germination Studies. Types of Seed Dormancy. Germination Ecology of Seeds with Nondeep Physiological Dormancy. Germination Ecology of Seeds with Morphophysiological Dormancy. Germination Ecology of Seeds with Physical Dormancy. Germination Ecology of Seeds in the Persistent Seed Bank. Causes of Within-Species Variations in Seed Dormancy and Germination Characteristics. A Geographical Perspective on Germination Ecology: Tropical and Subtropical Zones. A Geographical Perspective on Germination Ecology: Temperate and Arctic Zones. Germination Ecology of Plants with Specialized Life Cycles and/or Habitats. Biogeographical and Evolutionary Aspects of Seed Dormancy. Subject Index.

The Lace Expansion and its Applications

Abstract: Simple Random Walk.- The Self-Avoiding Walk.- The Lace Expansion for the Self-Avoiding Walk.- Diagrammatic Estimates for the Self-Avoiding Walk.- Convergence for the Self-Avoiding Walk.- Further Results for the Self-Avoiding Walk.- Lattice Trees.- The Lace Expansion for Lattice Trees.- Percolation.- The Expansion for Percolation.- Results for Percolation.- Oriented Percolation.- Expansions for Oriented Percolation.- The Contact Process.- Branching Random Walk.- Integrated Super-Brownian Excursion.- Super-Brownian Motion.

The Parabolic Anderson Model

Abstract: This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials We thereby consider examples of potentials studied in the literature In the particularly important case of an iid potential with double-exponential tails we formulate the asymptotic results in detail Furthermore, we explain that, under mild regularity assumptions, there are only four different universality classes of asymptotic behaviors Finally, we study the moment Lyapunov exponents for space-time homogeneous catalytic potentials generated by a Poisson field of random walks

Random walks on disordered media and their scaling limits

Abstract: The main theme of these lectures is to analyze heat conduction on disordered media such as fractals and percolation clusters using both probabilistic and analytic methods, and to study the scaling limits of Markov chains on the media.