The spatial embeddings of genealogies in models with fluctuating population sizes and local regulation are relatively complicated random walks in a space-time dependent random environment. They seem presently not well understood. We use the supercritical discrete-time contact process on Z^d as the simplest non-trivial example of a locally regulated population model and study the dynamics of an ancestral lineage sampled at stationarity, viz. a directed random walk on a supercritical directed percolation cluster. We prove a LLN and an annealed CLT for such a walk via a regeneration approach. Furthermore, we discuss approaches to larger samples and to more general models that allow multiple occupancy of sites. Based on joint work in progress with Jiri Cerny, Andrej Depperschmidt and Nina Gantert.
We describe the motion of multiple interfaces for the one-dimensional Cahn-Hilliard equation on a bounded interval perturbed by small additive noise. The approach is based on an approximate slow manifold introduced by Bates and Xun (1994/95), where the interface positions are the coordinates on the manifold. This is useful to verify metastable behaviour of solutions with multiple interfaces. We derive a stochastic differential equation for the motion of the interfaces. The approximation is (with high probability) valid until an interface breaks down or until very large times. This is joint work with Dimitra Antonopoulou and Georgia Karali.
We study models of the motion by mean curvature of an (1+1) dimensional interface with random forcing. For the well-posedness we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution equations in the variational framework of Krylov- Rozovskii, replacing the standard coercivity assumption by a Lyapunov type condition. We also study the long term behaviour, showing that the homogeneous normal noise model with periodic boundary conditions converges to a spatially constant profile whose height behaves like a Brownian motion. For the additive vertical noise model with Dirichlet boundary conditions we show ergodicity, using the lower bound technique for Markov semigroups by Komorowski, Peszat and Szarek.
We consider a continuous height version of the Abelian sandpile model with small amount of bulk dissipation gamma -> 0 on each toppling, in dimensions d = 2, 3. In the limit gamma -> 0, we give a power law upper bound, based on coupling, on the rate at which the stationary measure converges to the discrete critical sandpile measure. The proofs are based on a coding of the stationary measure by weighted spanning trees, and an analysis of the latter via Wilson's algorithm. In the course of the proof, we prove an estimate on coupling a geometrically killed loop-erased random walk to an unkilled loop-erased random walk.
It has been noticed that below te critical temperature the mixing properties of the Stochastic Ising model are strongly dependent on boundary condition. Indeed if one considers Heat-Bath Dynamics for Ising model in the cube of side L, the mixing time is exponential in L whereas it is believed that for all + boundary condition, it behaves like L^2 (conjecture called "Lifshitz law"). What we present here is a new step toward the verification of the conjecture, showing that in all dimension, the mixing time with + boundary condition is O(L^2 log L^c) for some appropriate c in any dimension. This generalizes a recent result by Caputo, Martinelli, Simenhaus and F.L. Toninelli (who proved it in two and three dimension). We will present the key results obtained by Caputo et al. for the three dimensional model and explain how they can be used to obtained the result for dimension larger than three.
I consider the evolution by mean curvature in a heterogeneous medium modeled by a periodic forcing term. I show existence of standing and travelling wave solutions, discuss the homogenization limit when the length of the periodicity cell tends to zero, and prove some properties of the effective speed.
Motivated by recent developments on solvable directed polymer models, we define a 'multi-layer' extension of the stochastic heat equation involving non-intersecting Brownian motions. This is joint work with Jon Warren.
The Allen-Cahn equation is a reaction-diffusion equation that arises as a model for phase separation processes. In the `sharp interface limit' solutions converge to the evolution of phase boundaries by mean curvature flow. We are interested in the effect of stochastic perturbations to the Allen-Cahn equation, in particular perturbations in form of a stochastic flow term. We show that solutions converge in the sharp interface limit to an evolution of phase indicator functions and discuss possibilities to identify the limiting motion as a stochastically perturbed mean curvature flow. This is joint work with Hendrik Weber, Warwick.
We consider a model for the propagation of a driven interface through a random field of obstacles. The evolution equation, commonly referred to as the Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with constant driving term $F \ge 0$ and random nonlinearity to model the influence of the obstacle field. We investigate transitions in the long time behaviour of the interface as a function of $F$. For a class of nonlinearities we show that one has pinning for small positive $F$ and depinning (with strictly positive speed) for large $F$. This is joint work with Patrick Dondl and Nicolas Dirr.
Lipid molecules consist of a hydrophilic head and a hydrophobic tail and spontaneously form bilayers when put into water in order to shield the tails. Such biomembranes are the basic component of cell membranes, and they reveal a fluid-like behaviour within the layers but also a solid-like behaviour as they resist strechting and bending. We will briefly review the established models and point out actual developments and problems which, often, involve additional phenomena such as proteins attached to the membrane. Exemplary, we will also look at phase separation of the lipid molecules and the impact on the membrane shape.
I will discuss recent results about the low-temperature Glauber dynamics of the 2D Ising model. The typical problem is to study equilibration in a square domain with boundary conditions such as to impose the presence of a single open interface between + and - spins. The expected equilibration time is then of the order L^2 times log L, with L the side of the square domain ("mean curvature motion heuristics"). In the Solid-on-Solid approximation, we prove this up to logarithmic corrections. For the full Ising model, the problem is considerably harder and via a recursive method we get a rougher upper bound of order L^(log L), which however improves considerably on previously known bounds. This is joint work with P. Caputo, E. Lubetzky, A. Sly, F. Martinelli.
We study the invariant measure of Allen-Cahn equation perturbed by an additive space-time white noise. This measure is absolutely continuous with respect to a Gaussian measure with an explicitly known density. We study an diagonal limit of vanishing temperature with a growing system size. We are interested in the asymptotic for the probability of emergence of kinks in this limit. The vanishing temperature makes kinks less probable, whereas the growing system size makes it easier for kinks to form. We obtain large deviation type bounds that correspond to this competition. We use symmetries of this measure, to make the problem amenable to large deviation techniques. This is a joint work with Felix Otto and Maria Westdickenberg.
The Robsinson-Schensted-Knuth algorithm maps a weight matrix onto a pair of semi-standard Young tableaux. This algorithm can be encoded in terms of the algebra of operators generated by (max; +). We will present a tropical version of this algorithm, encoded by the algebra of operators gen- erated by (+;x) and we will show how this tropical version can be used to identify the distribution of the partition function of a random directed polymer with inverse gamma disorder. This is joint work with I.Corwin, N. O'Connell, T. Seppalainen.