den WThF Frank Hollander

Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures

Abstract: We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following:(1) For all ν and μ, νS(t) is Gibbs for small t.(2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0.(3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t.(4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t.The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.

Intermittency in a catalytic random medium

In this paper, we study intermittency for the parabolic Anderson equation ∂u/∂t=κΔu+ξu, where u:ℤd×[0, ∞)→ℝ, κ is the diffusion constant, Δ is the discrete Laplacian and ξ:ℤd×[0, ∞)→ℝ is a space-time random medium. We focus on the case where ξ is γ times the random medium that is obtained by running independent simple random walks with diffusion constant ρ starting from a Poisson random field with intensity ν. Throughout the paper, we assume that κ, γ, ρ, ν∈(0, ∞). The solution of the equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of u, and show that they display an interesting dependence on the dimension d and on the parameters κ, γ, ρ, ν, with qualitatively different intermittency behavior in d=1, 2, in d=3 and in d≥4. Special attention is given to the asymptotics of these Lyapunov exponents for κ↓0 and κ→∞.

Phase transitions for the long-time behavior of interacting diffusions.

Let ({Xi(t)}i∈Zd)t≥0 be the system of interacting diffusions on [0,∞) defined by the following collection of coupled stochastic differential equations: dXi(t) = ∑ j∈Zd a(i, j)[Xj(t)−Xi(t)] dt+ √ bXi(t) dWi(t), i ∈ Z, t ≥ 0. Here, a(·, ·) is an irreducible random walk transition kernel on Z ×Z, b ∈ (0,∞) is a diffusion parameter, and ({Wi(t)}i∈Zd)t≥0 is a collection of independent standard Brownian motions on R. The initial condition is chosen such that {Xi(0)}i∈Zd is a shift-invariant and shift-ergodic random field on [0,∞) with mean Θ ∈ (0,∞) (the evolution preserves the mean). We show that the long-time behaviour of this system is the result of a delicate interplay between a(·, ·) and b, in contrast to systems where the diffusion function is subquadratic. In particular, let â(i, j) = 12 [a(i, j) + a(j, i)], i, j ∈ Z, denote the symmetrised transition kernel. We show that: (A) If â(·, ·) is recurrent, then for any b > 0 the system locally dies out. (B) If â(·, ·) is transient, then there exist b∗ ≥ b2 > 0 such that: (B1) The system converges to an equilibrium νΘ (with mean Θ) if 0 b∗. (B3) νΘ has a finite 2-nd moment if and only if 0 b2. The equilibrium νΘ is shown to be associated and mixing for all 0 b2. We further conjecture that the system locally dies out at b = b∗. For the case where a(·, ·) is symmetric and transient we further show that: (C) There exists a sequence b2 ≥ b3 ≥ b4 ≥ · · · > 0 such that: (C1) νΘ has a finite m-th moment if and only if 0 bm. (C3) b2 ≤ (m− 1)bm < 2. (C4) limm→∞(m− 1)bm = c = supm≥2(m− 1)bm. ∗Mathematisches Institut, Universitat Erlangen-Nurnberg, Bismarckstrasse 1 1 2 , D-91504 Erlangen, Germany, greven@mi.uni-erlangen.de †Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands, denholla@math.leidenuniv.nl ‡EURANDOM, P.O.Box 513, 5600 MB Eindhoven, The Netherlands

Mixing properties of the generalized T,T-1-process

Consider a general random walk on ℤd together with an i.i.d. random coloring of ℤd. TheT, T-1-process is the one where time is indexed by ℤ, and at each unit of time we see the step taken by the walk together with the color of the newly arrived at location. S. Kalikow proved that ifd = 1 and the random walk is simple, then this process is not Bernoulli. We generalize his result by proving that it is not Bernoulli ind = 2, Bernoulli but not Weak Bernoulli ind = 3 and 4, and Weak Bernoulli ind ≥ 5. These properties are related to the intersection behavior of the past and the future of simple random walk. We obtain similar results for general random walks on ℤd, leading to an almost complete classification. For example, ind = 1, if a step of sizex has probability proportional to l/|x|α(x ⊋ 0), then theT, T-1-process is not Bernoulli when α ≥2, Bernoulli but not Weak Bernoulli when 3/2 ≤α < 2, and Weak Bernoulli when 1 < α < 3/2.

Variational characterization of the critical curve for pinning of random polymers

In this paper we look at the pinning of a directed polymer by a one-dimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process (Birkner, Greven and den Hollander [6]), we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases. These variational formulas are used to obtain a necessary and sufficient criterion, stated in terms of relative entropies, for the two critical curves to be different at a given inverse temperature, a property referred to as relevance of the disorder. This criterion in turn is used to show that the regimes of relevant and irrelevant disorder are separated by a unique inverse critical temperature. Subsequently, upper and lower bounds are derived for the inverse critical temperature, from which sufficient conditions under which it is strictly positive, respectively, finite are obtained. The former condition is believed to be necessary as well, a problem that we will address in a forthcoming paper. Random pinning has been studied extensively in the literature. The present paper opens up a window with a variational view. Our variational formulas for the quenched and the annealed critical curve are new and provide valuable insight into the nature of the phase transition. Our results on the inverse critical temperature drawn from these variational formulas are not new, but they offer an alternative approach that is flexible enough to be extended to other models of random polymers with disorder.

Scaling of a random walk on a supercritical contact process

A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads to the existence of regeneration times at which the random walk forgets its past. The latter are used to prove a functional central limit theorem and a large deviation principle. The qualitative dependence of the speed, the volatility and the rate function on the infection parameter is investigated, and some open problems are mentioned.

Law of large numbers for non-elliptic random walks in dynamic random environments

We prove a law of large numbers for a class of Zd-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called conditional cone-mixing and that the random walk tends to stay inside wide enough space–time cones. The proof is based on a generalization of a regeneration scheme developed by Comets and Zeitouni (2004) [5] for static random environments and adapted by Avena et al. (2011) [2] to dynamic random environments. A number of one-dimensional examples are given. In some cases, the sign of the speed can be determined.

Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k be an integer For s let Ds IR be the set that is constructed iteratively as follows Take a regular open k gon with sides of unit length attach regular open k gons with sides of length s to the middles of the edges and so on At each stage of the iteration the k gons that are added are a factor s smaller than the previous generation and are attached to the outer edges of the family grown so far The set Ds is de ned to be the interior of the closure of the union of all the k gons It is easy to see that there must exist some sk such that no k gons overlap if and only if s sk We derive an explicit formula for sk The set Ds is open bounded connected and has a fractal polygonal boundary Let EDs t denote the heat content of Ds at time t when Ds initially has temperature and Ds is kept at temperature We derive the complete short time expansion of EDs t up to terms that are exponentially small in t It turns out that there are three regimes corresponding to s k s k respectively k s sk For s k the expansion has the formEDs t ps log t t ds Ast Bt O e rs t where ps is a log s periodic function ds log k log s is a similarity dimension As and B are constants related to the edges respectively vertices of Ds and rs is an error exponent For s k the t term carries an additional log t AMS subject classi cations K A J

Scaling for a Random Polymer

AbstractLetQnβ be the law of then-step random walk on ℤd obtained by weighting simple random walk with a factore−β for every self-intersection (Domb-Joyce model of “soft polymers”). It was proved by Greven and den Hollander (1993) that ind=1 and for every β∈(0, ∞) there exist θ*(β)∈(0,1) and such that under the lawQnβ asn→∞: