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In this paper we study metastable behaviour at low temperature of Glauber spin-flip dynamics on random graphs. We fix a large number of vertices and randomly allocate edges according to the Configuration Model with a prescribed degree distribution. Each vertex carries a spin that can point either up or down. Each spin interacts with a positive magnetic field, while spins at vertices that are connected by edges also interact with each other via a ferromagnetic pair potential. We start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In order to achieve this transition, the system needs to create a sufficiently large droplet of up-spins, called critical droplet, which triggers the crossover. In the limit as the temperature tends to zero, and subject to a certain \emph{key hypothesis} implying metastable behaviour, the average crossover time follows the classical \emph{Arrhenius law}, with an exponent and a prefactor that are controlled by the \emph{energy} and the \emph{entropy} of the critical droplet. The crossover time divided by its average is exponentially distributed. We study the scaling behaviour of the exponent as the number of vertices tends to infinity, deriving upper and lower bounds. We also identify a regime for the magnetic field and the pair potential in which the key hypothesis is satisfied. The critical droplets, representing the saddle points for the crossover, have a size that is of the order of the number of vertices. This is because the random graphs generated by the Configuration Model are expander graphs.

Let $T^m$ be the $m$-dimensional unit torus, $m \in N$. The torsional rigidity of an open set $\Omega \subset T^m$ is the integral with respect to Lebesgue measure over all starting points $x \in \Omega$ of the expected lifetime in $\Omega$ of a Brownian motion starting at $x$. In this paper we consider $\Omega = T^m \backslash \beta[0,t]$, the complement of the path $\beta[0,t]$ of an independent Brownian motion up to time $t$. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as $t \to \infty$. For $m=2$ the main contribution comes from the components in $T^2 \backslash \beta [0,t]$ whose inradius is comparable to the largest inradius, while for $m=3$ most of $T^3 \backslash \beta [0,t]$ contributes. A similar result holds for $m \geq 4$ after the Brownian path is replaced by a shrinking Wiener sausage $W_{r(t)}[0,t]$ of radius $r(t)=o(t^{-1/(m-2)})$, provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of $\beta[0,t]$ in $R^3$ and $W_1[0,t]$ in $R^m$, $m \geq 4$, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on $T^m$, which has received a lot of attention in the literature in past years., Comment: 26 pages, 1 figure

We prove 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 uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In 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 under the annealed law.The qualitative dependence of the asymptotic speed and the volatility on the infection parameter is investigated, and some open problems are mentioned.RésuméNous prouvons une loi forte des grands nombres pour une marche aléatoire dans un milieu aléatoire dynamique donné par un processus de contact sur-critique unidimensionnel en équilibre. La preuve utilise un argument de couplage basé sur l’observation que la marche est finalement confinée dans l’union de cônes spatio-temporels inclus dans les clusters d’infection générés par des infections individuelles. Si les taux locaux de saut de la marche sont plus petits que la vitesse de propagation de l’infection, la marche est finalement confinée dans un seul cône, ce qui entraîne l’existence de temps de régénération en lesquels la marche oublie son passé. Ces temps de régénération sont utilisés pour prouver un théorème central limite fonctionnel et un principe de grandes déviations sous la loi "annealed."La dépendance de la vitesse et de la variance asymptotiques par rapport au paramètre d’infection est étudiée, et quelques problèmes ouverts sont mentionnés.

In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \in (0,1)$. The copolymer in the emulsion has an energy that is minus $\alpha$ times the number of $AA$-matches minus $\beta$ times the number of $BB$-matches, where without loss of generality the interaction parameters can be taken from the cone $\{(\alpha,\beta)\in\R^2\colon \alpha\geq |\beta|\}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \cite{dHW06}, it was found that in the supercritical percolation regime $p \geq p_c$, with $p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase., Comment: 43 pages and 10 figures

In this paper we consider a two-dimensional copolymer consisting of a random concatenation of hydrophobic and hydrophilic monomers near a linear interface separating oil and water acting as solvents. The configurations of the copolymer are directed paths that can move above and below the interface. The interaction Hamiltonian, which rewards matches and penalizes mismatches of the monomers and the solvents, depends on two parameters: the interaction strength $\beta\geq 0$ and the interaction bias $h \geq 0$. The quenched excess free energy per monomer $(\beta,h) \mapsto g^\mathrm{que} (\beta,h)$ has a phase transition along a quenched critical curve $\beta \mapsto h^\mathrm{que}_c(\beta)$ separating a localized phase, where the copolymer stays close to the interface, from a delocalized phase, where the copolymer wanders away from the interface. We derive a variational expression for $g^\mathrm{que}(\beta,h)$ by applying the quenched large deviation principle for the empirical process of words cut out from a random letter sequence according to a random renewal process. We compare this variational expression with its annealed analogue, describing the annealed excess free energy $(\beta,h) \mapsto g^\mathrm{ann}(\beta,h)$, which has a phase transition along an annealed critical curve $\beta \mapsto h^\mathrm{ann}_c(\beta)$. Our results extend to a general class of disorder distributions and directed paths. We show that $g^\mathrm{que}(\beta,h)0$ when $\alpha>1$. This gap vanished when $\alpha=1$., Comment: 39 pages, 8 figures

In this paper we study the following system of reaction-diffusion equations: ∂�/∂t =∆ � − V� + λδ0 ,� (0 ,x ) ≡ 0, ∂V/∂t = −�V, V (0 ,x ) ≡ 1. Here � (t, x )a ndV (t, x) are functions of time t ∈ (0, ∞ )a nd spacex ∈ R d . This system describes a continuum version of a model in which particles are injected at the origin at rate λ, perform independent simple symmetric random walks on Z d , and are annihilated at rate 1 by traps located at the sites of Z d in such a way that the trap disappears with the particle. This lattice model was studied by a number of authors, who obtained the asymptotic size and shape of the front separating the zone of particles from the zone of traps as well as the asymptotic particle density profile to leading order, in the limit of large time. The continuum model has similar behavior but allows for a more detailed study. As t increases, the particle density � (t, · ) inflates and the trap density V (t, · ) deflates on a growing ball with radius R∗(t) centered at the origin. We derive the sharp asymptotics of the front position R∗(t), identify the shape of V (t, · )n ear the surface of the ball, and obtain the limiting profile of � (t, · ) inside the ball after appropriate scaling. We also identify the analogues of the total number and the age distribution of particles that are alive. It turns out that the cases d ≥ 3, d =2 , andd = 1 exhibit different behavior.

In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\xi u$, where $u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete Laplacian and $\xi:\mathbb{Z}^d\times[0,\infty)\to\mathbb {R}$ is a space-time random medium. We focus on the case where $\xi$ is $\gamma$ times the random medium that is obtained by running independent simple random walks with diffusion constant $\rho$ starting from a Poisson random field with intensity $\nu$. Throughout the paper, we assume that $\kappa,\gamma,\rho,\nu\in (0,\infty)$. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. 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 $\kappa,\gamma,\rho,\nu$, with qualitatively different intermittency behavior in $d=1,2$, in $d=3$ and in $d\geq4$. Special attention is given to the asymptotics of these Lyapunov exponents for $\kappa\downarrow0$ and $\kappa \to\infty$., Comment: Published at http://dx.doi.org/10.1214/009117906000000467 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In this paper we consider an arbitrary irreducible random walk on ℤd, d ≥ 1, with i.i.d. increments, together with an arbitrary i.i.d. random scenery. Walk and scenery are assumed to be independent. Random walk in random scenery (RWRS) is the random process where time is indexed by ℤ, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. Bad configurations for RWRS are the discontinuity points of the conditional probability distribution for the configuration at the origin of time given the configuration at all other times. We show that the set of bad configurations is non-empty. We give a complete description of this set and compute its probability under the random scenery measure. Depending on the type of random walk, this probability may be zero or positive. For simple symmetric random walk we get three different types of behavior depending on whether d = 1, 2, d = 3, 4 or d ≥ 5. Our classification is actually valid for a class of subshifts having a certain determinative property, which we call specifiable, of which RWRS is an example. We also consider bad configurations w.r.t. a finite time interval (replacing the origin) and obtain an almost complete generalization of our results. Remarkably, this extension turns out to be somewhat delicate.

In this paper we consider a standard Brownian motion in d, starting at 0 and observed until time t. The Brownian motion takes place in the presence of a Poisson random field of traps, whose centers have intensity t and whose shapes are drawn randomly and independently according to a probability distribution , on the set of closed subsets of d, subject to appropriate conditions. The Brownian motion is killed as soon as it hits one of the traps. With the help of a large deviation technique developed in an earlier paper, we find the tail of the probability St that the Brownian motion survives up to time t when where c (0,) is a parameter. This choice of intensity corresponds to a critical scaling. We give a detailed analysis of the rate constant in the tail of St as a function of c, including its limiting behaviour as c or c0. For d3, we find that there are two regimes, depending on the choice of . In one of the regimes there is a collapse transition at a critical value c* (0,), where the optimal survival strategy changes from being diffusive to being subdiffusive. At c*, the slope of the rate constant is discontinuous. For d=2, there is again a collapse transition, but the rate constant is independent of and its slope at c=c* is continuous.

For a>0 , let W a 1 (t) and W a 2 (t) be the a -neighbourhoods of two independent standard Brownian motions in R d starting at 0 and observed until time t . We prove that, for d=3 and c>0 , lim t¿8 1 t (d-2)/d logP(|W a 1 (ct)nW a 2 (ct)|=t)=-I ¿ a d (c) and derive a variational representation for the rate constant I ¿ a d (c) . Here, ¿ a is the Newtonian capacity of the ball with radius a . We show that the optimal strategy to realise the above large deviation is for W a 1 (ct) and W a 2 (ct) to "form a Swiss cheese": the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale t 1/d according to a certain optimal profile. We study in detail the function c¿I ¿ a d (c) . It turns out that I ¿ a d (c)=T d (¿ a c)/¿ a , where T d has the following properties: (1) For d=3 : T d (u)

This paper is a tutorial introduction to some of the mathematics behind metastable behavior of interacting particle systems. The main focus is on the formation of so-called critical droplets, in particular, on their geometry and the time of their appearance. Special attention is given to Ising spins subject to a Glauber spin-flip dynamics and lattice particles subject to a Kawasaki hopping dynamics. The latter is one of the hardest models that can be treated to date and therefore is representative for the current state of development of this research area.

The goal of this paper is to describe metastability and nucleation for a local version of the three-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Let $\Lambda\subseteq{\mathbb Z}^3$ be a large finite box. Particles perform simple exclusion on $\Lambda$, but when they occupy neighboring sites they feel a binding energy $-U 0$ is an activity parameter. Thus, the boundary of $\Lambda$ plays the role of an infinite gas reservoir with density $\rho$. We consider the regime where $\Delta\in (U,3U)$ and the initial configuration is such that $\Lambda$ is empty. For large $\beta$, the system wants to fill $\Lambda$ but is slow in doing so. We investigate how the transition from empty to full takes place under the dynamics. In particular, we identify the size and shape of the critical droplet\/ and the time of its creation in the limit as $\beta\to\infty$.

In this paper we study the parabolic Anderson equation \partial u(x,t)/\partial t=\kappa\Delta u(x,t)+\xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where the u-field and the \xi-field are \R-valued, \kappa \in [0,\infty) is the diffusion constant, and $\Delta$ is the discrete Laplacian. The initial condition u(x,0)=u_0(x), x\in\Z^d, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d\kappa, split into two at rate \xi\vee 0, and die at rate (-\xi)\vee 0. Our goal is to prove a number of basic properties of the solution u under assumptions on $\xi$ that are as weak as possible. Throughout the paper we assume that $\xi$ is stationary and ergodic under translations in space and time, is not constant and satisfies \E(|\xi(0,0)|), Comment: 50 pages. The comments of the referee are incorporated into the paper. A missing counting estimate was added in the proofs of Lemma 3.6 and Lemma 4.7

In this paper we consider $\beta[0; s]$, Brownian motion of time length $s > 0$, in $m$-dimensional Euclidean space $\mathbb R^m$ and on the $m$-dimensional torus $\mathbb T^m$. We compute the expectation of (i) the heat content at time $t$ of $\mathbb R^m\setminus \beta[0; s]$ for fixed $s$ and $m = 2,3$ in the limit $t \downarrow 0$, when $\beta[0; s]$ is kept at temperature 1 for all $t > 0$ and $\mathbb R^m\setminus \beta[0; s]$ has initial temperature 0, and (ii) the inradius of $\mathbb R^m\setminus \beta[0; s]$ for $m = 2,3,\cdots$ in the limit $s \rightarrow \infty$., Comment: 13 pages

For a>0,let W^a(t) be the a-neighbourhood of standard Brownian motion in R^d starting at 0 and observed until time t.It is well-known that E|W^a(t)|~kappa_a t (t->infty) for d >= 3,with kappa_a the Newtonian capacity of the ball with radius a. We prove that lim_{t->infty} 1/t^{(d-2)/d}log P(|W^a(t)|infty.This is markedly different from the optimal strategy for large deviations |W^a(t)|infty.We give a detailed analysis of the rate function I^{kappa_a},in particular,its behaviour near the boundary points of (0,kappa_a).It turns out that I^{kappa_a} has an infinite slope at kappa_a and,remarkably,for d>=5 is nonanalytic at some critical point in (0,kappa_a),above which it follows a pure power law.This crossover is associated with a collapse transition in the optimal strategy.We also derive the analogous moderate deviation result for d=2.In this case E|W^a(t)|~2pi t/log t (t->infty),and we prove that lim_{t->infty} 1/log t log P(|W^a(t), Comment: 52 pages, published version

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

We prove a law of large numbers for a class of $\Z^d$-valued random walks in dynamic random environments, including non-elliptic examples. We assume for the random environment a mixing property called \emph{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 for static random environments and adapted by Avena, den Hollander and Redig to dynamic random environments. A number of one-dimensional examples are given. In some cases, the sign of the speed can be determined., Comment: 36 pages, 4 figures

We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in Feng and Kurtz [11], we show that the trajectory under the spin-flip dynamics of the empirical measure of the spins in a large block in Z^d satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain non-linear generator, playing a role analogous to the moment-generating function in the Gartner-Ellis theorem of large deviation theory when this is applied to finite-dimensional Markov processes. This rate function is used to define the notion of "bad empirical measures", which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical mea- sures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a "nature-versus-nurture" transition., Key words and phrases: Stochastic spin-flip dynamics, Gibbs-non-Gibbs transition, empirical measure, non-linear generator, Nisio control semigroup, large deviation principle, bad configurations, bad empirical measures, nature versus nurture.

The present paper is a continuation of \cite{dHP07b}. The object of interest is a two-dimensional model of a directed copolymer, consisting of a random concatenation of hydrophobic and hydrophilic monomers, immersed in an emulsion, consisting of large blocks of oil and water arranged in a percolation-type fashion. The copolymer interacts with the emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water. The model has two regimes, supercritical and subcritical, depending on whether the oil blocks percolate or not. In \cite{dHP07b} we focussed on the supercritical regime and obtained a complete description of the phase diagram, which consists of two phases separated by a single critical curve. In the present paper we focus on the subcritical regime and show that the phase diagram consists of four phases separated by three critical curves meeting in two tricritical points., Comment: 31 pages

We prove an invariance principle for the random process (X n ) n≧1 given by $$\left\{ \begin{gathered} X_1 = x \in \mathbb{R} \hfill \\ X_{n + 1} = \max (X_{n,} \alpha _n X_n + Y_n ),{\text{ }}n \geqq 1 \hfill \\ \end{gathered} \right.$$ where (Y n ) n≧1 are i.i.d. random variables and (α n ) n≧ are nonrandom numbers tending upward to 1 (both in ℝ). This process interpolates between maxima (α n ≡0) and sums (α n ≡1). Depending on the distribution ofY n and on the rate at which α n →1 the scaling behaviour exhibits different regimes. Our techniques are flexible and are applicable to more general types of iterative schemes.

This paper studies countable systems of linearly and hierarchically interacting diffusions taking values in the positive quadrant. These systems arise in population dynamics for two types of individuals migrating between and interacting within colonies. Their large-scale space-time behavior can be studied by means of a renormalization program. This program, which has been carried out successfully in a number of other cases (mostly one-dimensional), is based on the construction and the analysis of a nonlinear renormalization transformation, acting on the diffusion function for the components of the system and connecting the evolution of successive block averages on successive time scales. We identify a general class of diffusion functions on the positive quadrant for which this renormalization transformation is well-defined and, subject to a conjecture on its boundary behavior, can be iterated. Within certain subclasses, we identify the fixed points for the transformation and investigate their domains of attraction. These domains of attraction constitute the universality classes of the system under space-time scaling., Comment: 48 pages, revised version, to appear in Ann. Inst. H. Poincare (B) Probab. Statist

Systems of hierarchically interacting diffusions allow for a rigorous renormalization analysis. By bringing into play the powerful machinery of stochastic analysis, it is possible to obtain a complete classification of the large space-time behavior of these systems into universality classes. The present paper outlines a general renormalization program that is being pursued since ten years and describes four examples where this program has been successfully carried through. The systems under consideration model the evolution of multi-type populations subject to migration and resampling.

In this paper we study the metastable behavior of the lattice gas in two and three dimensions subject to Kawasaki dynamics in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. Our approach combines geometric and potential theoretic arguments. In two dimensions, we identify the full geometry of the set of critical droplets for the nucleation, compute the average nucleation time up to a multiplicative factor that tends to one, show that the nucleation time divided by its average converges to an exponential random variable, express the proportionality constant for the average nucleation time in terms of certain capacities associated with simple random walk, and compute the asymptotic behavior of this constant as the system size tends to infinity. In three dimensions, we obtain similar results but with less control over the geometry and the constant. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet and the sharp asymptotics for the average nucleation time are highly sensitive to this motion.

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

We consider a quenched-disordered heteropolymer, consisting of hydrophobic and hydrophylic monomers, in the vicinity of an oil-water interface. The heteropolymer is modeled by a directed simple random walk$(i, S_i)_{i\epsilon\mathbb{N}}$ on $\mathbb{N} \times \mathbb{Z}$ with an interaction given by the Hamiltonians $H_n^{\omega}(S) = \lambda \Sigma_{i=1}^n(\omega_i + h)\text{sign}(S_i)(n \epsilon \mathbb{N})$. Here, $\lambda$ and h are parameters and $(\omega_i)_{i\epsilon\mathbb{N}}$ are i.i.d. $\pm1$-valued random variables. The sign $(S_i) = \pm1$ indicates whether the ith monomer is above or below the interface, the $\omega_i = \pm1$ indicates whether the ith monomer is hydrophobic or hydrophylic. It was shown by Bolthausen and den Hollander that the free energy exhibits a localization-delocalization phase transition at a curve in the $(\lambda, h)$-plane. ¶ In the present paper we show that the free-energy localization concept is equivalent to pathwise localization. In particular, we prove that free-energy localization implies exponential tightness of the polymer excursions away from the interface, strictly positive density of intersections with the interface and convergence of ergodic averages along the polymer. We include an argument due to G. Giacomin, showing that free-energy delocalization implies that there is pathwise delocalization in a certain weak sense.

Consider the directed process $(i, S_i)$ where the second component is simple random walk on $\mathbb{Z} (S_0 = 0)$. Define a transformed path measure by weighting each $n$-step path with a factor $\exp [\lambda \sum_{1 \leq i \leq n}(\omega_i + h)\sign (S_i)]$. Here, $(\omega_i)_{i \geq 1}$ is an i.i.d. sequence of random variables taking values $\pm 1$ with probability 1/2 (acting as a random medium) , while $\lambda \in [0, \infty)$ and $h \in [0, 1)$ are parameters. The weight factor has a tendency to pull the path towards the horizontal, because it favors the combinations $S_i > 0, \omega_i = +1$ and $S_i < 0, \omega_i = -1$. The transformed path measure describes a heteropolymer, consisting of hydrophylic and hydrophobic monomers, near an oil-water interface. ¶ We study the free energy of this model as $n \to \infty$ and show that there is a critical curve $\lambda \to h_c (\lambda)$ where a phase transition occurs between localized and delocalized behavior (in the vertical direction). We derive several properties of this curve, in particular, its behavior for $\lambda \downarrow 0$. To obtain this behavior, we prove that as $\lambda, h \downarrow 0$ the free energy scales to its Brownian motion analogue.

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.

Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤd, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant, and ξ = {ξ(x): x∈ℤ d } is an i.i.d.random field taking values in ℝ. Gartner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e s /θ] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result is that, for fixed x, y∈ℤ d and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w ρ∥−2 l2Σz ∈ℤd w ρ(x+z)w ρ(y+z). In this expression, ρ = θ/κ while w ρ:ℤd→ℝ+ is given by w ρ = (v ρ)⊗ d with v ρ: ℤ→ℝ+ the unique centered ground state (i.e., the solution in l2(ℤ) with minimal l 2-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞). empty It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u(x, t) and u(y, t) converges to δ x, y (resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure.

The Edwards model in one dimension is a transformed path measure for standard Brownian motion discouraging self-intersections. We prove a central limit theorem for the endpoint of the path, extending a law of large numbers proved byWestwater (1984). The scaled variance is characterized in terms of the largest eigenvalue of a one-parameter family of dierential operators, introduced and analyzed in van der Hofstad and den Hollander (1995). Interestingly, the scaled variance turns out to be independent of the strength of self-repellence and to be strictly smaller than one (the value for free Brownian motion).

A non-intrusive method is presented to determine the residence times of a single particle in a facility composed of four interconnected fluidized beds (IFB) with glass beads fluidized by air above minimum fluidization at ambient temperature. It uses a glass particle labelled with radioactive 24Na or 192Ir whose γ-ray emissions are monitored by two external NaI(Tl) scintillation detectors. It is shown that, within the batch particles' size range, the mean residence time of the radiotracer particle is practically independent of the radiotracer particle size and the beds can be considered as practically ideally mixed vessels. It is further demonstrated that the method can adequately measure the residence time distribution of the particles independent of the processes occurring within the system such as the generation of static electricity during fluidization. The presence of static electricity was found to substantially affect the mean residence time of the solids. By relating the bed mass and the mean residence time of the radiotracer particle, the solids circulation rate within the IFB can be obtained.

LetQ n β 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 lawQ n β asn→∞: $$\begin{array}{l} (i) \theta ^* (\beta ) is the \lim it empirical speed of the random walk; \\ (ii) \mu _\beta ^* is the limit empirical distribution of the local times. \\ \end{array}$$ A representation was given forθ *(β) andµ β β in terms of a largest eigenvalue problem for a certain family of ℕ x ℕ matrices. In the present paper we use this representation to prove the following scaling result as β⇂0: $$\begin{array}{l} (i) \beta ^{ - {\textstyle{1 \over 3}}} \theta ^* (\beta ) \to b^* ; \\ (ii) \beta ^{ - {\textstyle{1 \over 3}}} \mu _\beta ^* \left( {\left\lceil { \cdot \beta ^{ - {\textstyle{1 \over 3}}} } \right\rceil } \right) \to ^{L^1 } \eta ^* ( \cdot ) . \\ \end{array}$$ The limitsb *∈(0, ∞) and are identified in terms of a Sturm-Liouville problem, which turns out to have several interesting properties. The techniques that are used in the proof are functional analytic and revolve around the notion of epi-convergence of functionals onL 2(ℝ+). Our scaling result shows that the speed of soft polymers ind=1 is not right-differentiable at β=0, which precludes expansion techniques that have been used successfully ind≧5 (Hara and Slade (1992a, b)). In simulations the scaling limit is seen for β≦10−2.

A simple proof is pointed out for the asymptotic exponential decay of then-step survival probability of a random walk on a finite lattice with traps in the limit asn → ∞. Some bounds are mentioned, which are valid for finiten and for symmetric random walks.

In this paper we consider a two-dimensional lattice gas under Kawasaki dynamics, i.e., particles hop around randomly subject to hard-core repulsion and nearest-neighbor attraction. We show that, at fixed temperature and in the limit as the particle density tends to zero, such a gas evolves in a way that is close to an ideal gas, where particles have no interaction. In particular, we prove three theorems showing that particle trajectories are non-superdiffusive and have a diffusive spread-out property. We also consider the situation where the temperature and the particle density tend to zero simultaneously and focus on three regimes corresponding to the stable, the metastable and the unstable gas, respectively.Our results are formulated in the more general context of systems of “Quasi-Random Walks”, of which we show that the low-density lattice gas under Kawasaki dynamics is an example. We are able to deal with a large class of initial conditions having no anomalous concentration of particles and with time horizons that are much larger than the typical particle collision time. The results will be used in two forthcoming papers, dealing with metastable behavior of the two-dimensional lattice gas in large volumes at low temperature and low density.

We study tail $\sigma$-fields and loss of memory associated with sums of stationary integer-valued random variables. An application concerns convergence in distribution of interarrival times in zero-one sequences.