The scaling limits of the non critical strip wetting model

The strip wetting model is defined by giving a (continuous space) one dimensionnal random walk $S$ a reward \gb each time it hits the strip $\R^{+} \times [0,a]$ (where $a$ is a positive parameter), which plays the role of a defect line. We show that this model exhibits a phase transition between a delocalized regime (\gb \gb_{c}^{a}), where the critical point \gb_{c}^{a} > 0 depends on $S$ and on $a$. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. Our approach is based on Markov renewal theory.Comment: 32 pages. To appear in "Stochastic Processes and their Applications

Hierarchical pinning model: low disorder relevance in the b=sb=s case

We consider a hierarchical pinning model introduced by B.Derrida, V.Hakim and J.Vannimenus which undergoes a localization/delocalization phase transition. This model depends on two parameters $b$ and $s$. We show that in the particular case where $b=s$, the disorder is weakly relevant, in the sense that at any given temperature, the quenched and the annealed critical points coincide. This is in contrast with the case where $b \neq s$

Convergence to equilibrium for a directed (1+d)-dimensional polymer

We consider a flip dynamics for directed (1+d)-dimensional lattice paths with length L. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case d=1. We prove that the mixing time of the associated Markov chain scales like L^2\log L up to a d-dependent multiplicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scale L^2 for every fixed d, which we achieve by a suitable induction over the dimension together with an estimate for adjacent transpositions. The lower bound is obtained with a version of Wilson's argument for the one-dimensional case.Comment: 22 page

Finite size scaling for homogeneous pinning models

Pinning models are built from discrete renewal sequences by rewarding (or penalizing) the trajectories according to their number of renewal epochs up to time $N$, and $N$ is then sent to infinity. They are statistical mechanics models to which a lot of attention has been paid both because they are very relevant for applications and because of their {\sl exactly solvable character}, while displaying a non-trivial phase transition (in fact, a localization transition). The order of the transition depends on the tail of the inter-arrival law of the underlying renewal and the transition is continuous when such a tail is sufficiently heavy: this is the case on which we will focus. The main purpose of this work is to give a mathematical treatment of the {\sl finite size scaling limit} of pinning models, namely studying the limit (in law) of the process close to criticality when the system size is proportional to the correlation length

Scaling limits of a heavy tailed Markov renewal process

In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the \ga-stable regenerative set. We then apply these results to the strip wetting model which is a random walk $S$ constrained above a wall and rewarded or penalized when it hits the strip $[0,\infty) \times [0,a]$ where $a$ is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality

Residence time of symmetric random walkers in a strip with large reflective obstacles

We study the effect of a large obstacle on the so called residence time, i.e., the time that a particle performing a symmetric random walk in a rectangular (2D) domain needs to cross the strip. We observe a complex behavior, that is we find out that the residence time does not depend monotonically on the geometric properties of the obstacle, such as its width, length, and position. In some cases, due to the presence of the obstacle, the mean residence time is shorter with respect to the one measured for the obstacle--free strip. We explain the residence time behavior by developing a 1D analog of the 2D model where the role of the obstacle is played by two defect sites having a smaller probability to be crossed with respect to all the other regular sites. The 1D and 2D models behave similarly, but in the 1D case we are able to compute exactly the residence time finding a perfect match with the Monte Carlo simulations

A comparison between different cycle decompositions for Metropolis dynamics

In the last decades the problem of metastability has been attacked on rigorous grounds via many different approaches and techniques which are briefly reviewed in this paper. It is then useful to understand connections between different point of views. In view of this we consider irreducible, aperiodic and reversible Markov chains with exponentially small transition probabilities in the framework of Metropolis dynamics. We compare two different cycle decompositions and prove their equivalence

Metastability for General Dynamics with Rare Transitions: Escape Time and Critical Configurations

Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with Statistical Mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. In this paper, we provide a similar description in the general rare transitions setup. Beside their theoretical content, we believe that our results are a useful tool to approach metastability for non--Metropolis systems such as Probabilistic Cellular Automata

Change of scale strategy for the microstructural modelling of polymeric rohacell foams

International audienceIn this paper a numerical model dedicated to the simulation of the mechanical behaviour of polymeric Rohacell foams is presented. The finite elements model is developed at the scale of the microstructure idealized by a representative unit cell: the truncated octahedron. Observations made on micrographs of Rohacell lead to mesh this representative unit cell as a lattice of beam elements. Each beam is assigned a brittle linear elastic mechanical behaviour in tension and an elasto-plastic behaviour in compression. The plasticity in compression is introduced as a way to mimic the buckling of the edges of the cells observed in experimental crushing tests. A contact law introduced between the beams stands for densification. A change in scale is then realized by increasing the length of the edges of the unit cell. Several computations show the ability of the proposed approach to preserve the physical degradation phenomena and the loads while drastically decreasing the computational time