Decay of covariances, uniqueness of ergodic component and scaling limit for a class of \nabla\phi systems with non-convex potential

We consider a gradient interface model on the lattice with interaction potential which is a nonconvex perturbation of a convex potential. Using a technique which decouples the neighboring vertices sites into even and odd vertices, we show for a class of non-convex potentials: the uniqueness of ergodic component for \nabla\phi-Gibbs measures, the decay of covariances, the scaling limit and the strict convexity of the surface tension.Comment: 41 pages, 5 figure

Markovian perturbation, response and fluctuation dissipation theorem

We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of "linear response function" in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is, if the initial measure is invariant for the given Markov semi-group, then for any pair of times s<t and nice functions f,g, the dissipation, that is, the derivative in s of the covariance of g(X(t)) and f(X(s)) equals the infinitesimal response at time t and direction g to any Markovian perturbation that alters the invariant measure of X(.) in the direction of f at time s. The same applies in the so called FDT regime near equilibrium, i.e. in the limit s going to infinity with t-s fixed, provided X(s) converges in law to an invariant measure for its dynamics. We provide the response function of two generic Markovian perturbations which we then compare and contrast for pure jump processes on a discrete space, for finite dimensional diffusion processes, and for stochastic spin systems

Pinning and wetting transition for (1+1)-dimensional fields with Laplacian interaction

We consider a random field $\varphi:\{1,...,N\}\to\mathbb{R}$ as a model for a linear chain attracted to the defect line $\varphi=0$, that is, the x-axis. The free law of the field is specified by the density $\exp(-\sum_iV(\Delta\varphi_i))$ with respect to the Lebesgue measure on $\mathbb{R}^N$, where $\Delta$ is the discrete Laplacian and we allow for a very large class of potentials $V(\cdot)$. The interaction with the defect line is introduced by giving the field a reward $\varepsilon\ge0$ each time it touches the x-axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity $\varepsilon$ of the pinning reward varies: both in the pinning ($a=\mathrm{p}$) and in the wetting ($a=\mathrm{w}$) case, there exists a critical value $\varepsilon_c^a$ such that when $\varepsilon>\varepsilon_c^a$ the field touches the defect line a positive fraction of times (localization), while this does not happen for $\varepsilon<\varepsilon_c^a$ (delocalization). The two critical values are nontrivial and distinct: 0<\varepsilon_c^{\mat hrm{p}}<\varepsilon_c^{\mathrm{w}}<\infty, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at $\varepsilon=\varepsilon_c^{\mathrm{p}}$ is delocalized. On the other hand, the transition in the wetting model is of first order and for $\varepsilon=\varepsilon_c^{\mathrm{w}}$ the field is localized. The core of our approach is a Markov renewal theory description of the field.Comment: Published in at http://dx.doi.org/10.1214/08-AOP395 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The quenched invariance principle for random walks in random environments admitting a bounded cycle representation

We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields 129 (2004) 219--244) to the non-reversible setting.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP122 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local Central Limit Theorem for diffusions in a degenerate and unbounded Random Medium

We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local central limit theorem for $X$, under some moment conditions on the environment; the key tool is a local parabolic Harnack inequality obtained with Moser iteration technique.Comment: 25 page

Invariance Principle for the one-dimensional dynamic Random Conductance Model under Moment Conditions

Recent progress in the understanding of quenched invariance principles (QIP) for a continuous-time random walk on $\mathbb{Z}^d$ in an environment of dynamical random conductances is reviewed and extended to the $1$-dimensional case. The law of the conductances is assumed to be ergodic with respect to time-space shifts and satisfies certain integrability conditions.Comment: 15 pages, 1 figure; submitted to the Proceedings of the "RIMS Symposium on Stochastic Analysis on Large Scale Interacting Systems", October 2015, Kyoto, Japa