A quenched large deviation principle in a continuous scenario

We prove the analogue for continuous space-time of the quenched LDP derived in Birkner, Greven and den Hollander (2010) for discrete space-time. In particular, we consider a random environment given by Brownian increments, cut into pieces according to an independent continuous-time renewal process. We look at the empirical process obtained by recording both the length of and the increments in the successive pieces. For the case where the renewal time distribution has a Lebesgue density with a polynomial tail, we derive the quenched LDP for the empirical process, i.e., the LDP conditional on a typical environment. The rate function is a sum of two specific relative entropies, one for the pieces and one for the concatenation of the pieces. We also obtain a quenched LDP when the tail decays faster than algebraic. The proof uses coarse-graining and truncation arguments, involving various approximations of specific relative entropies that are not quite standard. In a companion paper we show how the quenched LDP and the techniques developed in the present paper can be applied to obtain a variational characterisation of the free energy and the phase transition line for the Brownian copolymer near a selective interface

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, Probab. Theory Related Fields 148 (2010) 403-456], 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.Comment: Published in at http://dx.doi.org/10.1214/11-AOP727 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Phase diagram for a copolymer in a micro-emulsion

In this paper we study a model describing a copolymer in a micro-emulsion. The copolymer consists of a random concatenation of hydrophobic and hydrophilic monomers, the micro-emulsion consists of large blocks of oil and water arranged in a percolation-type fashion. The interaction Hamiltonian assigns energy $-\alpha$ to hydrophobic monomers in oil and energy $-\beta$ to hydrophilic monomers in water, where $\alpha,\beta$ are parameters that without loss of generality are taken to lie in the cone $\{(\alpha,\beta) \in\mathbb{R}^2\colon\,\alpha \geq |\beta|\}$. Depending on the values of these parameters, the copolymer either stays close to the oil-water interface (localization) or wanders off into the oil and/or the water (delocalization). Based on an assumption about the strict concavity of the free energy of a copolymer near a linear interface, we derive a variational formula for the quenched free energy per monomer that is column-based, i.e., captures what the copolymer does in columns of different type. We subsequently transform this into a variational formula that is slope-based, i.e., captures what the polymer does as it travels at different slopes, and we use the latter to identify the phase diagram in the $(\alpha,\beta)$-cone. There are two regimes: supercritical (the oil blocks percolate) and subcritical (the oil blocks do not percolate). The supercritical and the subcritical phase diagram each have two localized phases and two delocalized phases, separated by four critical curves meeting at a quadruple critical point. The different phases correspond to the different ways in which the copolymer can move through the micro-emulsion. The analysis of the phase diagram is based on three hypotheses of percolation-type on the blocks. We show that these three hypotheses are plausible, but do not provide a proof.Comment: 100 pages, 16 figures. arXiv admin note: substantial text overlap with arXiv:1204.123

A general smoothing inequality for disordered polymers

This note sharpens the smoothing inequality of Giacomin and Toninelli for disordered polymers. This inequality is shown to be valid for any disorder distribution with locally finite exponential moments, and to provide an asymptotically sharp constant for weak disorder. A key tool in the proof is an estimate that compares the effect on the free energy of tilting, respectively, shifting the disorder distribution. This estimate holds in large generality (way beyond disordered polymers) and is of independent interest.Comment: 14 page

A large-deviation view on dynamical Gibbs-non-Gibbs transitions

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.Comment: 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

Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment

We continue our study of the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \gamma\xi u$ for the space-time field $u\colon\,\Z^d\times [0,\infty)\to\R$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in (0,\infty)$ is the coupling constant, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\xi$, both living on $\Z^d$. In earlier work we considered three choices for $\xi$: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $\xi$, and showed that these exponents display an interesting dependence on the diffusion constant $\kappa$, with qualitatively different behavior in different dimensions $d$. In the present paper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponential growth rate of $u$ conditional on $\xi$. We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general $\xi$ that is stationary and ergodic w.r.t.\ translations in $\Z^d$ and satisfies certain noisiness conditions. After that we focus on the three particular choices for $\xi$ mentioned above and derive some more detailed properties. We close by formulating a number of open problems.Comment: In honour of J\"urgen G\"artner on the occasion of his 60th birthday, 33 pages. Final revised versio