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 Mathematics , 2010, DOI: 10.1007/s10955-010-0110-x Abstract: We consider the continuous time version of the Random Walk Pinning Model (RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$on Z^d with jump rate rho (that plays the role of the random medium), we modify the law of a random walk X on Z^d with jump rate 1 by reweighting the paths, giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t \ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta L_t(X,Y)), beta in R. As beta increases, the system exhibits a delocalization/localization transition: there is a critical value beta_c, such that if beta>beta_c the two walks stick together for almost-all Y realizations. A natural question is that of disorder relevance, that is whether the quenched and annealed systems have the same behavior. In this paper we investigate how the disorder modifies the shape of the free energy curve: (1) We prove that, in dimension d larger or equal to three 3, the presence of disorder makes the phase transition at least of second order. This, in dimension larger or equal to 4, contrasts with the fact that the phase transition of the annealed system is of first order. (2) In any dimension, we prove that disorder modifies the low temperature asymptotic of the free energy.  Mathematics , 2012, DOI: 10.1214/13-AAP930 Abstract: We study the path properties of a random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense "tight in probability" as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.  Mathematics , 2012, DOI: 10.1007/s10955-013-0747-3 Abstract: This paper studies a polymer chain in the vicinity of a linear interface separating two immiscible solvents. The polymer consists of random monomer types, while the interface carries random charges. Both the monomer types and the charges are given by i.i.d. sequences of random variables. The configurations of the polymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The Hamiltonian has two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface interaction ("pinning"). The quenched and the annealed version of the model each undergo a transition from a localized phase (where the polymer stays close to the interface) to a delocalized phase (where the polymer wanders away from the interface). We exploit the approach developed in [5] and [3] to derive variational formulas for the quenched and the annealed free energy per monomer. These variational formulas are analyzed to obtain detailed information on the critical curves separating the two phases and on the typical behavior of the polymer in each of the two phases. Our main results settle a number of open questions.  Physics , 2001, DOI: 10.1103/PhysRevLett.88.117004 Abstract: We study the vortex glass transition in disordered high temperature superconductors using Monte Carlo simulations. We use a random pinning model with strong point-correlated quenched disorder, a net applied magnetic field, longrange vortex interactions, and periodic boundary conditions. From a finite size scaling study of the helicity modulus, the RMS current, and the resistivity, we obtain critical exponents at the phase transition. The new exponents differ substantially from those of the gauge glass model, but are consistent with those of the pure three-dimensional XY model.  Mathematics , 2008, DOI: 10.1214/08-AOP424 Abstract: We consider a random field$\varphi:\{1,...,N\}\to \mathbb{R}$with Laplacian interaction of the form$\sum_iV(\Delta\varphi_i)$, where$\Delta$is the discrete Laplacian and the potential$V(\cdot)$is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward$\varepsilon\ge0$each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime$(\varepsilon<\varepsilon_c)$and a localized one$(\varepsilon>\varepsilon_c)$, where$0<\varepsilon_c<\infty$. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show, in particular, that in the delocalized regime the field wanders away from the defect line at a typical distance$N^{3/2}$, while in the localized regime the distance is just$O((\log N)^2)$. A subtle scenario shows up in the critical regime$(\varepsilon=\varepsilon_c)$, where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable L\'evy process of index 2/5. Our approach is based on Markov renewal theory.  Mathematics , 2007, DOI: 10.1214/08-AOP395 Abstract: 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.