Related papers: Directed polymers in a random environment: a revie…
We discuss various aspects of the randomly interacting directed polymers with emphasis on the phases and phase transition. We also discuss the behaviour of overlaps of directed paths in a random medium.
In this paper, we consider directed polymers in random environment with long range jumps in discrete space and time. We extend to this case some techniques, results and classifications known in the usual short range case. However, some…
The transition from a weak-disorder (diffusive phase) to a strong-disorder (localized phase) for directed polymers in a random environment is a well studied phenomenon. In the most common setup, it is established that the phase transition…
This is a set of introductory lectures on the behaviour of a directed polymer in a random medium. Both the intuitive picture that helps in developing an understanding and systematic approaches for quantitative studies are discussed.
Modeling of polymer chains has received a lot of attention in mathematics. In fact, probabilistic models that naturally arise in statistical mechanics have been widely studied by mathematicians for the very challenging and novel problems…
The effects of two types of randomness on the behaviour of directed polymers are discussed in this chapter. The first part deals with the effect of randomness in medium so that a directed polymer feels a random external potential. The…
We study the directed polymer model for general graphs (beyond $\mathbb Z^d$) and random walks. We provide sufficient conditions for the existence or non-existence of a weak disorder phase, of an $L^2$ region, and of very strong disorder,…
For a directed polymer model in random environment, a characterization of the weak disorder phase in terms of the moment of the renormalized partition function has been proved in [S. Junk: Communications in Mathematical Physics 389,…
We study the relation between the directed polymer and the directed percolation models, for the case of a disordered energy landscape where the energies are taken from bimodal distribution. We find that at the critical concentration of the…
In this paper, we consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full weak disorder region, i.e., where the…
We study the Directed Polymer model subject to a particular form of disorder, $\eta(x,t)=\eta_X(x) \eta_T(t)$, recently proposed in biological applications. We find that two new universality classes arise, depending on the the lattice…
We study the long-range directed polymer model on $\mathbbm{Z}$ in a random environment, where the underlying random walk lies in the domain of attraction of an $\alpha$-stable process for some $\alpha\in(0,2]$. Similar to the more classic…
We provide a new short and elementary proof of diffusivity for directed polymer in a random environment in the weak disorder phase.
We show that choosing appropriate distributions of the randomness, the search for optimal paths links diverse problems of disordered media like directed percolation, invasion percolation, directed and non-directed spanning polymers. We also…
These notes are devoted to the statistical mechanics of directed polymers interacting with one-dimensional spatial defects. We are interested in particular in the situation where frozen disorder is present. These polymer models undergo a…
Very recently, Junk [11] showed that for directed polymers in bounded random environments, the weak disorder (uniform integrable) phase implies that the polymer martingale is bounded in $L^p$ for some $p>1$ and also in $L^q$ for some $q<0$.…
We consider directed polymers in 1+1 spatial dimension under action of an external repulsive potential along a line. Using the exact mapping onto imaginary time evolution of free fermions we find that for sufficiently strong potential the…
We study the directed polymer model in a bounded environment with bond disorder and show that, in the interior of the weak disorder phase, weak disorder continues to hold upon perturbation by a small bias. Using this stability result, we…
We introduce and study the planted directed polymer, in which the path of a random walker is inferred from noisy 'images' accumulated at each timestep. Formulated as a nonlinear problem of Bayesian inference for a hidden Markov model, this…
We present an approach to studying directed polymers in interaction with a defect line and subject to a force, which pulls them away from the line. We consider in particular the case of inhomogeneous interactions. We first give a formula…