Fluctuations of the winding number of a directed polymer in a random medium

The effect of ambient disorders and sequence heterogeneities on the reptation of a long polymer is studied with the aid of a disordered tube model. The dynamics of a random heteropolymer is found to be much slower than that of a homopolymer due to collective pinning effects. The asymptotic properties belong to the universality class of a directed path in (1+1)-dimensional random media.

The model of directed polymer in a random environment is a fundamental model of interaction between a simple random walk and ambient disorder. This interaction gives rise to complex phenomena and transitions from a central limit theory to novel statistical behaviours. Despite its intense study, there are still many aspects and phases which have not yet been identified. In this review we focus on the current status of our understanding of the transition between weak and strong...

We present systematic numerical simulations for directed polymers at finite temperatures in 1+1 and 2+1 dimensions. The transverse fluctuations and free energy fluctuations tend to the strong coupling limit at any temperature in both 1+1 and 2+1 dimensions for long time t. Two different definitions for energy fluctuations at finite temperatures, which are the ensemble energy fluctuations and the internal energy fluctuations, are investigated. Apart from zero temperature, the ...

For the KPZ equation on a torus with a $1+1$ spacetime white noise, it was shown in \cite{GK21,ADYGTK22} that the height function satisfies a central limit theorem, and the variance can be written as the expectation of a certain exponential functional of Brownian bridges. In this paper, we consider another physically relevant quantity, the winding number of the corresponding directed polymer on a cylinder, or equivalently, the displacement of the directed polymer endpoint in ...

In these lecture notes, which are based on the mini-course given at 2013 Prague School on Mathematical Statistical Physics, we discuss ballistic phase of quenched and annealed stretched polymers in random environment on ${\mathbb Z}^d$ with an emphasis on the natural renormalized renewal structures which appear in such models. In the ballistic regime an irreducible decomposition of typical polymers leads to an effectiverandom walk reinterpretation of the latter. In the anneal...

This review is devoted to the detailed consideration of the universal statistical properties of one-dimensional directed polymers in a random potential. In terms of the replica Bethe ansatz technique we derive several exact results for different types of the free energy probability distribution functions. In the second part of the review we discuss the problems which are still waiting for their solutions. Several mathematical appendices in the ending part of the review contai...

We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes till time t. Using a method based on the renewal properties of Brownian walker, we find exact analytical expressions of these distributions. This method serves as an alternative to the standard path integral techniques which are not always easily adaptable for certain observables. For large t, we...

The scaling behavior of a directed polymer in a two-dimensional (2D) random potential under confining force is investigated. The energy of a polymer with configuration $\{y(x)\}$ is given by $H\big(\{y(x)\}\big) = \sum_{x=1}^N \exyx + \epsilon \Wa^\alpha$, where $\eta(x,y)$ is an uncorrelated random potential and $\Wa$ is the width of the polymer. Using an energy argument, it is conjectured that the radius of gyration $R_g(N)$ and the energy fluctuation $\Delta E(N)$ of the p...

In this paper in terms of the replica method we consider the high temperature limit of (2+1) directed polymers in a random potential and propose an approach which allows to compute the scaling exponent $\theta$ of the free energy fluctuations as well as the left tail of its probability distribution function. It is argued that $\theta = 1/2$ which is different from the zero-temperature numerical value which is close to 0.241. This result implies that unlike the $(1+1)$ system ...

In this article we study from a non-perturbative point of view the entanglement of two directed polymers subjected to repulsive interactions given by a Dirac $\delta-$function potential. An exact formula of the so-called second moment of the winding angle is derived. This result is used to provide a thorough analysis of entanglement phenomena in the classical system of two polymers subjected to repulsive interactions and related problems. No approximation is made in treating ...