Polymer In Random Medium Research Articles

Front velocity and directed polymers in random medium

We consider a stochastic model of N evolving particles studied by Brunet and Derrida. This model can be seen as a directed polymer in random medium with N sites in the transverse direction. Cook and Derrida use heuristic arguments to obtain a formula for the ground state energy of the polymer. We formalize their argument and show that there is an additional term in the formula in the critical case. We also consider a generalization of the model, and show that in the noncritical case the behavior is basically the same, whereas in the critical case a new correction appears.

Free Energy Fluctuations for Directed Polymers in Random Media in 1 + 1 Dimension

We consider two models for directed polymers in space‐time independent random media (the O'Connell‐Yor semidiscrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions. In particular, we show that for large time τ, the probability distributions for the free energy fluctuations, when rescaled by τ1/3, converges to the GUE Tracy‐Widom distribution.We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics. For the semidiscrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik–Ben Arous–Péché distributions from spiked random matrix theory. For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm—the Kardar‐Parisi‐Zhang equation. The Laplace transform formula we prove can be inverted to give the one‐point probability distribution of the solution to these stochastic PDEs for the class of initial data. © 2014 Wiley Periodicals, Inc.

Magnetoresistance of an Anderson Insulator of Bosons

We study the magnetoresistance of two-dimensional bosonic Anderson insulators. We describe the change in spatial decay of localized excitations in response to a magnetic field, which is given by an interference sum over alternative tunneling trajectories. The excitations become more localized with increasing field (in sharp contrast to generic fermionic excitations which get weakly delocalized): the localization length ξ(B) is found to change as ξ(-1)(B)-ξ(-1)(0)~B(4/5). The quantum interference problem maps onto the classical statistical mechanics of directed polymers in random media (DPRM). We explain the observed scaling using a simplified droplet model which incorporates the nontrivial DPRM exponents. Our results have implications for a variety of experiments on magnetic-field-tuned superconductor-to-insulator transitions observed in disordered films, granular superconductors, and Josephson junction arrays, as well as for cold atoms in artificial gauge fields.

THE KARDAR–PARISI–ZHANG EQUATION AND UNIVERSALITY CLASS

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past 25 years a new universality class has emerged to describe a host of important physical and probabilistic models (including one-dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar–Parisi–Zhang (KPZ) universality class and underlying it is, again, a continuum object — a non-linear stochastic partial differential equation — known as the KPZ equation.The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media.As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.

Objective method for estimating asymptotic parameters, with an application to sequence alignment.

Sequence alignment is an indispensable computational tool in modern molecular biology. The model underlying biological sequence alignment is of interest to physicists because it approximates the statistical mechanics of DNA and protein annealing, while bearing an intimate relationship to models of directed polymers in random media. Recent methods for determining the statistics of random sequence alignments have reduced the computation time to less than 1 s, opening up some interesting possibilities for online computation with biological search engines. Before implementation, however, the methods required an objective technique for computing regression coefficients pertinent to an asymptotic regime. Typically, physicists estimate parameters pertinent to an asymptotic regime subjectively: They eyeball their data; estimate the asymptotic regime where the regression model holds with reasonable accuracy; and then regress data only within the estimated asymptotic regime. Our publicly available computer program ARRP replaces the subjective assessment of the asymptotic regime with an objective change-point detection method, increasing confidence in the scientific objectivity of the parameter estimates. Asymptotic regression has potential applications across most of physics.

The cavity method for quantum disordered systems: from transverse random fieldferromagnets to directed polymers in random media

After reviewing the basics of the cavity method in classical systems, we show how itsquantum version, with some appropriate approximation scheme, can be used to study asystem of spins with random ferromagnetic interactions and a random transverse field. Thequantum cavity equations describing the ferromagnetic–paramagnetic phase transitioncan be transformed into the well-known problem of a classical directed polymerin a random medium. The glass transition of this polymer problem translatesinto the existence of a ‘Griffiths phase’ close to the quantum phase transition ofthe quantum spin problem, where the physics is dominated by rare events. Thephysical behaviour of random transverse-field ferromagnets on the Bethe lattice isfound to be very similar to that found in finite-dimensional systems, and thequantum cavity method gets back the known exact results of the one-dimensionalproblem.

The phase diagram of a directed polymer in random media with p-spin ferromagnetic interactions

We consider a directed polymer model with an additive p-spin (p>2) ferromagnetic term in the Hamiltonian. We give a rigorous proof for the specific free energy and derive the phase diagram. This model was proposed previously, and a detailed proof was given in the case p = 2, while the main result was only stated for p > 2. We give a detailed proof of the main result and show the behavior of the model as p → ∞ by constructing the phase diagram also in this case. These results are important in many applications, for instance, in telecommunication and immunology. Our major finding is that in the phase diagram for p > 2, a new transition curve (absent for p = 2) emerges between the paramagnetic region and the so-called mixed region and that the ferromagnetic region diminishes as p → ∞.

Bethe ansatz solution for one-dimensional directed polymers in random media

We study the statistical properties of one-dimensional directed polymers in a short-rangerandom potential by mapping the replicated problem to a many-body quantumboson system with attractive interactions. We find the full set of eigenvalues andeigenfunctions of the many-body system and perform the summation over the entirespectrum of excited states. The analytic continuation of the obtained exact expressionfor the replica partition function from integer to non-integer replica parameterN turnsout to be ambiguous. Performing the analytic continuation simply by assuming that the parameterN can take arbitrary complex values, and going to the thermodynamic limit of the originaldirected polymer problem, we obtain the explicit universal expression for the probabilitydistribution function of free energy fluctuations.

Finite-temperature local protein sequence alignment: Percolation and free-energy distribution

Sequence alignment is a tool in bioinformatics that is used to find homological relationships in large molecular databases. It can be mapped on the physical model of directed polymers in random media. We consider the finite-temperature version of local sequence alignment for proteins and study the transition between the linear phase and the biologically relevant logarithmic phase, where the free energy grows linearly or logarithmically with the sequence length. By means of numerical simulations and finite-size-scaling analysis, we determine the phase diagram in the plane that is spanned by the gap costs and the temperature. We use the most frequently used parameter set for protein alignment. The critical exponents that describe the parameter-driven transition are found to be explicitly temperature dependent. Furthermore, we study the shape of the (free-) energy distribution close to the transition by rare-event simulations down to probabilities on the order 10(-64). It is well known that in the logarithmic region, the optimal score distribution (T=0) is described by a modified Gumbel distribution. We confirm that this also applies for the free-energy distribution (T>0). However, in the linear phase, the distribution crosses over to a modified Gaussian distribution.

Directed polymer in random media with a defect

We investigate a directed polymer in random media with an attractive defect at the center of the one-dimensional substrate. Without the defect, the end to end distance Deltax of the polymer follows Deltax approximately t;{1/z} , with z=3/2 and t is the polymer length. When the defect strength is weak, its contribution to Deltax is negligible. If > or =_{c} , then Deltax approaches a finite value Deltax_{sat}() in large t limit, and we find Deltax_{sat} approximately (-_{c});{-delta} , with delta approximately 3.0 . Such transition is related to the queuing phenomena of the asymmetric simple exclusion process. The polymer energy fluctuation is also discussed.

Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment

We first obtain exponential inequalities for martingales. Let ( X k ) ( 1 ≤ k ≤ n ) be a sequence of martingale differences relative to a filtration ( F k ) and set S n = X 1 + ⋯ + X n . We prove that if for some δ > 0 , Q ≥ 1 , K > 0 and all k , E [ e δ | X k | Q | F k − 1 ] ≤ K a.s., then for some constant c > 0 (depending only on δ , Q and K ) and all x > 0 , P [ | S n | > n x ] ≤ 2 e − n c ( x ) , where c ( x ) = c x 2 if x ∈ ] 0 , 1 ] and c ( x ) = c x Q if x > 1 ; the converse also holds if ( X i ) are independent and identically distributed. This extends Bernstein’s inequality for Q = 1 and Hoeffding’s inequality for Q = 2 . We then apply the preceding result to establish exponential concentration inequalities for the free energy of directed polymers in a random environment and obtain upper bounds for its rates of convergence (in probability, almost surely and in L p ); we also give an expression for the free energy in terms of those of some multiplicative cascades, which improves an inequality of Comets and Vargas [Francis Comets, Vincent Vargas, Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 267–277 (electronic)] to an equality.

Convergence of the Law of the Environment Seen by the Particle for Directed Polymers in Random Media in the L 2 Region

We consider the model of directed polymers in an i.i.d. Gaussian or bounded environment (Imbrie and Spencer in J. Stat. Phys. 52(3/4), 609–626, 1988; Carmona and Hu in Probab. Theory Relat. Fields 124(3), 431–457, 2002; Comets et al. in Adv. Stud. Pure Math. 39, 115–142, 2004) in the L2 region. We prove the convergence of the law of the environment seen by the particle.

Superdiffusivity for a Brownian polymer in a continuous Gaussian environment

This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field W on ℝ+×ℝ which is white noise in time and function-valued in space. According to the behavior of the spatial covariance of W, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdiffusive, with a wandering exponent exceeding any α<3/5.

Relation between directed polymers in random media and random-bond dimer models

We reassess the relation between classical lattice dimer models and the continuum elastic description of a lattice of fluctuating polymers. In the absence of randomness we determine the density and line tension of the polymers in terms of the bond weights of hard-core dimers on the square and the hexagonal lattice. For the latter, we demonstrate the equivalence of the canonical ensemble for the dimer model and the grand-canonical description for polymers by performing explicitly the continuum limit. Using this equivalence for the random bond dimer model on a square lattice, we resolve a previously observed discrepancy between numerical results for the random dimer model and a replica approach for polymers in random media. Further potential applications of the equivalence are briefly discussed.

Fracture Surfaces as Multiscaling Graphs

Fracture paths in quasi-two-dimensional (2D) media (e.g., thin layers of materials or paper) are analyzed as self-affine graphs h(x) of height h as a function of length x. We show that these are multiscaling, in the sense that nth order moments of the height fluctuations across any distance l scale with a characteristic exponent that depends nonlinearly on the order of the moment. Having demonstrated this, one rules out a widely held conjecture that fracture in 2D belongs to the universality class of directed polymers in random media. In fact, 2D fracture does not belong to any of the known kinetic roughening models. The presence of multiscaling offers a stringent test for any theoretical model; we show that a recently introduced model of quasistatic fracture passes this test.

A local limit theorem for directed polymers in random media: the continuous and the discrete case

In this article, we consider two models of directed polymers in random environment: a discrete model in a general random environment and a continuous model. We consider these models in dimension greater than or equal to 3 and we suppose that the normalized partition function is bounded in L 2 (the “high” temperature case). Under these assumptions, Sinai proved in [Y. Sinai, A remark concerning random walks with random potentials, Fund. Math. 147 (1995) 173–180] a local limit theorem for the discrete model, using a perturbation expansion. In this article, we give a new method for proving Sinai's local limit theorem. This new method can be transposed to the continuous setting in which we prove a similar local limit theorem.

Directed polymers in random media under confining force

The scaling behavior of a directed polymer in a two-dimensional random potential under confining force is investigated. The energy of a polymer with configuration {y(x)} is given by H({y(x)}) = sigma(x=1)(N) eta(x,y(x)) + epsilonW(alpha), where eta(x,y) is an uncorrelated random potential and W is the width of the polymer. Using an energy argument, it is conjectured that the radius of gyration Rg(N) and the energy fluctuation deltaE(N) of the polymer of length N in the ground state increase as Rg(N) approximately N(nu) and deltaE(N) approximately N(omega), respectively, with nu = 1/(1+alpha) and omega = (1+2alpha)/(4+4alpha) for alpha > or = 1/2. An algorithm of finding the exact ground state, with the effective time complexity of O(N3), is introduced and used to confirm the conjecture numerically.

Exactly soluble noisy traveling-wave equation appearing in the problem of directed polymers in a random medium.

We calculate exactly the velocity and diffusion constant of a microscopic stochastic model of N evolving particles which can be described by a noisy traveling-wave equation with a noise of order N(-1/2). Our model can be viewed as the infinite range limit of a directed polymer in random medium with N sites in the transverse direction. Despite some peculiarities of the traveling-wave equations in the absence of noise, our exact solution allows us to test the validity of a simple cutoff approximation and to show that, in the weak noise limit, the position of the front can be completely described by the effect of the noise on the first particle.

Queuing transitions in the asymmetric simple exclusion process.

Stochastic driven flow along a channel can be modeled by the asymmetric simple exclusion process. We confirm numerically the presence of a dynamic queuing phase transition at a nonzero obstruction strength, and establish its scaling properties. Below the transition, the traffic jam is macroscopic in the sense that the length of the queue scales linearly with system size. Above the transition, only a power-law shaped queue remains. Its density profile scales as deltarho approximately x(-nu) with nu=1/3, and x is the distance from the obstacle. We construct a heuristic argument, indicating that the exponent nu=1/3 is universal and independent of the dynamic exponent of the underlying dynamic process. Fast bonds create only power-law shaped depletion queues, and with an exponent that could be equal to nu=2/3, but the numerical results yield consistently somewhat smaller values nu approximately 0.63(3). The implications of these results to faceting of growing interfaces and localization of directed polymers in random media, both in the presence of a columnar defect are pointed out as well.

Dynamics of linear polymers in random media

We study phenomenological scaling theories of the polymer dynamics in random media, employing the existing scaling theories of polymer chains and the percolation statistics. We investigate both the Rouse and the Zimm model for Brownian dynamics and estimate the diffusion constant of the center-of-mass of the chain in such disordered media. For internal dynamics of the chain, we estimate the dynamic exponents. We propose similar scaling theory for the reptation dynamics of the chain in the framework of Flory theory for the disordered medium. The modifications in the case of correlated disorders are also discussed.