Random walk in a two-dimensional self-affine random potential: Properties of the anomalous diffusion phase at small external force.

We study the dynamical response to an external force F for a particle performing a random walk in a two-dimensional quenched random potential of Hurst exponent H=1/2 . We present numerical results on the statistics of first-passage times that satisfy closed backward master equations. We find that there exists a zero-velocity phase in a finite region of the external force 0 View Article and Find Full Text PDF

Random walk in two-dimensional self-affine random potentials: strong-disorder renormalization approach.

We consider the continuous-time random walk of a particle in a two-dimensional self-affine quenched random potential of Hurst exponent H>0 . The corresponding master equation is studied via the strong disorder renormalization procedure introduced in Monthus and Garel [J. Phys.

Driven interfaces in random media at finite temperature: existence of an anomalous zero-velocity phase at small external force.

The motion of driven interfaces in random media at finite temperature T and small external force F is usually described by a linear displacement h{G}(t) approximately V(F,T)t at large times, where the velocity vanishes according to the creep formula as V(F,T) approximately e;{-K(T)F;{mu}} for F-->0 . In this paper, we question this picture on the specific example of the directed polymer in a two-dimensional random medium. We have recently shown [C.

Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices.

We study the wetting transition and the directed polymer delocalization transition on diamond hierarchical lattices. These two phase transitions with frozen disorder correspond to the critical points of quadratic renormalizations of the partition function. (These exact renormalizations on diamond lattices can also be considered as approximate Migdal-Kadanoff renormalizations for hypercubic lattices.

Multifractal statistics of the local order parameter at random critical points: application to wetting transitions with disorder.

Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.

Directed polymer in a random medium of dimension 1+3: multifractal properties at the localization-delocalization transition.

We consider the model of the directed polymer in a random medium of dimension 1+3 , and investigate its multifractal properties at the localization-delocalization transition. In close analogy with models of the quantum Anderson localization transition, where the multifractality of critical wavefunctions is well established, we analyze the statistics of the position weights w{L}(r[over]) of the endpoint of the polymer of length L via the moments [equation: see text]. We measure the generalized exponents tau(q) and tau[over](q) governing the decay of the typical values [equation: see text] and disorder-averaged values Y{q}(L)[over] approximately L{-tau[over](q)} , respectively.

Critical weight statistics of the random energy model and of the directed polymer on the Cayley tree.

We consider the critical point of two mean-field disordered models: (i) the random energy model (REM), introduced by Derrida as a mean-field spin-glass model of N spins and (ii) the directed polymer of length N on a Cayley Tree (DPCT) with random bond energies. Both models are known to exhibit a freezing transition between a high-temperature phase where the entropy is extensive and a low-temperature phase of finite entropy, where the weight statistics coincides with the weight statistics of Lévy sums with index mu=TT{c}<1 . In this paper, we study the weight statistics at criticality via the entropy S=Sigma w{i}lnw{i} and the generalized moments Y{k}= Sigma w{i}{k} , where the w{i} are the Boltzmann weights of the 2{N} configurations.

Freezing transition of the random bond RNA model: Statistical properties of the pairing weights.

To characterize the pairing specificity of RNA secondary structures as a function of temperature, we analyze the statistics of the pairing weights as follows: for each base (i) of the sequence of length N , we consider the (N-1) pairing weights w{i}(j) with the other bases (j not equal i) of the sequence. We numerically compute the probability distributions P1(w) of the maximal weight w[{i}{max}=max{j}[w{i}(j)] , the probability distribution Pi(Y(2)) of the parameter Y2(i)= summation operator{j}w{i}{2}(j) , as well as the average values of the moments Y{k}(i)= summation operator_{j}w_{i}{k}(j) . We find that there are two important temperatures T_{c} View Article and Find Full Text PDF

Probing the tails of the ground-state energy distribution for the directed polymer in a random medium of dimension d=1,2,3 via a Monte Carlo procedure in the disorder.

In order to probe with high precision the tails of the ground-state energy distribution of disordered spin systems, Körner, Katzgraber, and Hartmann have recently proposed an importance-sampling Monte Carlo Markov chain in the disorder. In this paper, we combine their Monte Carlo procedure in the disorder with exact transfer matrix calculations in each sample to measure the negative tail of ground-state energy distribution Pd(E0) for the directed polymer in a random medium of dimension d=1,2,3. In d=1, we check the validity of the algorithm by a direct comparison with the exact result, namely, the Tracy-Widom distribution.

Freezing transition of the directed polymer in a 1 + d random medium: location of the critical temperature and unusual critical properties.

In dimension d > or =3, the directed polymer in a random medium undergoes a phase transition between a free phase at high temperature and a low-temperature disorder-dominated phase. For the latter phase, Fisher and Huse have proposed a droplet theory based on the scaling of the free-energy fluctuations Delta F(l) approximately l theta at scale l. On the other hand, in related growth models belonging to the Kardar-Parisi-Zhang universality class, Forrest and Tang have found that the height-height correlation function is logarithmic at the transition.

Statistics of low energy excitations for the directed polymer in a random medium.

We consider a directed polymer of length L in a random medium of space dimension d = 1,2,3. The statistics of low energy excitations as a function of their size l is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities rho(bulk)(L) (E = 0,l) and rho(boundary)(L)(E=0,l).

Generalized Poland-Scheraga model for DNA hybridization.

The Poland-Scheraga (PS) model for the helix-coil transition of DNA considers the statistical mechanics of the binding (or hybridization) of two complementary strands of DNA of equal length, with the restriction that only bases with the same index along the strands are allowed to bind. In this article, we extend this model by relaxing these constraints: We propose a generalization of the PS model that allows for the binding of two strands of unequal lengths N1 and N2 with unrelated sequences. We study in particular (i) the effect of mismatches on the hybridization of complementary strands, (ii) the hybridization of noncomplementary strands (as resulting from point mutations) of unequal lengths N1 and N2.

Directed polymers and interfaces in random media: free-energy optimization via confinement in a wandering tube.

We analyze, via Imry-Ma scaling arguments, the strong disorder phases that exist in low dimensions at all temperatures for directed polymers and interfaces in random media. For the uncorrelated Gaussian disorder, we obtain that the optimal strategy for the polymer in dimension 1+d with 01/2 for the wandering of the best favorable tube available. The corresponding free energy then scales as F approximately Lomega with omega=2nu-1 and the left tail of the probability distribution involves a stretched exponential of exponent eta=(4-d)/2.