Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures

According to recent progress in the finite size scaling theory of critical disordered systems, the nature of the phase transition is reflected in the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. In this paper, we apply this analysis to the delocalization transition of an heteropolymeric chain at a selective fluid-fluid interface. The width $\Delta T_c(L)$ and the shift $[T_c(\infty)-T_c^{av}(L)]$ are found to decay with the same exponent $L^{-1/\nu_{R}}$, where $1/\nu_{R} \sim 0.26$. The distribution of pseudo-critical temperatures $T_c(i,L)$ is clearly asymmetric, and is well fitted by a generalized Gumbel distribution of parameter $m \sim 3$. We also consider the free energy distribution, which can also be fitted by a generalized Gumbel distribution with a temperature dependent parameter, of order $m \sim 0.7$ in the critical region. Finally, the disorder averaged number of contacts with the interface scales at $T_c$ like $L^{\rho}$ with $\rho \sim 0.26 \sim 1/\nu_R$.Comment: 9 pages,6 figure

Smoothening of Depinning Transitions for Directed Polymers with Quenched Disorder

We consider disordered models of pinning of directed polymers on a defect line, including (1+1)-dimensional interface wetting models, disordered Poland--Scheraga models of DNA denaturation and other (1+d)-dimensional polymers in interaction with columnar defects. We consider also random copolymers at a selective interface. These models are known to have a (de)pinning transition at some critical line in the phase diagram. In this work we prove that, as soon as disorder is present, the transition is at least of second order: the free energy is differentiable at the critical line, and the order parameter (contact fraction) vanishes continuously at the transition. On the other hand, it is known that the corresponding non-disordered models can have a first order (de)pinning transition, with a jump in the order parameter. Our results confirm predictions based on the Harris criterion.Comment: 4 pages, 1 figure. Version 2: references added, minor changes made. To appear on Phys. Rev. Let

Numerical study of the disordered Poland-Scheraga model of DNA denaturation

We numerically study the binary disordered Poland-Scheraga model of DNA denaturation, in the regime where the pure model displays a first order transition (loop exponent $c=2.15>2$). We use a Fixman-Freire scheme for the entropy of loops and consider chain length up to $N=4 \cdot 10^5$, with averages over $10^4$ samples. We present in parallel the results of various observables for two boundary conditions, namely bound-bound (bb) and bound-unbound (bu), because they present very different finite-size behaviors, both in the pure case and in the disordered case. Our main conclusion is that the transition remains first order in the disordered case: in the (bu) case, the disorder averaged energy and contact densities present crossings for different values of $N$ without rescaling. In addition, we obtain that these disorder averaged observables do not satisfy finite size scaling, as a consequence of strong sample to sample fluctuations of the pseudo-critical temperature. For a given sample, we propose a procedure to identify its pseudo-critical temperature, and show that this sample then obeys first order transition finite size scaling behavior. Finally, we obtain that the disorder averaged critical loop distribution is still governed by $P(l) \sim 1/l^c$ in the regime $l \ll N$, as in the pure case.Comment: 12 pages, 13 figures. Revised versio

Directed polymer in a random medium of dimension 1+1 and 1+3: weights statistics in the low-temperature phase

We consider the low-temperature $T<T_c$ disorder-dominated phase of the directed polymer in a random potentiel in dimension 1+1 (where $T_c=\infty$) and 1+3 (where $T_c<\infty$). To characterize the localization properties of the polymer of length $L$, we analyse the statistics of the weights $w_L(\vec r)$ of the last monomer as follows. We numerically compute the probability distributions $P_1(w)$ of the maximal weight $w_L^{max}= max_{\vec r} [w_L(\vec r)]$, the probability distribution $\Pi(Y_2)$ of the parameter $Y_2(L)= \sum_{\vec r} w_L^2(\vec r)$ as well as the average values of the higher order moments $Y_k(L)= \sum_{\vec r} w_L^k(\vec r)$. We find that there exists a temperature $T_{gap}<T_c$ such that (i) for $T<T_{gap}$, the distributions $P_1(w)$ and $\Pi(Y_2)$ present the characteristic Derrida-Flyvbjerg singularities at $w=1/n$ and $Y_2=1/n$ for $n=1,2..$. In particular, there exists a temperature-dependent exponent $\mu(T)$ that governs the main singularities $P_1(w) \sim (1-w)^{\mu(T)-1}$ and $\Pi(Y_2) \sim (1-Y_2)^{\mu(T)-1}$ as well as the power-law decay of the moments $\bar{Y_k(i)} \sim 1/k^{\mu(T)}$. The exponent $\mu(T)$ grows from the value $\mu(T=0)=0$ up to $\mu(T_{gap}) \sim 2$. (ii) for $T_{gap}<T<T_c$, the distribution $P_1(w)$ vanishes at some value $w_0(T)<1$, and accordingly the moments $\bar{Y_k(i)}$ decay exponentially as $(w_0(T))^k$ in $k$. The histograms of spatial correlations also display Derrida-Flyvbjerg singularities for $T<T_{gap}$. Both below and above $T_{gap}$, the study of typical and averaged correlations is in full agreement with the droplet scaling theory.Comment: 13 pages, 29 figure

Statistics of first-passage times in disordered systems using backward master equations and their exact renormalization rules

We consider the non-equilibrium dynamics of disordered systems as defined by a master equation involving transition rates between configurations (detailed balance is not assumed). To compute the important dynamical time scales in finite-size systems without simulating the actual time evolution which can be extremely slow, we propose to focus on first-passage times that satisfy 'backward master equations'. Upon the iterative elimination of configurations, we obtain the exact renormalization rules that can be followed numerically. To test this approach, we study the statistics of some first-passage times for two disordered models : (i) for the random walk in a two-dimensional self-affine random potential of Hurst exponent $H$, we focus on the first exit time from a square of size $L \times L$ if one starts at the square center. (ii) for the dynamics of the ferromagnetic Sherrington-Kirkpatrick model of $N$ spins, we consider the first passage time $t_f$ to zero-magnetization when starting from a fully magnetized configuration. Besides the expected linear growth of the averaged barrier $\bar{\ln t_{f}} \sim N$, we find that the rescaled distribution of the barrier $(\ln t_{f})$ decays as $e^{- u^{\eta}}$ for large $u$ with a tail exponent of order $\eta \simeq 1.72$. This value can be simply interpreted in terms of rare events if the sample-to-sample fluctuation exponent for the barrier is $\psi_{width}=1/3$.Comment: 8 pages, 4 figure

Two-dimensional wetting with binary disorder: a numerical study of the loop statistics

We numerically study the wetting (adsorption) transition of a polymer chain on a disordered substrate in 1+1 dimension.Following the Poland-Scheraga model of DNA denaturation, we use a Fixman-Freire scheme for the entropy of loops. This allows us to consider chain lengths of order $N \sim 10^5$ to $10^6$, with $10^4$ disorder realizations. Our study is based on the statistics of loops between two contacts with the substrate, from which we define Binder-like parameters: their crossings for various sizes $N$ allow a precise determination of the critical temperature, and their finite size properties yields a crossover exponent $\phi=1/(2-\alpha) \simeq 0.5$.We then analyse at criticality the distribution of loop length $l$ in both regimes $l \sim O(N)$ and $1 \ll l \ll N$, as well as the finite-size properties of the contact density and energy. Our conclusion is that the critical exponents for the thermodynamics are the same as those of the pure case, except for strong logarithmic corrections to scaling. The presence of these logarithmic corrections in the thermodynamics is related to a disorder-dependent logarithmic singularity that appears in the critical loop distribution in the rescaled variable $\lambda=l/N$ as $\lambda \to 1$.Comment: 12 pages, 13 figure

A simple model for DNA denaturation

Following Poland and Scheraga, we consider a simplified model for the denaturation transition of DNA. The two strands are modeled as interacting polymer chains. The attractive interactions, which mimic the pairing between the four bases, are reduced to a single short range binding term. Furthermore, base-pair misalignments are forbidden, implying that this binding term exists only for corresponding (same curvilinear abscissae) monomers of the two chains. We take into account the excluded volume repulsion between monomers of the two chains, but neglect intra-chain repulsion. We find that the excluded volume term generates an effective repulsive interaction between the chains, which decays as $1/r^{d-2}$. Due to this long-range repulsion between the chains, the denaturation transition is first order in any dimension, in agreement with previous studies.Comment: 10 page

Probing the tails of the ground state energy distribution for the directed polymer in a random medium of dimension d=1,2,3d=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, K\"orner, Katzgraber and Hartmann \cite{Ko_Ka_Ha} 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 $P_d(E_0)$ 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. In dimensions $d=2$ and $d=3$, we measure the negative tail up to ten standard deviations, which correspond to probabilities of order $P_d(E_0) \sim 10^{-22}$. Our results are in agreement with Zhang's argument, stating that the negative tail exponent $\eta(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{\eta(d)}$ as $E_0 \to -\infty$ is directly related to the fluctuation exponent $\theta(d)$ (which governs the fluctuations $\Delta E_0(L) \sim L^{\theta(d)}$ of the ground state energy $E_0$ for polymers of length $L$) via the simple formula $\eta(d)=1/(1-\theta(d))$. Along the paper, we comment on the similarities and differences with spin-glasses.Comment: 13 pages, 16 figure

A Solvable Model of Secondary Structure Formation in Random Hetero-Polymers

We propose and solve a simple model describing secondary structure formation in random hetero-polymers. It describes monomers with a combination of one-dimensional short-range interactions (representing steric forces and hydrogen bonds) and infinite range interactions (representing polarity forces). We solve our model using a combination of mean field and random field techniques, leading to phase diagrams exhibiting second-order transitions between folded, partially folded and unfolded states, including regions where folding depends on initial conditions. Our theoretical results, which are in excellent agreement with numerical simulations, lead to an appealing physical picture of the folding process: the polarity forces drive the transition to a collapsed state, the steric forces introduce monomer specificity, and the hydrogen bonds stabilise the conformation by damping the frustration-induced multiplicity of states.Comment: 24 pages, 14 figure