41 research outputs found

### 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

### 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

### 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

### 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

### Non equilibrium dynamics of disordered systems : understanding the broad continuum of relevant time scales via a strong-disorder RG in configuration space

We show that an appropriate description of the non-equilibrium dynamics of
disordered systems is obtained through a strong disorder renormalization
procedure in {\it configuration space}, that we define for any master equation
with transitions rates $W ({\cal C} \to {\cal C}')$ between configurations. The
idea is to eliminate iteratively the configuration with the highest exit rate
$W_{out} ({\cal C})= \sum_{{\cal C}'} W ({\cal C} \to {\cal C}')$ to obtain
renormalized transition rates between the remaining configurations. The
multiplicative structure of the new generated transition rates suggests that,
for a very broad class of disordered systems, the distribution of renormalized
exit barriers defined as $B_{out} ({\cal C}) \equiv - \ln W_{out}({\cal C})$
will become broader and broader upon iteration, so that the strong disorder
renormalization procedure should become asymptotically exact at large time
scales. We have checked numerically this scenario for the non-equilibrium
dynamics of a directed polymer in a two dimensional random medium.Comment: v2=final versio

### 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

### Anderson localization transition with long-ranged hoppings : analysis of the strong multifractality regime in terms of weighted Levy sums

For Anderson tight-binding models in dimension $d$ with random on-site
energies $\epsilon_{\vec r}$ and critical long-ranged hoppings decaying
typically as $V^{typ}(r) \sim V/r^d$, we show that the strong multifractality
regime corresponding to small $V$ can be studied via the standard perturbation
theory for eigenvectors in quantum mechanics. The Inverse Participation Ratios
$Y_q(L)$, which are the order parameters of Anderson transitions, can be
written in terms of weighted L\'evy sums of broadly distributed variables (as a
consequence of the presence of on-site random energies in the denominators of
the perturbation theory). We compute at leading order the typical and
disorder-averaged multifractal spectra $\tau_{typ}(q)$ and $\tau_{av}(q)$ as a
function of $q$. For $q<1/2$, we obtain the non-vanishing limiting spectrum
$\tau_{typ}(q)=\tau_{av}(q)=d(2q-1)$ as $V \to 0^+$. For $q>1/2$, this method
yields the same disorder-averaged spectrum $\tau_{av}(q)$ of order $O(V)$ as
obtained previously via the Levitov renormalization method by Mirlin and Evers
[Phys. Rev. B 62, 7920 (2000)]. In addition, it allows to compute explicitly
the typical spectrum, also of order $O(V)$, but with a different $q$-dependence
$\tau_{typ}(q) \ne \tau_{av}(q)$ for all $q>q_c=1/2$. As a consequence, we find
that the corresponding singularity spectra $f_{typ}(\alpha)$ and
$f_{av}(\alpha)$ differ even in the positive region $f>0$, and vanish at
different values $\alpha_+^{typ} > \alpha_+^{av}$, in contrast to the standard
picture. We also obtain that the saddle value $\alpha_{typ}(q)$ of the Legendre
transform reaches the termination point $\alpha_+^{typ}$ where
$f_{typ}(\alpha_+^{typ})=0$ only in the limit $q \to +\infty$.Comment: 13 pages, 2 figures, v2=final versio

### Random wetting transition on the Cayley tree : a disordered first-order transition with two correlation length exponents

We consider the random wetting transition on the Cayley tree, i.e. the
problem of a directed polymer on the Cayley tree in the presence of random
energies along the left-most bonds. In the pure case, there exists a
first-order transition between a localized phase and a delocalized phase, with
a correlation length exponent $\nu_{pure}=1$. In the disordered case, we find
that the transition remains first-order, but that there exists two diverging
length scales in the critical region : the typical correlation length diverges
with the exponent $\nu_{typ}=1$, whereas the averaged correlation length
diverges with the bigger exponent $\nu_{av}=2$ and governs the finite-size
scaling properties. We describe the relations with previously studied models
that are governed by the same "Infinite Disorder Fixed Point". For the present
model, where the order parameter is the contact density $\theta_L=l_a/L$
(defined as the ratio of the number $l_a$ of contacts over the total length
$L$), the notion of "infinite disorder fixed point" means that the thermal
fluctuations of $\theta_L$ within a given sample, become negligeable at large
scale with respect to sample-to-sample fluctuations. We characterize the
statistics over the samples of the free-energy and of the contact density. In
particular, exactly at criticality, we obtain that the contact density is not
self-averaging but remains distributed over the samples in the thermodynamic
limit, with the distribution ${\cal P}_{T_c}(\theta) = 1/(\pi \sqrt{\theta
(1-\theta)})$.Comment: 15 pages, 1 figur