arXiv subjects

Cecile Monthus

Publications and source records attributed to Cecile Monthus.

At least 127 records · Page 7Linked to original sources

arXiv2008

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.

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arXiv2008

Driven interfaces in random media at finite temperature : is there 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 $h_G(t) \sim V(F,T) t$ at large times, where the velocity vanishes according to the creep formula as $V(F,T) \sim e^{-K(T)/F^μ}$ for $F \to 0$. 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. Monthus and T. Garel, arxiv:0802.2502) that its dynamics for F=0 can be analyzed in terms of a strong disorder renormalization procedure, where the distribution of renormalized barriers flows towards some "infinite disorder fixed point". In the present paper, we obtain that for small $F$, this "infinite disorder fixed point" becomes a "strong disorder fixed point" with an exponential distribution of renormalized barriers. The corresponding distribution of trapping times then only decays as a power-law $P(τ) \sim 1/τ^{1+α}$, where the exponent $α(F,T)$ vanishes as $α(F,T) \propto F^μ$ as $F \to 0$. Our conclusion is that in the small force region $α(F,T)<1$, the divergence of the averaged trapping time $\barτ=+\infty$ induces strong non-self-averaging effects that invalidate the usual creep formula obtained by replacing all trapping times by the typical value. We find instead that the motion is only sub-linearly in time $h_G(t) \sim t^{α(F,T)}$, i.e. the asymptotic velocity vanishes V=0. This analysis is confirmed by numerical simulations of a directed polymer with a metric constraint driven in a traps landscape. We moreover obtain that the roughness exponent, which is governed by the equilibrium value $ζ_{eq}=2/3$ up to some large scale, becomes equal to $ζ=1$ at the largest scales.

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arXiv2008

Equilibrium of disordered systems : constructing the appropriate valleys in each sample via strong disorder renormalization in configuration space

To describe the equilibrium properties of disordered systems and the possible emergence of various 'phases' at low temperature, we adopt here the 'broken ergodicity' point of view advocated in particular by Palmer [Adv. Phys. 31, 669 (1982)] : the aim is then to construct the valleys of configurations that become separated by diverging barriers and to study their relative weights, as well as their internal properties. To characterize the slow non-equilibrium dynamics of disordered systems, we have recently introduced in [C. Monthus and T. Garel, J. Phys. A 41, 255002 (2008) and arxiv:0804.1847] a strong disorder renormalization procedure in configuration space, based on the iterative elimination of the smallest barrier remaining in the system. In the present paper, we show how this renormalization procedure allows to construct the longest-lived valleys in each disordered sample, and to obtain their free-energies, energies and entropies. This explicit RG formulation is very general since it can be defined for any master equation, and it gives new insights into the main ingredients of the droplet scaling picture. As an application, we have followed numerically the RG flow for the case of a directed polymer in a two-dimensional random medium to obtain histograms of the free-energy, entropy and energy differences between the two longest-lived valleys in each sample.

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arXiv2008

Critical behavior of interfaces in disordered Potts ferromagnets : statistics of free-energy, energy and interfacial adsorption

A convenient way to study phase transitions of finite spins systems of linear size $L$ is to fix boundary conditions that impose the presence of a system-size interface. In this paper, we study the statistical properties of such an interface in a disordered Potts ferromagnet in dimension $d=2$ within Migdal-Kadanoff real space renormalization. We first focus on the interface free-energy and energy to measure the singularities of the average and random contributions, as well as the corresponding histograms, both in the low-temperature phase and at criticality. We then consider the critical behavior of the interfacial adsorption of non-boundary states. Our main conclusion is that all singularities involve the correlation length $ξ_{av}(T) \sim (T_c-T)^{-ν}$ appearing in the average free-energy $\bar{F} \sim (L/ξ_{av}(T))^{d_s}$ of the interface of dimension $d_s=d-1$, except for the free-energy width $ΔF \sim (L/ξ_{var}(T))^θ$ that involves the droplet exponent $θ$ and another correlation length $ξ_{var}(T)$ which diverges more rapidly than $ξ_{av}(T)$. We compare with the spin-glass transition in $d=3$, where $ξ_{var}(T)$ is the 'true' correlation length, and where the interface energy presents unconventional scaling with a chaos critical exponent $ζ_c>1/ν$ [Nifle and Hilhorst, Phys. Rev. Lett. 68, 2992 (1992)]. The common feature is that in both cases, the characteristic length scale $L_{ch}(T)$ associated with the chaotic nature of the low-temperature phase, diverges more slowly than the correlation length.

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arXiv2007

Non-equilibrium dynamics of polymers and interfaces in random media : conjecture $ψ=d_s/2$ for the barrier exponent

We consider various random models (directed polymer, random ferromagnets, spin-glasses) in their disorder-dominated phases, where the free-energy cost $F(L)$ of an excitation of length $L$ presents fluctuations that grow as a power-law $ΔF(L) \sim L^θ$ with the 'droplet' exponent $θ$. Within the droplet theory, the energy and entropy of such excitations present fluctuations that grow as $ΔE(L) \sim ΔS(L) \sim L^{d_s/2}$ where $d_s$ is the dimension of the surface of the excitation. These systems usually present a positive 'chaos' exponent $ζ=d_s/2-θ>0$, meaning that the free-energy fluctuation of order $L^θ$ is a near-cancellation of much bigger energy and entropy fluctuations of order $L^{d_s/2}$. Within the standard droplet theory, the dynamics is characterized by a barrier exponent $ψ$ satisfying the bounds $θ\leq ψ\leq d-1$. In this paper, we argue that a natural value for this barrier exponent is $ψ=d_s/2$ : (i) for the directed polymer where $d_s=1$, this corresponds to $ψ=1/2$ in all dimensions; (ii) for disordered ferromagnets where $d_s=d-1$, this corresponds to $ψ=(d-1)/2$; (iii) for spin-glasses where interfaces have a non-trivial dimension $d_s$ known numerically, our conjecture $ψ=d_s/2$ gives numerical predictions in $d=2$ and $d=3$. We compare these values with the available numerical results for each case, in particular with the measure $ψ\simeq 0.49$ of Kolton, Rosso, Giamarchi, Phys. Rev. Lett. 95, 180604 (2005) for the non-equilibrium dynamics of a directed elastic string.

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arXiv2007

Disorder-dominated phases of random systems : relations between tails exponents and scaling exponents

We consider various random models (directed polymer, ferromagnetic random Potts model, Ising spin-glasses) in their disorder-dominated phases, where the free-energy cost $F(L)$ of an excitation of length $L$ present fluctuations that grow as a power-law $ΔF(L) \sim L^ω$ with the so-called droplet exponent $ω>0$. We study the tails of the probability distribution $Π(x)$ of the rescaled free-energy cost $x= \frac{F_L- \bar{F_L}}{L^ω}$, which are governed by two exponents $(η_-,η_+)$ defined by $\ln Π(x \to \pm \infty) \sim - | x |^{η_{\pm}}$. The aim of this paper is to establish simple relations between these tail exponents $(η_-,η_+) $ and the droplet exponent $ω$. We first prove these relations for disordered models on diamond hierarchical lattices where exact renormalizations exist for the probability distribution $Π(x)$. We then interpret these relations via an analysis of the measure of the rare disorder configurations governing the tails. Our conclusion is that these relations, when expressed in terms of the dimensions of the bulk and of the excitation surface are actually valid for general lattices.

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arXiv2007

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). In terms of the rescaled partition function $z=Z/Z_{typ}$,we find that the critical point corresponds to a fixed point distribution with a power-law tail $P_c(z) \sim Φ(\ln z)/z^{1+μ}$ as $z \to +\infty$ (up to some sub-leading logarithmic correction $Φ(\ln z)$), so that all moments $z^{n}$ with $n>μ$ diverge. For the wetting transition, the first moment diverges $\bar{z}=+\infty$ (case $0<μ<1$), and the critical temperature is strictly below the annealed temperature $T_c<T_{ann}$. For the directed polymer case, the second moment diverges $\bar{z^2}=+\infty$ (case $1<μ<2$), and the critical temperature is strictly below the exactly known transition temperature $T_2$ of the second moment.We then consider the correlation length exponent $ν$ :the linearized renormalization around the fixed point distribution coincides with the transfer matrix describing a directed polymer on the Cayley tree, but the random weights determined by the fixed point distribution $P_c(z)$ are broadly distributed. This induces some changes in the travelling wave solutions with respect to the usual case of more narrow distributions.

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arXiv2007

On the 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.W. Ludwig, Nucl. Phys. B 330, 639 (1990)) on the case of diluted two-dimensional Potts model, the moments $\bar{ρ^q(r)}$ of the local order parameter $ρ(r)$ scale with a set $x(q)$ of non-trivial exponents $x(q) \neq q x(1)$. In this paper, we revisit these ideas to incorporate more recent findings: (i) whenever a multifractal measure $w(r)$ normalized over space $ \sum_r w(r)=1$ occurs in a random system, it is crucial to distinguish between the typical values and the disorder averaged values of the generalized moments $Y_q =\sum_r w^q(r)$, since they may scale with different generalized dimensions $D(q)$ and $\tilde D(q)$ (ii) as discovered by Wiseman and Domany (S. Wiseman and E. Domany, Phys Rev E {\bf 52}, 3469 (1995)), the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponent $c=1.5$ (marginal disorder) and $c=1.75$ (relevant disorder). Finally, we argue that the presence of finite Griffiths ordered clusters at criticality determines the asymptotic value $x(q \to \infty) =d$ and the minimal value $ α_{min}=D(q \to \infty)=d-x(1) $ of the typical multifractal spectrum $f(α)$.

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arXiv2007

On the 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 (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. In this paper, we study the weight statistics at criticality via the entropy $S=-\sum w_i \ln w_i$ and the generalized moments $Y_k=\sum w_i^k$, where the $w_i$ are the Boltzmann weights of the $2^N$ configurations. In the REM, we find that the critical weight statistics is governed by the finite-size exponent $ν=2$ : the entropy scales as $\bar{S}_N(T_c) \sim N^{1/2}$, the typical values $e^{\bar{\ln Y_k}}$ decay as $N^{-k/2}$, and the disorder-averaged values $\bar{Y_k}$ are governed by rare events and decay as $N^{-1/2}$ for any $k>1$. For the DPCT, we find that the entropy scales similarly as $\bar{S}_N(T_c) \sim N^{1/2}$, whereas another exponent $ν'=1$ governs the $Y_k$ statistics : the typical values $e^{\bar{\ln Y_k}}$ decay as $N^{-k}$, the disorder-averaged values $\bar{Y_k}$ decay as $N^{-1}$ for any $k>1$. As a consequence, the asymptotic probability distribution $\barπ_{N=\infty}(q)$ of the overlap $q$, beside the delta function $δ(q)$ which bears the whole normalization, contains an isolated point at $q=1$, as a memory of the delta peak $(1-T/T_c) δ(q-1)$ of the low-temperature phase $T<T_c$. The associated value $\barπ_{N=\infty}(q=1)$ is finite for the DPCT, and diverges as $\barπ_{N=\infty}(q=1) \sim N^{1/2}$ for the REM.

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arXiv2007

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 $Π(Y_2)$ 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 $Π(Y_2)$ 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 $μ(T)$ that governs the main singularities $P_1(w) \sim (1-w)^{μ(T)-1}$ and $Π(Y_2) \sim (1-Y_2)^{μ(T)-1}$ as well as the power-law decay of the moments $ \bar{Y_k(i)} \sim 1/k^{μ(T)}$. The exponent $μ(T)$ grows from the value $μ(T=0)=0$ up to $μ(T_{gap}) \sim 2$. (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.

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arXiv2007

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 analyse the statistics of the position weights $w_L(\vec r)$ of the end-point of the polymer of length $L$ via the moments $Y_q(L) = \sum_{\vec r} [w_L(\vec r)]^q$. We measure the generalized exponents $τ(q)$ and $\tilde τ(q)$ governing the decay of the typical values $Y^{typ}_q(L) = e^{\bar{\ln Y_q(L)}} \sim L^{- τ(q)} $ and disorder-averaged values $\bar{Y_q(L)} \sim L^{- \tilde τ(q)} $ respectively. To understand the difference between these exponents, $ τ(q) \neq \tilde τ(q)$ above some threshold $q>q_c \sim 2$, we compute the probability distributions of $y=Y_q(L)/Y^{typ}_q(L) $ over the samples : we find that these distributions becomes scale invariant with a power-law tail $1/y^{1+x_q}$. These results thus correspond to the Ever-Mirlin scenario [Phys. Rev. Lett. 84, 3690 (2000)] for the statistics of Inverse Participation Ratios at the Anderson localization transitions. Finally, the finite-size scaling analysis in the critical region yields the correlation length exponent $ν\sim 2$.

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arXiv2006

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 analyse 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 $(j \neq i)$ of the sequence. We numerically compute the probability distributions $P_1(w)$ of the maximal weight, the probability distribution $Π(Y_2)$ of the parameter $Y_2(i)= \sum_j w_i^2(j)$, as well as the average values of the moments $Y_k(i)= \sum_j w_i^k(j)$. We find that there are two important temperatures $T_c T_{gap}$, the distribution $P_1(w)$ vanishes at some value $w_0(T)<1$, and accordingly the moments $\bar{Y_k(i)}$ decay exponentially in $k$. For $T<T_{gap}$, the distributions $P_1(w)$ and $Π(Y_2)$ present the characteristic Derrida-Flyvbjerg singularities at $w,Y_2=1/n$ for $n=1,2..$. In particular, there exists a temperature-dependent exponent $μ(T)$ that governs these singularities and the decay of the moments $ \bar{Y_k(i)} \sim 1/k^{μ(T)}$. The exponent $μ(T)$ grows from $μ(T=0)=0$ up to $μ(T_{gap}) \sim 2$. The study of spatial properties indicates that the critical temperature $T_c$ where the roughness exponent changes from the low temperature value $ζ\sim 0.67$ to the high temperature value $ζ\sim 0.5$ corresponds to the exponent $μ(T_c)=1$. For $T<T_c$, there exists frozen pairs of all sizes, whereas for $T_c< T <T_{gap}$, there exists frozen pairs, but only up to some characteristic length diverging as $ξ(T) \sim 1/(T_c-T)^ν$ with $ν\simeq 2$. The similarities and differences with the weight statistics in Lévy sums and in Derrida's Random Energy Model are discussed.

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arXiv2006

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 \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 $η(d)$ of the asymptotic behavior $\ln P_d (E_0) \sim - | E_0 |^{η(d)}$ as $E_0 \to -\infty$ is directly related to the fluctuation exponent $θ(d)$ (which governs the fluctuations $ΔE_0(L) \sim L^{θ(d)}$ of the ground state energy $E_0$ for polymers of length $L$) via the simple formula $η(d)=1/(1-θ(d))$. Along the paper, we comment on the similarities and differences with spin-glasses.

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arXiv2006

Numerical study of the directed polymer in a 1+3 dimensional random medium

The directed polymer in a 1+3 dimensional random medium is known to present a disorder-induced phase transition. For a polymer of length $L$, the high temperature phase is characterized by a diffusive behavior for the end-point displacement $R^2 \sim L$ and by free-energy fluctuations of order $ΔF(L) \sim O(1)$. The low-temperature phase is characterized by an anomalous wandering exponent $R^2/L \sim L^ω$ and by free-energy fluctuations of order $ΔF(L) \sim L^ω$ where $ω\sim 0.18$. In this paper, we first study the scaling behavior of various properties to localize the critical temperature $T_c$. Our results concerning $R^2/L$ and $ΔF(L)$ point towards $0.76 < T_c \leq T_2=0.79$, so our conclusion is that $T_c$ is equal or very close to the upper bound $T_2$ derived by Derrida and coworkers ($T_2$ corresponds to the temperature above which the ratio $\bar{Z_L^2}/(\bar{Z_L})^2$ remains finite as $L \to \infty$). We then present histograms for the free-energy, energy and entropy over disorder samples. For $T \gg T_c$, the free-energy distribution is found to be Gaussian. For $T \ll T_c$, the free-energy distribution coincides with the ground state energy distribution, in agreement with the zero-temperature fixed point picture. Moreover the entropy fluctuations are of order $ΔS \sim L^{1/2}$ and follow a Gaussian distribution, in agreement with the droplet predictions, where the free-energy term $ΔF \sim L^ω$ is a near cancellation of energy and entropy contributions of order $L^{1/2}$.

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arXiv2006

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

In dimension $d \geq 3$, the directed polymer in a random medium undergoes a phase transition between a free phase and a disorder dominated phase. For the latter, Fisher and Huse have proposed a droplet theory based on the scaling of the free energy fluctuations $ΔF(l) \sim l^θ$. On the other hand, in related growth models belonging to the KPZ universality class, Forrest and Tang have found that the height-height correlation function is logarithmic at the transition. For the directed polymer model at criticality, this translates into logarithmic free energy fluctuations $ΔF_{T_c}(l) \sim (\ln l)^σ$ with $σ=1/2$. In this paper, we propose a droplet scaling analysis exactly at criticality based on this logarithmic scaling. Our main conclusion is that the typical correlation length $ξ(T)$ of the low temperature phase, diverges as $ \ln ξ(T) \sim (- \ln (T_c-T))^{1/σ} \sim (- \ln (T_c-T))^{2} $. Furthermore, the logarithmic dependence of $ΔF_{T_c}(l)$ leads to the conclusion that the critical temperature $T_c$ actually coincides with the explicit upper bound $T_2$ derived by Derrida and coworkers, where $T_2$ corresponds to the temperature below which the ratio $\bar{Z_L^2}/(\bar{Z_L})^2$ diverges exponentially in $L$. Finally, since the Fisher-Huse droplet theory was initially introduced for the spin-glass phase, we briefly mention the similarities and differences with the directed polymer model. If one speculates that the free energy of droplet excitations for spin-glasses is also logarithmic at $T_c$, one obtains a logarithmic decay for the mean square correlation function at criticality $\bar{C^2(r)} \sim 1/(\ln r )^σ$.

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arXiv2006

Random polymers and delocalization transitions

In these proceedings, we first summarize some general properties of phase transitions in the presence of quenched disorder, with emphasis on the following points: the need to distinguish typical and averaged correlations, the possible existence of two correlation length exponents $ν$, the general bound $ν_{FS} \geq 2/d$, the lack of self-averaging of thermodynamic observables at criticality, the scaling properties of the distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples of size $L$. We then review our recent works on the critical properties of various delocalization transitions involving random polymers, namely (i) the bidimensional wetting (ii) the Poland-Scheraga model of DNA denaturation (iii) the depinning transition of the selective interface model (iv) the freezing transition of the directed polymer in a random medium.

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arXiv2006

Statistics of low energy excitations for the directed polymer in a $1+d$ random medium ($d=1,2,3$)

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 $ρ^{bulk}_L(E=0,l)$ and $ρ^{boundary}_L(E=0,l)$. We find that both densities follow the scaling behavior $ρ^{bulk,boundary}_L(E=0,l) = L^{-1-θ_d} R^{bulk,boundary}(x=l/L)$, where $θ_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value $θ_1=1/3$ in one dimension). In the limit $x=l/L \to 0$, both scaling functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x) \sim x^{-1-θ_d}$, leading to the droplet power law $ρ^{bulk,boundary}_L(E=0,l)\sim l^{-1-θ_d} $ in the regime $1 \ll l \ll L$. Beyond their common singularity near $x \to 0$, the two scaling functions $R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decays monotonically for $0<x<1$, the function $R^{boundary}(x)$ first decays for $0<x<x_{min}$, then grows for $x_{min}<x<1$, and finally presents a power law singularity $R^{boundary}(x)\sim (1-x)^{-σ_d}$ near $x \to 1$. The density of excitations of length $l=L$ accordingly decays as $ρ^{boundary}_L(E=0,l=L)\sim L^{- λ_d} $ where $λ_d=1+θ_d-σ_d$. We obtain $λ_1 \simeq 0.67$, $λ_2 \simeq 0.53$ and $λ_3 \simeq 0.39$, suggesting the possible relation $λ_d= 2 θ_d$.

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arXiv2006

Random walks and polymers in the presence of quenched disorder

After a general introduction to the field, we describe some recent results concerning disorder effects on both `random walk models', where the random walk is a dynamical process generated by local transition rules, and on `polymer models', where each random walk trajectory representing the configuration of a polymer chain is associated to a global Boltzmann weight. For random walk models, we explain, on the specific examples of the Sinai model and of the trap model, how disorder induces anomalous diffusion, aging behaviours and Golosov localization, and how these properties can be understood via a strong disorder renormalization approach. For polymer models, we discuss the critical properties of various delocalization transitions involving random polymers. We first summarize some recent progresses in the general theory of random critical points : thermodynamic observables are not self-averaging at criticality whenever disorder is relevant, and this lack of self-averaging is directly related to the probability distribution of pseudo-critical temperatures $T_c(i,L)$ over the ensemble of samples $(i)$ of size $L$. We describe the results of this analysis for the bidimensional wetting and for the Poland-Scheraga model of DNA denaturation.

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