arXiv subjects

Pierre Le Doussal

Publications and source records attributed to Pierre Le Doussal.

At least 145 records · Page 8Linked to original sources

arXiv2016

Universality in the mean spatial shape of avalanches

Quantifying the universality of avalanche observables beyond critical exponents is of current great interest in theory and experiments. Here, we improve the characterization of the spatio-temporal process inside avalanches in the universality class of the depinning of elastic interfaces in random media. Surprisingly, at variance with the temporal shape, the spatial shape of avalanches has not yet been predicted. In part this is due to a lack of an analytically tractable definition: how should the shapes be centered? Here we introduce such a definition, accessible in experiments, and study the mean spatial shape of avalanches at fixed size centered around their starting point (seed). We calculate the associated universal scaling functions, both in a mean-field model and beyond. Notably, they are predicted to exhibit a cusp singularity near the seed. The results are in good agreement with a numerical simulation of an elastic line.

cond-mat.dis-nn↗

arXiv2016

Exact solution for a random walk in a time-dependent 1D random environment: the point-to-point Beta polymer

We consider the Beta polymer, an exactly solvable model of directed polymer on the square lattice, introduced by Barraquand and Corwin. We study the statistical properties of its point to point partition sum. The problem is equivalent to a model of a random walk in a time-dependent (and in general biased) 1D random environment. In this formulation, we study the sample to sample fluctuations of the transition probability distribution function (PDF) of the random walk. Using the Bethe ansatz we obtain exact formulas for the integer moments, and Fredholm determinant formulas for the Laplace transform of the directed polymer partition sum/random walk transition probability. The asymptotic analysis of these formulas at large time $t$ is performed both (i) in a diffusive vicinity, $x \sim t^{1/2}$, of the optimal direction (in space-time) chosen by the random walk, where the fluctuations of the PDF are found to be Gamma distributed; (ii) in the large deviations regime, $x \sim t$, of the random walk, where the fluctuations of the logarithm of the PDF are found to grow with time as $t^{1/3}$ and to be distributed according to the Tracy-Widom GUE distribution. Our exact results complement those of BC for the cumulative distribution function of the random walk in regime (ii), and in regime (i) they unveil a novel fluctuation behavior. We also discuss the crossover regime between (i) and (ii), identified as $x \sim t^{3/4}$. Our results are confronted to extensive numerical simulations of the model.

cond-mat.dis-nn↗

arXiv2016

The crossing probability for directed polymers in random media

We study the probability that two directed polymers in the same random potential do not intersect. We use the replica method to map the problem onto the attractive Lieb-Liniger model with generalized statistics between particles. We obtain analytical expressions for the first few moments of this probability, and compare them to a numerical simulation of a discrete model at high-temperature. From these observations, several large time properties of the non-crossing probabilities are conjectured. Extensions of our formalism to more general observables are discussed.

cond-mat.dis-nn↗

arXiv2016

Joint min-max distribution and Edwards-Anderson's order parameter of the circular $1 / f$-noise model

We calculate the joint min--max distribution and the Edwards-Anderson's order parameter for the circular model of $1 / f$-noise. Both quantities, as well as generalisations, are obtained exactly by combining the freezing-duality conjecture and Jack-polynomial techniques. Numerical checks come with significantly improved control of finite-size effects in the glassy phase, and the results convincingly validate the freezing-duality conjecture. Application to diffusive dynamics is discussed. We also provide a formula for the pre-factor ratio of the joint/marginal Carpentier-Le Doussal tail for minimum/maximum which applies to any logarithmic random energy model.

cond-mat.stat-mech↗

arXiv2016

Large deviations for the height in 1D Kardar-Parisi-Zhang growth at late times

We study the atypically large deviations of the height $H \sim {\cal O}(t)$ at the origin at late times in $1+1$-dimensional growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. We present exact results for the rate functions for the discrete single step growth model, as well as for the continuum KPZ equation in a droplet geometry. Based on our exact calculation of the rate functions we argue that models in the KPZ class undergo a third order phase transition from a strong coupling to a weak coupling phase, at late times.

cond-mat.stat-mech↗

arXiv2016

Distribution of joint local and total size and of extension for avalanches in the Brownian force model

The Brownian force model (BFM) is a mean-field model for the local velocities during avalanches in elastic interfaces of internal space dimension $d$, driven in a random medium. It is exactly solvable via a non-linear differential equation. We study avalanches following a kick, i.e. a step in the driving force. We first recall the calculation of the distributions of the global size (total swept area) and of the local jump size for an arbitrary kick amplitude. We extend this calculation to the joint density of local and global sizes within a single avalanche, in the limit of an infinitesimal kick. When the interface is driven by a single point we find new exponents $τ_0=5/3$ and $τ=7/4$, depending on whether the force or the displacement is imposed. We show that the extension of a single avalanche along one internal direction (i.e. the total length in $d=1$) is finite and we calculate its distribution, following either a local or a global kick. In all cases it exhibits a divergence $P(\ell) \sim \ell^{-3}$ at small $\ell$. Most of our results are tested in a numerical simulation in dimension $d=1$.

cond-mat.dis-nn↗

arXiv2016

Universal ground state properties of free fermions in a $d$-dimensional trap

The ground state properties of $N$ spinless free fermions in a $d$-dimensional confining potential are studied. We find that any $n$-point correlation function has a simple determinantal structure that allows us to compute several properties exactly for large $N$. We show that the average density has a finite support with an edge, and near this edge the density exhibits a universal (valid for a wide class of potentials) scaling behavior for large $N$. The associated edge scaling function is computed exactly and generalizes to any $d$ the edge electron gas result of Kohn and Mattsson in $d=3$ [Phys. Rev. Lett. 81, 3487 (1998)]. In addition, we calculate the kernel (that characterizes any $n$-point correlation function) for large $N$ and show that, when appropriately scaled, it depends only on dimension $d$, but has otherwise universal scaling forms, at the edges. The edge kernel, for higher $d$, generalizes the Airy kernel in one dimension, well known from random matrix theory.

cond-mat.stat-mech↗

arXiv2016

On integrable directed polymer models on the square lattice

In a recent work Povolotsky provided a three-parameter family of stochastic particle systems with zero-range interactions in one dimension which are integrable by coordinate Bethe ansatz. Using these results we obtain the corresponding condition for integrability of a class of directed polymer models with random weights on the square lattice. Analyzing the solutions we find, besides known cases, a new two-parameter family of integrable DP model, which we call the Inverse-Beta polymer, and provide its Bethe ansatz solution.

cond-mat.dis-nn↗

arXiv2015

Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time

We consider the system of $N$ one-dimensional free fermions confined by a harmonic well $V(x) = mω^2 {x^2}/{2}$ at finite inverse temperature $β= 1/T$. The average density of fermions $ρ_N(x,T)$ at position $x$ is derived. For $N \gg 1$ and $β\sim {\cal O}(1/N)$, $ρ_N(x,T)$ is given by a scaling function interpolating between a Gaussian at high temperature, for $β\ll 1/N$, and the Wigner semi-circle law at low temperature, for $β\gg N^{-1}$. In the latter regime, we unveil a scaling limit, for $β{\hbar ω}= b N^{-1/3}$, where the fluctuations close to the edge of the support, at $x \sim \pm \sqrt{2\hbar N/(mω)}$, are described by a limiting kernel $K^{\rm ff}_b(s,s')$ that depends continuously on $b$ and is a generalization of the Airy kernel, found in the Gaussian Unitary Ensemble of random matrices. Remarkably, exactly the same kernel $K^{\rm ff}_b(s,s')$ arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time $t$, with the correspondence $t= b^3$.

cond-mat.stat-mech↗

arXiv2015

Crossing probability for directed polymers in random media: exact tail of the distribution

We study the probability $p \equiv p_η(t)$ that two directed polymers in a given random potential $η$ and with fixed and nearby endpoints, do not cross until time $t$. This probability is itself a random variable (over samples $η$) which, as we show, acquires a very broad probability distribution at large time. In particular the moments of $p$ are found to be dominated by atypical samples where $p$ is of order unity. Building on a formula established by us in a previous work using nested Bethe Ansatz and Macdonald process methods, we obtain analytically the leading large time behavior of {\it all moments} $\overline{p^m}\simeq γ_m/t$. From this, we extract the exact tail $\sim ρ(p)/t$ of the probability distribution of the non-crossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.

cond-mat.stat-mech↗

arXiv2015

Spatial shape of avalanches in the Brownian force model

We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.

cond-mat.dis-nn↗

arXiv2015

Experimental evidence for three universality classes for reaction fronts in disordered flows

Self-sustained reaction fronts in a disordered medium subject to an external flow display self-affine roughening, pinning and depinning transitions. We measure spatial and temporal fluctuations of the front in $1+1$ dimensions, controlled by a single parameter, the mean flow velocity. Three distinct universality classes are observed, consistent with the Kardar-Parisi-Zhang (KPZ) class for fast advancing or receding fronts, the quenched KPZ class (positive-qKPZ) when the mean flow approximately cancels the reaction rate, and the negative-qKPZ class for slowly receding fronts. Both quenched KPZ classes exhibit distinct depinning transitions, in agreement with the theory.

cond-mat.stat-mech↗

arXiv2014

Random matrices and entanglement entropy of trapped Fermi gases

We exploit and clarify the use of random matrix theory for the calculation of the entanglement entropy of free Fermi gases. We apply this method to obtain analytic predictions for Renyi entanglement entropies of a one-dimensional gas trapped by a harmonic potential in all the relevant scaling regimes. We confirm our findings with accurate numerical calculations obtained by means of an ingenious discretisation of the reduced correlation matrix.

cond-mat.stat-mech↗

arXiv2014

Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz

We study the model of a discrete directed polymer (DP) on the square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen. The integer moments of the partition sum, $\overline{Z^n}$, are studied using a transfer matrix formulation, which appears as a generalization of the Lieb-Liniger quantum mechanics of bosons to discrete time and space. In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet. Using the Brunet-Bethe eigenstates we obtain an exact expression for the integer moments of $\overline{Z^n}$ for polymers of arbitrary lengths and fixed endpoint positions. Although these moments do not exist for all integer n, we are nevertheless able to construct a generating function which reproduces all existing integer moments, and which takes the form of a Fredholm determinant (FD). This suggests an analytic continuation via a Mellin-Barnes transform and we thereby propose a FD ansatz representation for the probability distribution function (PDF) of $Z$ and its Laplace transform. In the limit of very long DP, this ansatz yields that the distribution of the free energy converges to the GUE Tracy-Widom distribution up to a non-trivial average and variance that we calculate. Our asymptotic predictions coincide with a result by Borodin et al. based on a formula obtained by Seppalainen using the gRSK correspondence. In addition we obtain the dependence on the endpoint position and the exact elastic coefficient at large time. We argue the equivalence between our formula and the one of Borodin et al. As we discuss, this open the way to explore the connections between quantum integrability and tropical geometry.

cond-mat.dis-nn↗

arXiv2014

Revisiting Directed Polymers with heavy-tailed disorder

In this mostly numerical study, we revisit the statistical properties of the ground state of a directed polymer in a $d=1+1$ "hilly" disorder landscape, i.e. when the quenched disorder has power-law tails. When disorder is Gaussian, the polymer minimizes its total energy through a collective optimization, where the energy of each visited site only weakly contributes to the total. Conversely, a hilly landscape forces the polymer to distort and explore a larger portion of space to reach some particularly deep energy sites. As soon as the fifth moment of the disorder diverges, this mechanism radically changes the standard "KPZ" scaling behaviour of the directed polymer, and new exponents prevail. After confirming again that the Flory argument accurately predicts these exponent in the tail-dominated phase, we investigate several other statistical features of the ground state that shed light on this unusual transition and on the accuracy of the Flory argument. We underline the theoretical challenge posed by this situation, which paradoxically becomes even more acute above the upper critical dimension.

cond-mat.dis-nn↗

arXiv2014

An exact mapping of the stochastic field theory for Manna sandpiles to interfaces in random media

We show that the stochastic field theory for directed percolation in presence of an additional conservation law (the C-DP class) can be mapped exactly to the continuum theory for the depinning of an elastic interface in short-range correlated quenched disorder. On one line of parameters commonly studied, this mapping leads to the simplest overdamped dynamics. Away from this line, an additional memory term arises in the interface dynamics; we argue that it does not change the universality class. Since C-DP is believed to describe the Manna class of self-organized criticality, this shows that Manna stochastic sandpiles and disordered elastic interfaces (i.e. the quenched Edwards-Wilkinson model) share the same universal large-scale behavior.

cond-mat.dis-nn↗

arXiv2014

Superfluid/Bose-glass transition in one dimension

We consider a one-dimensional system of interacting bosons in a random potential. At zero temperature, it can be either in the superfluid or in the insulating phase. We study the transition at weak disorder and moderate interaction. Using a systematic approach, we derive the renormalization group equations at two-loop order and discuss the phase diagram. We find the universal form of the correlation functions at the transitions and compute the logarithmic corrections to the main universal power-law behavior. In order to mimic large density fluctuations on a single site, we study a simplified model of disordered two-leg bosonic ladders with correlated disorder across the rung. Contrarily to the single-chain case, the latter system exhibits a transition between a superfluid and a localized phase where the exponents of the correlation functions at the transition do not take universal values.

cond-mat.quant-gas↗

arXiv2014

Avalanche shape and exponents beyond mean-field theory

Elastic systems, such as magnetic domain walls, density waves, contact lines, and cracks, are all pinned by substrate disorder. When driven, they move via successive jumps called avalanches, with power law distributions of size, duration and velocity. Their exponents, and the shape of an avalanche, defined as its mean velocity as function of time, have recently been studied. They are known approximatively from experiments and simulations, and were predicted from mean-field models, such as the Brownian force model (BFM), where each point of the elastic interface sees a force field which itself is a random walk. As we showed in EPL 97 (2012) 46004, the BFM is the starting point for an $ε= d_{\rm c}-d$ expansion around the upper critical dimension, with $d_{\rm c}=4$ for short-ranged elasticity, and $d_{\rm c}=2$ for long-ranged elasticity. Here we calculate analytically the ${\cal O}(ε)$, i.e. 1-loop, correction to the avalanche shape at fixed duration $T$, for both types of elasticity. The exact expression is well approximated by $\left< \dot u(t=x T)\right>_T\simeq [ Tx(1-x)]^{γ-1} \exp\left( {\cal A}\left[\frac12-x\right]\right)$, $0<x<1$. The asymmetry ${\cal A}\approx - 0.336 (1-d/d_{\rm c})$ is negative for $d$ close to $d_{\rm c}$, skewing the avalanche towards its end, as observed in numerical simulations in $d=2$ and $3$. The exponent $γ=(d+ζ)/z$ is given by the two independent exponents at depinning, the roughness $ζ$ and the dynamical exponent $z$. We propose a general procedure to predict other avalanche exponents in terms of $ζ$ and $z$. We finally introduce and calculate the shape at fixed avalanche size, not yet measured in experiments or simulations.

cond-mat.dis-nn↗