Distribution of velocities in an avalanche

For a driven elastic object near depinning, we derive from first principles the distribution of instantaneous velocities in an avalanche. We prove that above the upper critical dimension, d >= d_uc, the n-times distribution of the center-of-mass velocity is equivalent to the prediction from the ABBM stochastic equation. Our method allows to compute space and time dependence from an instanton equation. We extend the calculation beyond mean field, to lowest order in epsilon=d_uc-d.Comment: 4 pages, 2 figure

Height fluctuations of a contact line: a direct measurement of the renormalized disorder correlator

We have measured the center-of-mass fluctuations of the height of a contact line at depinning for two different systems: liquid hydrogen on a rough cesium substrate and isopropanol on a silicon wafer grafted with silanized patches. The contact line is subject to a confining quadratic well, provided by gravity. From the second cumulant of the height fluctuations, we measure the renormalized disorder correlator Delta(u), predicted by the Functional RG theory to attain a fixed point, as soon as the capillary length is large compared to the Larkin length set by the microscopic disorder. The experiments are consistent with the asymptotic form for Delta(u) predicted by Functional RG, including a linear cusp at u=0. The observed small deviations could be used as a probe of the underlying physical processes. The third moment, as well as avalanche-size distributions are measured and compared to predictions from Functional RG.Comment: 6 pages, 14 figure

Shock statistics in higher-dimensional Burgers turbulence

We conjecture the exact shock statistics in the inviscid decaying Burgers equation in D>1 dimensions, with a special class of correlated initial velocities, which reduce to Brownian for D=1. The prediction is based on a field-theory argument, and receives support from our numerical calculations. We find that, along any given direction, shocks sizes and locations are uncorrelated.Comment: 4 pages, 8 figure

Avalanches in mean-field models and the Barkhausen noise in spin-glasses

We obtain a general formula for the distribution of sizes of "static avalanches", or shocks, in generic mean-field glasses with replica-symmetry-breaking saddle points. For the Sherrington-Kirkpatrick (SK) spin-glass it yields the density rho(S) of the sizes of magnetization jumps S along the equilibrium magnetization curve at zero temperature. Continuous replica-symmetry breaking allows for a power-law behavior rho(S) ~ 1/(S)^tau with exponent tau=1 for SK, related to the criticality (marginal stability) of the spin-glass phase. All scales of the ultrametric phase space are implicated in jump events. Similar results are obtained for the sizes S of static jumps of pinned elastic systems, or of shocks in Burgers turbulence in large dimension. In all cases with a one-step solution, rho(S) ~ S exp(-A S^2). A simple interpretation relating droplets to shocks, and a scaling theory for the equilibrium analog of Barkhausen noise in finite-dimensional spin glasses are discussed.Comment: 6 pages, 1 figur

Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities

We reveal a phase transition with decreasing viscosity $\nu$ at \nu=\nu_c>0 in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities \sim|x-x'|^{-2}. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities, continuously dependent on \nu, reflecting a spontaneous one step replica symmetry breaking (RSB) in the associated statistical mechanics problem. We obtain the low orders cumulants analytically. Our results, which are checked numerically, are based on combining insights in the mechanism of the freezing transition in random logarithmic potentials with an extension of duality relations discovered recently in Random Matrix Theory. They are essentially non mean-field in nature as also demonstrated by the shock size distribution computed numerically and different from the short range correlated Kida model, itself well described by a mean field one step RSB ansatz. We also provide some insights for the finite viscosity behaviour of velocities in the latter model.Comment: Published version, essentially restructured & misprints corrected. 6 pages, 5 figure

Free-energy distribution of the directed polymer at high temperature

We study the directed polymer of length $t$ in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods. The universal regime of high temperature $T$ is described, upon scaling 'time' $t \sim T^5/\kappa$ and space $x = T^3/\kappa$ (with $\kappa=T$ for the discrete model) by a continuum model with $\delta$-function disorder correlation. Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function. The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics. We then obtain the exact expression at any time for the generating function of the free energy distribution, in terms of a Fredholm determinant. At large time we find that it crosses over to the Tracy Widom distribution (TW) which describes the fixed $T$ infinite $t$ limit. The exact free energy distribution is obtained for any time and compared with very recent results on growth and exclusion models.Comment: 6 pages, 3 figures large time limit corrected and convergence to Tracy Widom established, 1 figure changed

Interacting crumpled manifolds

In this article we study the effect of a delta-interaction on a polymerized membrane of arbitrary internal dimension D. Depending on the dimensionality of membrane and embedding space, different physical scenarios are observed. We emphasize on the difference of polymers from membranes. For the latter, non-trivial contributions appear at the 2-loop level. We also exploit a ``massive scheme'' inspired by calculations in fixed dimensions for scalar field theories. Despite the fact that these calculations are only amenable numerically, we found that in the limit of D to 2 each diagram can be evaluated analytically. This property extends in fact to any order in perturbation theory, allowing for a summation of all orders. This is a novel and quite surprising result. Finally, an attempt to go beyond D=2 is presented. Applications to the case of self-avoiding membranes are mentioned

Glassy trapping of manifolds in nonpotential random flows

We study the dynamics of polymers and elastic manifolds in non potential static random flows. We find that barriers are generated from combined effects of elasticity, disorder and thermal fluctuations. This leads to glassy trapping even in pure barrier-free divergenceless flows $v {f \to 0}{\sim} f^\phi$ ($\phi > 1$). The physics is described by a new RG fixed point at finite temperature. We compute the anomalous roughness $R \sim L^\zeta$ and dynamical $t\sim L^z$ exponents for directed and isotropic manifolds.Comment: 5 pages, 3 figures, RevTe