Finite temperature Functional RG, droplets and decaying Burgers Turbulence

The functional RG (FRG) approach to pinning of $d$-dimensional manifolds is reexamined at any temperature $T$. A simple relation between the coupling function $R(u)$ and a physical observable is shown in any $d$. In $d=0$ its beta function is displayed to a high order, ambiguities resolved; for random field disorder (Sinai model) we obtain exactly the T=0 fixed point $R(u)$ as well as its thermal boundary layer (TBL) form (i.e. for $u \sim T$) at $T>0$. Connection between FRG in $d=0$ and decaying Burgers is discussed. An exact solution to the functional RG hierarchy in the TBL is obtained for any $d$ and related to droplet probabilities.Comment: 8 pages 1 figur

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

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

Disorder induced transitions in layered Coulomb gases and application to flux lattices in superconductors

A layered system of charges with logarithmic interaction parallel to the layers and random dipoles in each layer is studied via a variational method and an energy rationale. These methods reproduce the known phase diagram for a single layer where charges unbind by increasing either temperature or disorder, as well as a freezing first order transition within the ordered phase. Increasing interlayer coupling leads to successive transitions in which charge rods correlated in N>1 neighboring layers are unbounded by weaker disorder. Increasing disorder leads to transitions between different N phases. The method is applied to flux lattices in layered superconductors in the limit of vanishing Josephson coupling. The unbinding charges are point defects in the flux lattice, i.e. vacancies or interstitials. We show that short range disorder generates random dipoles for these defects. We predict and accurately locate a disorder-induced defect-unbinding transition with loss of superconducting order, upon increase of disorder. While N=1 charges dominate for most system parameters, we propose that in multi-layer superconductors defect rods can be realized.Comment: 26 pages, 6 figure

SLE on doubly-connected domains and the winding of loop-erased random walks

Two-dimensional loop-erased random walks (LERWs) are random planar curves whose scaling limit is known to be a Schramm-Loewner evolution SLE_k with parameter k = 2. In this note, some properties of an SLE_k trace on doubly-connected domains are studied and a connection to passive scalar diffusion in a Burgers flow is emphasised. In particular, the endpoint probability distribution and winding probabilities for SLE_2 on a cylinder, starting from one boundary component and stopped when hitting the other, are found. A relation of the result to conditioned one-dimensional Brownian motion is pointed out. Moreover, this result permits to study the statistics of the winding number for SLE_2 with fixed endpoints. A solution for the endpoint distribution of SLE_4 on the cylinder is obtained and a relation to reflected Brownian motion pointed out.Comment: 22 pages, 4 figure