Comment on ``Roughening Transition of Interfaces in Disordered Media''

Emig and Nattermann (Phys. Rev. Lett. 81, 1469 (1998)) have recently investigated the competition between lattice pinning and impurity pinning using a Renormalisation Group (RG) approach. For elastic objects of internal dimensions $2 < D < 4$, they find, at zero temperature, an interesting second order phase transition between a flat phase for small disorder and a rough phase for large disorder. These results contrast with those obtained using the replica variational approach for the same problem, where a first order transition between flat and rough phases was predicted. In this comment, we show that these results can be reconciled by analysing the RG flow for an arbitrary dimension $N$ for the displacement field.Comment: Submitted to Phys. Rev. Let

Anomalous price impact and the critical nature of liquidity in financial markets

We propose a dynamical theory of market liquidity that predicts that the average supply/demand profile is V-shaped and {\it vanishes} around the current price. This result is generic, and only relies on mild assumptions about the order flow and on the fact that prices are (to a first approximation) diffusive. This naturally accounts for two striking stylized facts: first, large metaorders have to be fragmented in order to be digested by the liquidity funnel, leading to long-memory in the sign of the order flow. Second, the anomalously small local liquidity induces a breakdown of linear response and a diverging impact of small orders, explaining the "square-root" impact law, for which we provide additional empirical support. Finally, we test our arguments quantitatively using a numerical model of order flow based on the same minimal ingredients.Comment: 16 pages, 7 figure

Evidence of Deep Water Penetration in Silica during Stress Corrosion Fracture

We measure the thickness of the heavy water layer trapped under the stress corrosion fracture surface of silica using neutron reflectivity experiments. We show that the penetration depth is 65–85 Å, suggesting the presence of a damaged zone of ~100 Å extending ahead of the crack tip during its propagation. This estimate of the size of the damaged zone is compatible with other recent results

MOBILITY IN A ONE-DIMENSIONAL DISORDER POTENTIAL

In this article the one-dimensional, overdamped motion of a classical particle is considered, which is coupled to a thermal bath and is drifting in a quenched disorder potential. The mobility of the particle is examined as a function of temperature and driving force acting on the particle. A framework is presented, which reveals the dependence of mobility on spatial correlations of the disorder potential. Mobility is then calculated explicitly for new models of disorder, in particular with spatial correlations. It exhibits interesting dynamical phenomena. Most markedly, the temperature dependence of mobility may deviate qualitatively from Arrhenius formula and a localization transition from zero to finite mobility may occur at finite temperature. Examples show a suppression of this transition by disorder correlations.Comment: 10 pages, latex, with 3 figures, to be published in Z. Phys.

Rejuvenation in the Random Energy Model

We show that the Random Energy Model has interesting rejuvenation properties in its frozen phase. Different `susceptibilities' to temperature changes, for the free-energy and for other (`magnetic') observables, can be computed exactly. These susceptibilities diverge at the transition temperature, as (1-T/T_c)^-3 for the free-energy.Comment: 9 pages, 1 eps figur

Multiple scaling regimes in simple aging models

We investigate aging in glassy systems based on a simple model, where a point in configuration space performs thermally activated jumps between the minima of a random energy landscape. The model allows us to show explicitly a subaging behavior and multiple scaling regimes for the correlation function. Both the exponents characterizing the scaling of the different relaxation times with the waiting time and those characterizing the asymptotic decay of the scaling functions are obtained analytically by invoking a `partial equilibrium' concept.Comment: 4 pages, 3 figure

Energy exponents and corrections to scaling in Ising spin glasses

We study the probability distribution P(E) of the ground state energy E in various Ising spin glasses. In most models, P(E) seems to become Gaussian with a variance growing as the system's volume V. Exceptions include the Sherrington-Kirkpatrick model (where the variance grows more slowly, perhaps as the square root of the volume), and mean field diluted spin glasses having +/-J couplings. We also find that the corrections to the extensive part of the disorder averaged energy grow as a power of the system size; for finite dimensional lattices, this exponent is equal, within numerical precision, to the domain-wall exponent theta_DW. We also show how a systematic expansion of theta_DW in powers of exp(-d) can be obtained for Migdal-Kadanoff lattices. Some physical arguments are given to rationalize our findings.Comment: 12 pages, RevTex, 9 figure

Dynamical ultrametricity in the critical trap model

We show that the trap model at its critical temperature presents dynamical ultrametricity in the sense of Cugliandolo and Kurchan [CuKu94]. We use the explicit analytic solution of this model to discuss several issues that arise in the context of mean-field glassy dynamics, such as the scaling form of the correlation function, and the finite time (or finite forcing) corrections to ultrametricity, that are found to decay only logarithmically with the associated time scale, as well as the fluctuation dissipation ratio. We also argue that in the multilevel trap model, the short time dynamics is dominated by the level which is at its critical temperature, so that dynamical ultrametricity should hold in the whole glassy temperature range. We revisit some experimental data on spin-glasses in light of these results.Comment: 7 pages, 4 .eps figures. submitted to J. Phys.

Statistical properties of stock order books: empirical results and models

We investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse. The results are to a large degree independent of the stock studied. The most interesting features concern (i) the statistics of incoming limit order prices, which follows a power-law around the current price with a diverging mean; and (ii) the humped shape of the average order book, which can be quantitatively reproduced using a `zero intelligence' numerical model, and qualitatively predicted using a simple approximation.Comment: Revised version, 10 pages, 4 .eps figures. to appear in Quantitative Financ