Experimental realizations of a one-dimensional (1D) interface always exhibit a finite microscopic width ξ > 0; its influence is erased by thermal fluctuations at sufficiently high temperatures, but turns out to be a crucial ingredient for the description of the interface fluctuations below a characteristic temperature T c (ξ ). Exploiting the exact mapping between the static 1D interface and a 1 + 1 directed polymer (DP) growing in a continuous space, we study analytically both the free-energy and geometrical fluctuations of a DP, at finite temperature T , with a short-range elasticity and submitted to a quenched random-bond Gaussian disorder of finite correlation length ξ . We derive the exact time-evolution equations of the disorder free energyF (t,y), which encodes the microscopic disorder integrated by the DP up to a growing time t and an endpoint position y, its derivative η(t,y), and their respective two-point correlatorsC(t,y) andR(t,y). We compute the exact solution of its linearized evolution R lin (t,y) and we combine its qualitative behavior and the asymptotic properties known for an uncorrelated disorder (ξ = 0) to justify the construction of a "toy model" leading to a simple description of the DP properties. This model is characterized by Gaussian Brownian-type free-energy fluctuations, correlated at small |y| ξ , and of amplitude D ∞ (T ,ξ). We present an extended scaling analysis of the roughness, supported by saddle-point arguments on its path-integral representation, which predicts D ∞ ∼ 1/T at high temperatures and D ∞ ∼ 1/T c (ξ ) at low temperatures. We identify the connection between the temperature-induced crossover of D ∞ (T ,ξ) and the full replica symmetry breaking in previous Gaussian variational method (GVM) computations. In order to refine our toy model with respect to finite-time geometrical fluctuations, we propose an effective time-dependent amplitude D t . Finally, we discuss the consequences of the low-temperature regime for two experimental realizations of Kardar-Parisi-Zhang interfaces, namely, the static and quasistatic behavior of magnetic domain walls and the high-velocity steady-state dynamics of interfaces in liquid crystals.

We briefly introduce the generic framework of Disordered Elastic Systems (DES), giving a short 'recipe' of a DES modeling and presenting the quantities of interest in order to probe the static and dynamical disorder-induced properties of such systems. We then focus on a particular low-dimensional DES, namely the one-dimensional interface in shortranged elasticity and short-ranged quenched disorder. Illustrating different elements given in the introductory sections, we discuss specifically the consequences of the interplay between a finite temperature T > 0 and a finite interface width ξ > 0 on the static geometrical fluctuations at different lengthscales, and the implications on the quasistatic dynamics.

We consider the Langevin dynamics of a many-body system of interacting particles in d dimensions, in a very general setting suitable to model several out-of-equilibrium situations, such as liquid and glass rheology, active self-propelled particles, and glassy aging dynamics. The pair interaction potential is generic, and can be chosen to model colloids, atomic liquids, and granular materials. In the limit d → ∞, we show that the dynamics can be exactly reduced to a single one-dimensional effective stochastic equation, with an effective thermal bath described by kernels that have to be determined self-consistently. We present two complementary derivations, via a dynamical cavity method and via a path-integral approach. From the effective stochastic equation, one can compute dynamical observables such as pressure, shear stress, particle mean-square displacement, and the associated response function. As an application of our results, we derive dynamically the 'state-following' equations that describe the response of a glass to quasistatic perturbations, thus bypassing the use of replicas. The article is written in a modular way, that allows the reader to skip the details of the derivations and focus on the physical setting and the main results.

At finite temperature and in presence of disorder, a one-dimensional elastic interface displays different scaling regimes at small and large lengthscales. Using a replica approach and a Gaussian variational method ͑GVM͒, we explore the consequences of a finite interface width on the small-lengthscale fluctuations. We compute analytically the static roughness B͑r͒ of the interface as a function of the distance r between two points on the interface. We focus on the case of short-range elasticity and random-bond disorder. We show that for a finite width two temperature regimes exist. At low temperature, the expected thermal and randommanifold regimes, respectively, for small and large scales, connect via an intermediate "modified" Larkin regime, that we determine. This regime ends at a temperature-independent characteristic "Larkin" length. Above a certain characteristic temperature that we identify, this intermediate regime disappears. The thermal and random-manifold regimes connect at a single crossover lengthscale, that we compute. This is also the expected behavior for zero width. Using a directed polymer description, we also study via a second GVM procedure and generic scaling arguments, a modified toy model that provides further insights on this crossover. We discuss the relevance of the two GVM procedures for the roughness at large lengthscale in those regimes. In particular, we analyze the scaling of the temperature-dependent prefactor in the roughness B͑r͒ϳT 2þ r 2 and its corresponding thorn exponent þ. We briefly discuss the consequences of those results for the quasistatic creep law of a driven interface, in connection with previous experimental and numerical studies.

We show that, at least at a mean-field level, the effect of structural disorder in sheared amorphous media is very dissimilar depending on the thermal or athermal nature of their underlying dynamics. We first introduce a toy model, including explicitly two types of noise (thermal versus athermal). Within this interpretation framework, we argue that mean-field athermal dynamics can be accounted for by the so-called Hébraud-Lequeux (HL) model, in which the mechanical noise stems explicitly from the plastic activity in the sheared medium. Then, we show that the inclusion of structural disorder, by means of a distribution of yield energy barriers, has no qualitative effect in the HL model, while such a disorder is known to be one of the key ingredients leading kinematically to a finite macroscopic yield stress in other mean-field descriptions, such as the Soft-Glassy-Rheology model. We conclude that the statistical mechanisms at play in the emergence of a macroscopic yield stress, and a complex stationary dynamics at low shear rate, are different in thermal and athermal amorphous systems.