We describe a directed avalanche model; a slowly unloading sandbox driven by lowering a retaining wall. The directness of the dynamics allows us to interpret the stable sand surfaces as world sheets of fluctuating interfaces in one lower dimension. In our specific case, the interface growth dynamics belongs to the Kardar-Parisi-Zhang (KPZ) universality class. We formulate relations between the ...

The KPZ fixed point is a 2d random field, conjectured to be the universal limiting fluctuation field for height function of models in universality class. Similarly, periodic spatially models. For both fields, their multi-point distributions space-time domain have been computed recently. We show that case narrow-wedge initial condition, these can expressed terms so-called integrable operators. t...

We explore linear control of the one-dimensional non-linear Kardar--Parisi--Zhang (KPZ) equation with goal to understand effects process has on dynamics and stationary state resulting stochastic growth kinetics. In control, intrinsic non-linearity system is maintained at all times. our protocol, applied only a small number $n_c$ Fourier modes. The stationary-state roughness obtained analyticall...

– We study the scaling regimes for the Kardar–Parisi–Zhang equation with noise correlator R(q) ∝ (1 + w q) in Fourier space, as a function of ρ and the spatial dimension d. By means of a stochastic Cole–Hopf transformation, the critical and correction-to-scaling exponents at the roughening transition are determined to all orders in a (d−dc) expansion. We also argue that there is a intriguing po...

These lecture notes grew out of a series of lectures given by the second named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main aim is to explain some aspects of the theory of “Regularity structures” developed recently by Hairer in [27]. This theory gives a way to study wellposedness for a class of stochastic PDEs that could not be treated previously. Prominent examples i...

This work considers the behavior of the height distributions of the equipotential lines in a region confined by two interfaces: a cathode with an irregular interface and a distant flat anode. Both boundaries, which are maintained at distinct and constant potential values, are assumed to be conductors. The morphology of the cathode interface results from the deposit of 2 × 10(4) monolayers that ...

In this paper I study a model for a growing surface in the presence of anomalous diffusion, also known as the fractal Kardar-Parisi-Zhang equation (FKPZ). This equation includes a fractional Laplacian that accounts for the possibility that surface transport is caused by a hopping mechanism of a Levy flight. It is shown that for a specific choice of parameters of the FKPZ equation, the equation ...