We present a toy charge density wave (CDW) model in 1d exhibiting a depinning transition with threshold force and configurations that are explicit. Due to the periodic boundary conditions imposed, the threshold configuration has a set of topological defects whose location and number depend on the realization of the random phases. Approaching threshold, these defects are relocated by avalanches whose size dependence on the external driving force F is described by a recordbreaking process. We find that the depinning transition in this model is a critical phenomenon, with the cumulative avalanche size diverging near threshold as (F th − F ) −2 . The exact avalanche size distributions and their dependence on the control parameter (F th −F ) are obtained. Remarkably, the scaling exponents associated with the critical behavior depend on (1) the initial conditions and (2) the relationship between the system size and the pinning strength.

We discuss a one-dimensional, periodic charge density wave model, and a toy model introduced in Kaspar and Mungan (EPL 103:46002, 2013) which is intended to approximate the former in the case of strong pinning. For both systems an external force may be applied, driving it in one of two (±) directions, and we describe an avalanche algorithm producing an ordered sequence of static configurations leading to the depinning thresholds. For the toy model these threshold configurations are explicit functions of the underlying quenched disorder, as is the threshold-to-threshold evolution via iteration of the algorithm. These explicit descriptions are used to study the law of the random polarization P for the toy model in two cases. Evolving from a macroscopically flat initial state to threshold, we give a scaling limit characterization determines the final value of P . Evolving from (−)-threshold to (+)-threshold, we use an identification with record sequences to study P as a function of the difference between the current force F and the threshold force F th . The results presented are rigorous and give strong evidence that the depinning transition in the toy model is a dynamic critical phenomenon.

Abstract. In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions ρ to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe ρ(x, t) as a stochastic process in x with t fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of ρ(x, t) over 0 ≤ x ≤ L for L > 0, with a suitable random boundary condition at x = L.