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.
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