Crossover scaling functions in the asymmetric avalanche process

TL;DR

This paper analyzes the particle current in the asymmetric avalanche process on a ring, revealing how it transitions from intermittent to continuous flow at a critical density, with detailed asymptotic results and crossover scaling functions.

Contribution

It provides exact large-time expressions for the first two cumulants of the current using Bethe ansatz, and introduces crossover scaling functions unifying different regimes.

Findings

Average current is finite below critical density and grows at or above it.

Diffusion coefficient decays as N^{-1/2} below critical density and grows as N^{3/2} above.

Crossover scaling functions are derived and compared to avalanche size statistics.

Abstract

We consider the particle current in the asymmetric avalanche process on a ring. It is known to exhibit a transition from the intermittent to continuous flow at the critical density of particles. The exact expressions for the first two scaled cumulants of the particle current are obtained in the large time limit $t\to\infty$ via the Bethe ansatz and a perturbative solution of the TQ-equation. The results are presented in an integral form suitable for the asymptotic analysis in the large system size limit $N\to\infty$. In this limit the first cumulant, the average current per site or the average velocity of the associated interface, is asymptotically finite below the critical density and grows linearly and exponentially times power law prefactor at the critical density and above, respectively. The scaled second cumulant per site, i.e. the diffusion coefficient or the scaled variance of the associated interface height, shows the $O(N^{-1/2})$ decay expected for models in the Kardar-Parisi-Zhang universality class below the critical density, while it is growing as $O(N^{3/2})$ and exponentially times power law prefactor at the critical point and above. Also, we identify the crossover regime and obtain the scaling functions for the uniform asymptotics unifying the three regimes. These functions are compared to the scaling functions describing crossover of the cumulants of the avalanche size, obtained as statistics of the first return area under the time space trajectory of the Vasicek random process.