Crossover scaling functions in the asymmetric avalanche process

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.

arXiv comment34 pages, 5 figures

References in corpus (17)

• Exact Large Deviation Function in the Asymmetric Exclusion Process
• Phase Transitions in one-dimensional nonequilibrium systems
• Nonequilibrium Dynamics of Interfaces and Lines
• Exact Current Statistics of the ASEP with Open Boundaries
• Driven Depinning in Anisotropic Media
• Probability distribution of the free energy of a directed polymer in a random medium
• Current large deviation function for the open asymmetric simple exclusion process
• An Exact Formula for the Statistics of the Current in the TASEP with Open Boundaries
• Large Deviation Function of the Partially Asymmetric Exclusion Process
• Bethe ansatz solution of zero-range process with nonuniform stationary state
• Current Fluctuations in the exclusion process and Bethe Ansatz
• The Asymmetric Avalanche Process
• Bethe Ansatz and Q-operator for the open ASEP
• Bethe ansatz solution of discrete time stochastic processes with fully parallel update
• Statistics of the first passage area functional for an Ornstein-Uhlenbeck process
• Transition from KPZ to Tilted Interface Critical Behavior in a Solvable Asymmetric Avalanche Model
• Current statistics in the q-boson zero range process