Title: Crossover scaling functions in the asymmetric avalanche process Show Chinese title

Title: 不对称雪崩过程中的交叉缩放函数

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. Show Chinese abstract

Abstract: 我们考虑环状非对称雪崩过程中的粒子流。已知在粒子的临界密度处，它会从间歇流过渡到连续流。通过Bethe假设和TQ方程的微扰解，在大时间极限$t\to\infty$下得到了粒子流的前两个缩放累积量的精确表达式。结果以适合在大系统尺寸极限$N\to\infty$下进行渐近分析的积分形式呈现。在此极限下，第一个累积量，即每个站点的平均电流或相关界面的平均速度，在临界密度以下渐近为有限值，并在临界密度及以上分别线性增长和指数乘以幂律因子。每个站点的缩放第二累积量，即扩散系数或相关界面高度的缩放方差，在临界密度以下表现出Kardar-Parisi-Zhang普适类模型预期的$O(N^{-1/2})$衰减，而在临界点及以上则表现为$O(N^{3/2})$的增长以及指数乘以幂律因子。此外，我们确定了交叉区域，并获得了统一三个区域的均匀渐近的标度函数。这些函数与描述雪崩大小累积量交叉的标度函数进行了比较，这些标度函数作为Vasicek随机过程的时间空间轨迹下的首次返回面积的统计量得到。