We consider the one-dimensional totally asymmetric simple exclusion process with an arbitrary initial condition in a spatially periodic domain, and obtain explicit formulas for the multi-point distributions in the space-time plane. The formulas are given in terms of an integral involving a Fredholm determinant. We then evaluate the large-time, large-period limit in the relaxation time scale, which is the scale such that the system size starts to affect the height fluctuations. The limit is obtained assuming certain conditions on the initial condition, which we show that the step, flat, and step-flat initial conditions satisfy. Hence, we obtain the limit theorem for these three initial conditions in the periodic model, extending the previous work on the step initial condition. We also consider uniform random and uniform-step random initial conditions.

中文翻译：

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具有一般初始条件的周期性 TASEP

我们考虑在空间周期域中具有任意初始条件的一维完全非对称简单排除过程，并获得时空平面上多点分布的显式公式。这些公式是根据涉及 Fredholm 行列式的积分给出的。然后我们评估弛豫时间尺度中的大时间、大周期限制，这是系统大小开始影响高度波动的尺度。极限是假设初始条件的某些条件而获得的，我们证明阶梯、平坦和阶梯平坦初始条件满足。因此，我们在周期模型中获得了这三个初始条件的极限定理，扩展了先前关于阶跃初始条件的工作。