TASEP的多层推广
Multi-floor generalization of TASEP
AI总结：
研究每个站点可容纳多个粒子的TASEP推广模型，通过背压算法控制粒子移动，证明了c>1时存在非平凡相变，并给出了通量的极限行为。
摘要：
我们考虑一个相互作用粒子系统，它推广了经典的全不对称简单排斥过程（TASEP），其中每个站点最多可容纳固定数量的粒子，粒子运动由{\em背压}（BP）算法（通常也称为{\em MaxWeight}）控制。有$N$个站点（$N$有限或无限），每个站点最多容纳$c$个粒子，$1 \le c < \infty$。新粒子以速率$\alpha\le 1$的泊松过程进入最左侧站点$1$，除非站点$1$已有$c$个粒子。粒子（如果有）以速率$\beta \le 1$的泊松过程从最右侧站点$N$移除。相邻站点间从左到右的粒子运动由BP规则控制：当站点$n$的粒子数严格多于站点$n+1$时，粒子以速率$1$的泊松过程从$n$移动到$n+1$。当$c=1$时，这就是标准的TASEP。我们的主要结果涉及有限系统平稳分布的渐近性，特别是当$N\to\infty$时通量（流）的极限。特别地，我们证明了在$c>1$的系统中会发生有趣的非平凡相变。例如，如果$c>1$且$1/2 \le \beta \le 1$，只要$\alpha \ge \alpha_c^*$，最大极限通量$1/4$就能达到，其中$\alpha_c^* < 1/2$是某个非平凡阈值。（对于标准TASEP，阈值为$1/2$。）我们还提出了关于任意参数设置下平稳分布渐近性的一般猜想。我们通过模拟说明了我们的形式结果和猜想，并指出了进一步研究的有趣方向。
英文摘要：
We consider an interacting particle system, which generalizes the classical totally asymmetric simple exclusion process (TASEP), in that each site can contain up to a fixed finite number of particles, and the particle movement is governed by a {\em back-pressure} (BP) algorithm (also often called {\em MaxWeight}). There are $N$ sites (with $N$ finite or infinite), each may contain at most $c$ particles, $1 \le c < \infty$. New particles enter the system at the left-most site $1$ as a Poisson process of rate $α\le 1$, unless site $1$ has $c$ particles. Particles (if any) are removed from the right-most site $N$ as a Poisson process of rate $β\le 1$. The left-to-right movement of particles between neighboring sites is governed by the BP rule: one particle moves from site $n$ to $n+1$ at epochs of a rate $1$ Poisson process, as long as the former site has strictly more particles than the latter. When $c=1$, this is the standard TASEP. Our main results address the asymptotics of the stationary distribution of a finite system, and especially the limit of the flux (current) as $N\to\infty$. In particular, we prove that interesting non-trivial phase transitions take place in a system with $c>1$. For example, if $c>1$ and $1/2 \le β\le 1$, the maximum limiting flux $1/4$ is achieved as long as $α\ge α_c^*$, where $α_c^* < 1/2$ is some non-trivial threshold. (For the standard TASEP the threshold is $1/2$.) We also put forward a general conjecture about the stationary distribution asymptotics under an arbitrary parameter setting. We illustrate our formal results and the conjecture by simulations, and identify interesting directions for further research.
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