Local interactions of uncoordinated individuals produce the collective behaviors of many biological systems, inspiring much of the current research in programmable matter. A striking example is the spontaneous assembly of fire ants into "bridges" comprising their own bodies to traverse obstacles and reach sources of food. Experiments and simulations suggest that, remarkably, these ants always form one bridge -- instead of multiple, competing bridges -- despite a lack of central coordination. We argue that the reliable formation of a single bridge does not require sophistication on behalf of the individuals by provably reproducing this behavior in a self-organizing particle system. We show that the formation of a single bridge by the particles is a statistical inevitability of their preferences to move in a particular direction, such as toward a food source, and their preference for more neighbors. Two parameters, $\eta$ and $\beta$, reflect the strengths of these preferences and determine the Gibbs stationary measure of the corresponding particle system's Markov chain dynamics. We show that a single bridge almost certainly forms when $\eta$ and $\beta$ are sufficiently large. Our proof introduces an auxiliary Markov chain, called an "occupancy chain," that captures only the significant, global changes to the system. Through the occupancy chain, we abstract away information about the motion of individual particles, but we gain a more direct means of analyzing their collective behavior. Such abstractions provide a promising new direction for understanding many other systems of programmable matter.

翻译：无协调个体的局部相互作用产生了许多生物系统的集体行为，这启发了当前可编程物质研究的大部分工作。一个引人注目的例子是火蚁自发组装成由自身身体构成的"桥梁"，以穿越障碍物并到达食物源。实验和模拟表明，尽管缺乏中央协调，这些蚂蚁总是形成一个桥梁——而不是多个相互竞争的桥梁。我们通过在一个自组织粒子系统中可证明地重现这种行为，论证了可靠形成单一桥梁并不需要个体具备复杂性。我们证明，粒子形成单一桥梁是其偏好朝特定方向（例如朝向食物源）移动以及偏好更多邻居的统计必然性。两个参数$\eta$和$\beta$反映了这些偏好的强度，并决定了相应粒子系统马尔可夫链动力学的吉布斯平稳测度。我们证明当$\eta$和$\beta$足够大时，几乎必然形成单一桥梁。我们的证明引入了一个称为"占据链"的辅助马尔可夫链，它仅捕捉系统显著的全局变化。通过占据链，我们抽象掉了关于单个粒子运动的信息，但获得了分析其集体行为的更直接方法。此类抽象为理解许多其他可编程物质系统提供了一个有前景的新方向。