Local Stochastic Algorithms for Alignment in Self-Organizing Particle Systems

The work studies how many simple particles, each with limited memory and only local communication, can collectively agree on a common direction of orientation. The particles live on a 2D lattice with at most one particle per site, and during each move they may either change their direction or move to a neighboring empty site. The authors consider “oriented” particles, where each particle’s direction is one of q discrete options, and each particle can compare its direction to those of neighboring particles without knowing any global reference orientation. They analyze two movement/connectivity regimes: one where configurations must stay simply connected at all times, and one where spatial moves can disconnect the configuration. Using ideas from statistical physics (Potts/clock models), the paper proves that for any q ≥ 2 the system can be driven toward either an aligned state (a single dominant direction) or a non-aligned state (roughly equal fractions in each direction). It also shows that parameter choices can control both how tightly the particles gather (compression/expansion) and whether they align or remain non-aligned.