Threshold criterion for trend formation

Establish whether trend formation in the centroid-seeking sparse-sampling particle system occurs only when the turning-early measure satisfies the inequality \(\eta(v,n)>\eta_c(n)\), where \(v\) is the normalized particle speed, \(n\) is the sampling size, and \(\eta_c(n)\) is the empirically determined critical threshold.

Background

The paper studies point particles moving at constant speed in a bounded two-dimensional domain. At discrete updates, each particle samples $n$ other particles, estimates their centroid, and reorients toward that estimated centroid when the estimated centroid lies beyond the turning threshold. Sampling fluctuations, combined with the geometry of the confining domain, can produce a self-reinforcing nematic alignment mechanism.

To quantify whether particles turn sufficiently early to participate in this mechanism, the paper defines $\eta$ as a measure comparing the conditional expected distance of the estimated centroid at turning with the expected distance of the actual centroid. The authors derive $\eta(v,n)$ from an effectively one-dimensional invariant state and compare it with a critical value $\eta_c(n)$, which is determined empirically by matching the first no-trend boundary in simulations. They explicitly formulate the condition $\eta(v,n)>\eta_c(n)$ as a conjecture rather than a proved result.