Fluctuation induced equality of multi-particle eccentricities for four or more particles

We discuss eccentricities (ellipticity and triangularity) generated in nucleus-nucleus and proton-nucleus collisions. We define multi-particle eccentricities $\epsilon_n\{m\}$ which are associated with the $n'th$ angular multipole moment for $m$ particles. We show that in the limit of fluctuation dominance all of the $\epsilon_n\{m\}$'s are approximately equal for $m \ge 4$. For dynamics linearly responding to these eccentricities such as hydrodynamics or proposed in this paper weakly interacting field theory, these relations among eccentricities are translated into relations among flow moments $v_n\{m\}$. We explicitly demonstrate it with hydrodynamic calculations.