Exact scaling exponents of two-dimensional compressible polar flocks
In plain words
Flocks of self-propelled particles that align with neighbours can move together in two dimensions, which equilibrium physics forbids for comparable systems. The exact laws governing the fluctuations of such a flock are disputed.
Precise statement
In the ordered phase of compressible dry polar active matter (Vicsek model, Toner-Tu hydrodynamics) in $d = 2$, determine the roughness exponent $\chi$, anisotropy exponent $\zeta$ and dynamic exponent $z$ of velocity and density fluctuations. The incompressible 2D case is solved exactly by mapping to the 1D KPZ equation (Kardar-Parisi-Zhang, the standard equation for a randomly growing interface; Chen, Lee, Toner 2016). The 1995 conjecture $\chi = -1/5, \zeta = 3/5, z = 6/5$ for the compressible case is inconsistent with large-scale simulations published in 2019. An answer gives exact or controlled values confirmed by simulations that reach the asymptotic regime.
What would settle it
An analytic determination of $\chi$, $\zeta$ and $z$ for $2\mathrm{D}$ compressible flocks, matched by Vicsek-model simulations large enough to rule out crossovers.
Status in the literature
Unverified note
In 2024 Jentsch and Lee (PRL 133, 128301), Ikeda (PRL 133, 258301) and Amoretti et al. (PRE 110, 054108) each proposed exact exponents or a new universality class for the Vicsek ordered phase; these proposals are mutually inconsistent and none is established numerically (2026).