Determine the universality class of Vicsek’s flocking phase

Determine the universality class that governs the scaling behavior of the ordered (flocking) phase of the Vicsek model, specifically identifying the roughness, dynamic, and anisotropy exponents that characterize this phase in physical dimensions (two and three spatial dimensions).

Background

The Vicsek model (1995) and the Toner–Tu hydrodynamic theory laid the groundwork for active matter. Subsequent work revealed that the Toner–Tu renormalization group analysis neglected important nonlinearities, and simulations reported exponents that differ from the original predictions. As a result, despite extensive progress, the universality class controlling the scaling of Vicsek’s ordered phase has been debated.

The paper motivates resolving this by developing a functional renormalization group analysis of a simplified Toner–Tu model and compares the resulting exponents with recent simulations. The authors present evidence for a novel universality class but frame the determination of the governing class as an open question in the introductory context.

References

As a result, the question of { i what universality class (UC) actually describes Vicsek's flocking phase remains open}.

— A new universality class describes Vicsek's flocking phase in physical dimensions  (2402.01316 - Jentsch et al., 2024) in Opening paragraphs before “Simplified Toner-Tu model”, first page

Unfortunately in additional non-linearities were shown to be present and it seemed impossible to determine the exact values of such scaling exponents (beyond the fact that $\chi<0$ and $\xi<\leq1$ which, by Eqs. real_space-nlnSperp, implies long range order in $d=2$).

real_space:

⟨∣v⊥(r,t)∣2⟩=∫ddq(2π)dSv(q)\langle|\mathbf{v}_\perp(\mathbf{r},t)|^2\rangle = \int \frac{d^dq}{(2\pi)^d}S_v(\mathbf{q})

nlnSperp:

Sv(q⊥,q∥,μ∗)∼Sρ(q⊥,q∥,μ∗)∼q⊥−ζ .S_v(q_\perp, q_\parallel, {\bm \mu}^*)\sim S_\rho(q_\perp, q_\parallel, {\bm \mu}^*) \sim q_\perp^{-\zeta} \,.

— Boundary and bulk perturbations in vectorial active matter  (2609.28593 - Fava, 23 Sep 2026) in Section 1, subsection “On the value of the scaling exponents”