scaling behavior outlined above, indicating that the ap- proach to the homogeneous state is no longer controlled by the single energy scale associated with the cost of a do- main wall. Convective fluxes induced by self-propulsion lead to correlations on longer length scales and hence ac- celerate the coarsening dynamics. This picture can be substantiated by extracting a length scale L ( t ) from the correlation functions. This is shown in Fig. 6 where we see that the dynamical exponent z is smaller than the equilibrium value, indicating that the coarsening dynam- ics in polar active systems is faster than that of both equilibrium systems with vector order parameters and active nematics. This may be due to the fact that, in a polar self-propelled system, there is true long range or- der in 2D [16, 17] and the associated suppression of the Goldstone mode by the nonlinear couplings changes the dynamics of the system as the orientational order builds up towards the homogeneous ordered state. At long times, for v$_{0}$ < v$_{c}$ , the system reaches the “fluctuating flocking state”, a steady state with finite mean polarization and anomalous fluctuations. To char- acterize the properties of this state we have evaluated the two-point correlation function of fluctuations in the orientational order parameter, with δ P ( r , t ) = P ( r , t ) − P ( t ). This is shown in the right frame of Fig. 8 and it decays logarithmically as expected for vector order in 2D. The correlation function shown in Fig. 8 has been averaged over r$_{0}$ , hence it represents only the isotropic (angular-averaged) part of the order parameter correlations. In general we expect the corre- lation function to be anisotropic and its spatial decay to be described by different length scales in the directions longitudinal and transverse to the direction of mean mo- tion, as described in Ref. [17]. We have calculated the spatial decay of the order parameter correlation in each direction and find it indeed to be anisotropic, as shown in Fig. (7). The theoretical analysis of Ref. [17] predicts a power law behavior with different exponents character- izing the decay along and orthogonal to the direction of broken symmetry. Given the large spatial and temporal fluctuations in our system, the system sizes considered here are too small to obtain reliable statistics to quantify this behavior and extract scaling exponents. In contrast to the order parameter correlations, which decay logarithmically, the two point density correlation shown in the in the left frame of Fig. 8 displays correla- tions over longer length scales than expected in equilib- rium. In fact the density correlation functions exhibits cuspy behavior of the form C$_{ρ}$ ( r, t ) = 1 − ( r L ( t ) ) α with α ≃ 0 . 6 typically characteristic of a state with growing domains. Furthermore, C$_{ρ}$ ( r , t ) depends very weakly on the self propulsion speed. These large correlations in the density arise because of the coupling of this conserved field to the fluctuations in the underlying order parame- ter field that is an intensive variable. This leads to what The behavior of polar active system appears to be somewhat different. Figure 5 shows the two-point cor- relation function for the density and the order parameter for our flock. Both exhibit a growing correlation length as a function of time, but they do not exhibit the simple In the case of self propelled particles, the orientational fluctuations that drive the coarsening of the ordered state also induce mass fluxes and hence couple to density fluc- tuations. As a result, density correlations are essentially slaved to the order parameter correlations and both C$_{ρ}$ and C$_{P}$ are expected to exhibit coarsening behavior [38]. This is indeed the behavior that has been observed in active nematic liquid crystals, where both density and orientational correlations have been shown to coarsen on a characteristic length scale that grows like t 1 $^{/z}$, with z ∼ 2 [39]. ferent ordered regions. This implies that the time evo- lution of the system occurs via the growth of a single length scale L ( t ) that characterizes the size of a typical ordered region in the system at a time t . The order pa- rameter correlation function will then depend on time only through L ( t ), i.e., C$_{P}$ ( r, t ) = C ( r L ( t ) ) . Scaling analysis indicates that the length scale L ( t ) grows with the dynamical critical exponent z , L ( t ) ∼ t 1 $^{/z}$. For a vector order parameter in two dimensions in equilibrium one expects z ∼ 2 [37]. C$_{δP}$ ( r , t ) = < δ P ( r 0 + r , t ) · δ P ( r 0 , t ) >, (25) FIG. 4: (color online) The mean magnitude of the polar order parameter as a function of system size L for different values of v 0 and ρ 0 = 0 . 7, corresponding to v$_{c}$ ≃ 0 . 42. The curves appear to saturate as the system size increases, suggesting that the system remains macroscopically ordered in the ther- modynamic limit. 9 0 500 1000 System size L 0.92 0.94 0.96 0.98 1  <P> v$_{0}$ = 0.1 Ordered Initial state v$_{0}$ = 0.2 Ordered Initial state v$_{0}$ = 0.3 Ordered Initial state v$_{0}$ = 0.4 Ordered Initial state v$_{0}$ = 0.1 Isotropic Initial state