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for all v$_{0}$ and ρ < ρ$_{c}$ . For ρ > ρ$_{c}$ and v$_{0}$ < v$_{c}$ ( ρ$_{0}$ ) the system is in the fluctuating flocking state, charac- terized by finite polarization and large spatial and tem- poral fluctuations of both density and order parameter. For v > v$_{c}$ ( ρ$_{0}$ ) we find a striped phase, with alternating ordered high density bands and disordered low density bands, propagating in the direction of order. In the nu- merics the value of v$_{c}$ is identified as the self propulsion velocity where the density histograms shown in Fig. (9) change from unimodal to bimodal. The histograms are constructed by recording the local density at each spatial grid point for fixed mean density ρ$_{0}$ averaged over many initial conditions. We have also verified that histogram of local polarization magnitude change from unimodal to bimodal as the same value of v$_{0}$ . The numerical bound- ary for the onset of the stripe regime vanishes with v$_{0}$ for ρ$_{0}$ → ρ$_{c}$ and is in qualitative agreement with the bound- ary calculated in section for the onset of the longitudinal instability. The theoretical curve, however, grows much faster with v$_{0}$ than the numerical boundary. Surprisingly, at high density the theoretical boundary for the linear in- stability of splay fluctuations agrees very well with the numerical onset of stripes. 

In this subsection we characterize the fluctuating flock- ing state that exists in the region ρ > ρ$_{c}$ and v$_{0}$ < v$_{c}$ of the phase diagram in Fig. 1. As noted earlier, this state is characterized by large fluctuations in the den- sity. These fluctuations do not, however, destroy the underlying orientational order of the system. This is dis- played in Figs. 3 and 4. Figure 3 shows the time evo- lution of the magnitude squared of the order parameter, 〈 P $^{2}$( t ) 〉 = 〈 P 2 $_{x}$( t ) + P 2 $_{y}$( t ) 〉 , where here and below the brackets denote a spatial average over the system and an average over different realizations of initial conditions, for both an initial ordered ( 〈 P $^{2}$( t = 0) 〉 = 1) and an initial disordered ( 〈 P $^{2}$( t = 0) 〉 = 0) state. Both states reach the same ordered state at long times, although on very different time scales. The asymptotic state is or- dered (i.e., 〈 P $^{2}$〉 ̸ = 0)and this does not appear to be an artifact of the finite system size, as shown in Fig. 4 where the magnitude of the order parameter is displayed as function of system size for various values of v$_{0}$ and dif- ferent initial states. The numerics suggest that P is finite and close to unity (note the narrow range of 〈 P 〉 on the vertical axis of Fig. 4) in the fluctuating flocking state. The large difference in the relaxation time from an ini- tial disordered/ordered state to the asymptotic ordered state seen in Fig. 3 is not unexpected. When starting in a disordered state, the system locally finds different degenerate ordered states, which subsequently coarsen towards the homogeneous ordered state. Below we char- acterize both the coarsening behavior as the system seeks out its asymptotic steady state and the properties of the asymptotic fluctuating flocking state. 

## A. Fluctuating flocking state 

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To quantify the coarsening behavior we have measured the two-point correlation function of both the density and the order parameter, defined as 

Before discussing the behavior of these correlation func- tions, it is useful to recall the dynamics of phase order- ing developed in the context of equilibrium second order phase transitions [37]. Phase ordering theories consider a system in an initially disordered state that is rapidly quenched below the order-disorder transition point and describe the time evolution following the quench. Imme- diately after the quench the system consists of finite-size ordered regions, each in one of the continuum of degen- erate ground states that correspond to one choice of the spontaneously broken continuous symmetry. The sys- tem then evolves in time and “coarsens” with some of the ordered regions growing at the expense of others and eventually taking over the entire system. The coarsening process is typically controlled by a single energy scale, namely the energy cost of the domain walls between dif- 

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