*3.1. Coarsening from Two-Sided Initial Conditions*

In the absence of transversal splitting, changes in the population of each state only comes from transversal collisions. As documented in [14], when starting from an initial condition with two similarly represented orientations, collisions lead to the formation of domains uniformly populated by one of each species, following from a majority rule, with interactions limited to the domain boundaries. A coarsening takes place with one species progressively disappearing to the benefit of the other, leaving a single-sided state at large times. The process is illustrated here using simulations of the model with *p*<sup>5</sup> = 0.9, *p*<sup>1</sup> = 0.1, *p*<sup>2</sup> = 0.07, values known from the preliminary study to produce a sustained nontrivial final state.

The decay from a fully active state populated with a random distribution of *B* and *R* states in equal proportions is scrutinized in a 256 × 256 domain with periodic boundary conditions. Figure 8 illustrates a particularly long transient displaying the different stages observed during a typical experiment.

The upper panel displays the time series of the turbulent fractions for each species, *B* and *R*, for a two-sided high-density initial condition, *F*t(*B*) + *F*t(*R*) = 1, *F*t(*B*) *F*t(*R*) 0.5. Contrasting with the monotonic variation observed when starting from one-sided initial conditions, either increasing from a low density of active states (*F*<sup>t</sup> = 0.05) or decreasing from a fully active configuration (*F*<sup>t</sup> = 1), the turbulent fractions change in a more complicated way that is easily understood when looking at the bottom line of snapshots. The total turbulent fraction first decreases due to the dominant effect of collisions. These collisions tend to favor a spatial modulation of the activity amplifying inhomogeneities in the initial conditions. This distribution results from the majority effect expressing the local stability of one-sided states predicted by the mean-field analysis. A periodic pattern already appears at *t* = 100, with bands oriented parallel to the second diagonal of the square domain. *B* states move right along the horizontal axis, and *R* states up along the vertical axis, at the same average speed so that the pattern drifts along the first diagonal of the domain. Regions where *B* or *R* dominate are locally stable against destructive collisions and activity is limited to *B*/*R* interfaces. After a while, splittings begin to counteract collisions and an overall activity recovers, here for *t* ≈ 250. The local density of *B* and *R* states increases inside bands that become better defined, reaching a sustained regime with two *R*–*B* alternations, wide and narrow, at *t* 1500. This configuration is nearly stable and slowly evolves only due to the erosion of narrowest bands at the *R*/*B* interfaces. At *t* ≈ 5500 these bands disappear by merging, leaving two bands, *B* wide and *R* narrow. The same slow erosion process leads to the final homogeneous *B* regime by decay of the *R* band at *t* ∼96,000. The two successive band decays take place at roughly constant total turbulent fraction with fast adjustment at the band decay, up to the final single-sided turbulent fraction. The asymptotic state is independent of the way it has been obtained, from one-sided or two-sided initial conditions.

**Figure 8.** (**Top**): Time series of the turbulent fractions for a simulation from a fully active initial configuration with *B* and *R* states in equal proportions—blue and red in graphs, respectively; the dotted black trace is for the total turbulent fraction. Two simulations starting from low (*F*<sup>t</sup> = 0.05, cyan) and high (*F*<sup>t</sup> = 1, magenta) density one-sided states are displayed for comparison. (**Bottom**): Snapshot of state during the simulations from the two-sided initial condition, at *t* = 100 during initial decay, at *t* = 5500 with two pairs of active bands of each color, at *t* = 13,000 when the narrowest bands merge and disappear, at *t* = 96,000 when the *R* active band disappears, leaving a uniform *B* state.

The long duration of the transient taken as an example is due to the near stability of the rather regular pattern building up after the initial fast decay. This property is in fact the result of a geometrical peculiarity of the square domain: *B* and *R* states travel statistically at the same speed through the domain, horizontally and vertically, respectively, so that the band integrity is maintained despite propagation and the evolution controlled by collisions at the *B*–*R* and *R*–*B* interfaces only. The observed slow erosion process only results from large deviations among collisions. In rectangular domains, the propagation times become different and the symmetry of the two interfaces is lost. A bias results, which induces a systematic erosion of bands and a shorter transient duration. Whatever the aspect ratio, one of the states is always ultimately eliminated and the last stage of the transient corresponds to a trend toward a statistically uniform saturated one-sided regime with a turbulent fraction strictly independent of the shape. Accordingly, to save the time corresponding to the transient, in the next section we will study the decay of the one-sided regime by starting from random one-sided initial conditions.

All these features nicely fit the empirical observations discussed at length in [14] where similar transients were obtained below the onset of transversal splitting—in much smaller effective domains and with far fewer interacting LTBs, however (Figure 2, right panel, and Figure 4, left panels).
