*2.4. Numerical Simulations*

While serving as a guide to the exploration of a vast range of parameters, the simplified mean-field theory developed above is not expected to give realistic results relative to the critical properties expected near the transition point, whether decay at Reg or symmetry restoration above Re2. For example, observations suggest that LTB propagation is a dominant feature; hence, *p*<sup>5</sup> 1 and {*p*1, *p*2} is small, leading us to expect stable one-sided solutions systematically. This conclusion, however, strongly relies on neglecting all terms beyond second degree in (4) and (5) in the evaluation of the contribution of densely populated configurations, leading to (7) and (8). This is legitimate only when *S<sup>n</sup> S*2, i.e., *S* 1, that is, close to decay in the case of a continuous (second-order) transition but not necessarily elsewhere in the parameter space, in particular at the one-sided/two-sided bifurcation where both *R* and *B* are of the same order of magnitude but may be large. Even when keeping the assumption of independence of contributions to the future state at a given lattice node, this problem is not easily addressed and, at any rate, has to be properly accounted for in the presence of stochastic fluctuations, which will be done numerically.

The translation of the probabilistic rules introduced in Section 2.2 using Matlab® is straightforward once the "*B*/*R*/∅" convention is appropriately translated into "+1/−1/0". No assumption is made other than the independence of the contributions of the different configurations to the outcome at a given lattice node, by strict application of the rules expressed through (2) and (3). In particular, computations involve the contribution of all configurations and not only the unary or binary ones, as presumed to derive the mean-field equations. Periodic boundary conditions have been applied to 2D lattices of various dimensions (*NB* × *NR*), where *NB* (*NR*) is the number of sites in the propagation direction of *B* (*R*) active states, with ordinarily *NB* = *NR*. At each simulation step, we shall measure the mean activity of *B* and *R* states denoted *B* and *R* above and from now on called turbulent fractions, as *F*t(*B*)=(*NBNR*)−1#(*B*) and *F*t(*R*)=(*NBNR*)−1#(*R*) where #(*B*) and #(*R*) are the numbers of sites in the corresponding active state.

A preliminary study of the model in a small domain has shown that the different transitional regimes and the symmetry-breaking bifurcation were indeed present as expected from the simplified mean-field approach. (We remind that the model contains nothing appropriate for organized laminar–turbulent regimes for Re > 1200 and is relevant only for the strongly intermittent sparse LTB networks pictured in Figures 2 and 3). In [14], we argued that the onset of transversal splitting was the source of genuinely 2D behavior. Accordingly we shall consider the stochastic model in two steps, below and above the onset of transversal splitting, here associated with *p* <sup>4</sup> ≡ 0 and *p* <sup>4</sup> > 0 respectively. Furthermore, in the simulations the LTBs were seen to propagate obliquely with respect to the background downstream current. This propagation is nearly all contained in the probability attached to configuration C<sup>5</sup> (*p*<sup>5</sup> for propagation and 1 − *p*<sup>5</sup> for decay or slowing-down), and to a lesser extent influenced by the contribution of configuration C1, mostly associated with in-line longitudinal splitting. We shall account for the limited sensitivity of the propagation speed to the value of Re to fix *p*<sup>5</sup> constant and close to 1, more specifically *p*<sup>5</sup> = 0.9, and let other parameters vary. The role of *p*<sup>2</sup> and *p* <sup>4</sup>, both related to 2D features, will be studied separately in the two next sections.

#### **3. Before Onset of Transversal Splitting,** *P-* **<sup>4</sup> = 0**
