*2.2. Design of the Model*

Let us first give a brief description of the processes to be accounted for. Below Re ≈ 800 (event A) only decay and longitudinal splittings are possible. Not visible in the snapshots of Figure 4 (left) but observable in the movies is the fact that a daughter LTB resulting from longitudinal splitting runs behind its mother along a track that may be slightly shifted upstream. This shift is negligible when Re is small (in-line longitudinal splitting) but as Re increases it becomes more and more visible while the general propagation direction is unchanged (off-aligned longitudinal splitting). On the other hand, Figure 4 clearly illustrates the fact that, upon transversal splitting, the new-born LTB

systematically develops on the downstream side of its parent. Importantly, the propagation of LTBs is a dynamical feature different from advection treated as a deterministic step in [22]. Accordingly, it will be understood as a statistical propensity to move in a given direction resulting from an imbalance of stochastic "forward" and "backward" processes along their directions of motion. Other complex processes also seen in the simulations, such as fluctuating propagation with acceleration, slowing down, or lateral wandering, will be included only in so far as they can be decomposed into such more elementary events. All the events to be included in the model can be translated into the language of reaction–diffusion processes, persistence or death, offspring production, and coalescence, common in the field of DP theory [2].

On general grounds the governing equation reads:

$$S'(i,j) = \sum\_{\mathcal{C}'} R\_{\mathcal{C}'} \delta\_{\mathcal{C}'\mathcal{C}(i,j)} \, \prime \, \tag{2}$$

where C(*i*, *j*) is the neighborhood configuration of site (*i*, *j*) at time *t*, C one of the possible configurations, and *R*<sup>C</sup> a stochastic variable taking value 1 with probability *p*<sup>C</sup> corresponding to configuration C and value 0 with probability 1 − *p*<sup>C</sup> . The Kronecker symbol *δ*C<sup>C</sup> is here to select the configuration C that matches C. Depending on C and C , the output *S* (*i*, *j*) can be *B* or *R*.

Figure 6 illustrates the set of possible single-colored neighborhoods, either *B* (upper line) or *R* (lower line). Following the indexation in Figure 5c, the order of the columns is based on the physical condition and respects the upstream/downstream distinction illustrated in Figure 5a, making configurations with the same index physically equivalent.

**Figure 6.** Single-color configurations: from the overall geometry depicted in Figure 5a, the downstream side of a state is to the top for *B* states and to the right for *R* states. Each colored square indicates the active state in the configuration at time *t* of site (*i*, *j*) at the center. The question mark features the probabilistic outcome (time *t* + 1).

These single-color elementary configurations will be denoted as C*<sup>i</sup>* with *i* ∈ [1:5]. They will be described as [*SSSSS*] with *S* = *B*, *R*, or ∅. Hence C<sup>3</sup> ≡ [∅ ∅ *B* ∅ ∅] or [∅ *R* ∅∅∅]. Later, more complicated configurations will not be given a name but just a description following the same rule, e.g., [∅ *BBR* ∅].

Importantly, we make the assumption that the future state at a given node, the question marks in Figure 6, is the result of the probabilistic combination of the independent contributions of elementary configurations involving a single active state in its neighborhood.

First of all, the void configuration C<sup>0</sup> ≡ [∅∅∅∅∅] obviously generates an empty site with probability 1, hence an occupied site with probability *p*C<sup>0</sup> = 0, in order to preserve the absorbing character of the dynamics. All the other configurations evolves according to probabilities that are free parameters just constrained by empirical observations. Let us now interpret probabilities associated with the five situations depicted in Figure 6, focusing on the case of *B* states:


To summarize, as it stands the model involves four parameters: *p*<sup>1</sup> mainly governs longitudinal splitting and *p*<sup>2</sup> additional lateral diffusion, *p*<sup>5</sup> is for propagation, and *p* <sup>4</sup> for transversal splitting. The propagation of active states along their own direction involves probabilities associated with elementary configurations C<sup>1</sup> and C<sup>5</sup> while the overwhelming contribution of *p*<sup>5</sup> favors one direction. Configuration C<sup>3</sup> that could have contributed to the balance is empirically found negligible, saving one parameter as indicated above.

Neighborhoods with more than one active site are treated by assuming that the future state *S* of the central node (*i*, *j*) is the combined output of its elementary ingredients, each contribution being considered as independent of the others, i.e., without memory of the anterior evolution, of which the considered configuration is the outcome. The computation of the probability attached to the output of a given single-colored neighborhood is then straightforward. The argument follows the lines given for directed percolation, bearing on the probability that the state at the node will be absorbing (empty) and leading to Equation (1) in the mean-field approximation [20,24]. Things are a little more complicated when the neighborhood is two-colored since in all mixed-colored cases some configurations correspond to collisions and others allow for the nucleation of a differently colored offspring when *p* <sup>4</sup> = 0.

For an elementary configuration, non-contamination of site (*i*, *j*) from an active neighboring state in position *k* ∈ [1:5] takes place with probability (1 − *pk*) and of course with probability 1 if the corresponding site is empty. This gives the general formula (1 − *pkSk*), where *Sk* = 1, when the site is active, either *B* or *R*, and *Sk* = 0 when it is absorbing (∅). For a configuration C*<sup>x</sup>* = [*S*1, *S*2, *S*3, *S*4, *S*5], where *S* = *B*, *R*, or ∅, the probability to get an absorbing state is (1 − *p*C*<sup>x</sup>* ) = ∏*k*(1 − *pkSk*) hence for the node to be activated *p*C*<sup>x</sup>* = 1 − ∏*k*(1 − *pkSk*). To deal with two-colored neighborhoods properly, we must be a little more specific and write the probability of the state *S* of a given color *S* as

$$p\_{[\mathcal{S}\_1, \mathcal{S}\_2, \mathcal{S}\_4, \mathcal{S}\_5]} = 1 - (1 - p\_1 \mathcal{S}\_1)(1 - p\_2 \mathcal{S}\_2)(1 - p\_4' \bar{\mathcal{S}}\_4)(1 - p\_5 \mathcal{S}\_5) \tag{3}$$

where it is understood that if *S* = *B*, then *S*¯ = *R* or the reverse, and *Sj* = 0 for *j* = 1, 2, 5, or *S*¯ <sup>4</sup> = 0 if the corresponding states are ∅. Figure 7 (right) illustrates the most interesting two-state configurations with different colors corresponding to collisions (C1) and offspring generation (C2). Such a situation is dealt with by adding a supplementary rule:

**Figure 7.** Modeling of transversal splitting for states of type (B) propagating horizontally and (R) propagating vertically, the base flow being along the diagonal (). Heavy colors indicate states present at time *t* and, playing the role attributed to question marks in Figure 6; light colors stand for states possibly present at time *t* + 1 according to probabilities *p*<sup>5</sup> (propagation) and *p* <sup>4</sup> (transversal splitting). Conflicting configurations are (C1) ([*SSSRB*] corresponding to propagation leading to a collision and (C2) [*SSSBR*] corresponding to simultaneous transversal splittings, respectively (here *S* = ∅ for clarity).

7. When the general expression (3) gives non-zero probabilities to *S* and *S*¯ the resulting superposition of states is not allowed and a choice has to be made. It might seem natural to keep the state with the maximum probability but, depending on circumstances hard to decipher, collisions sometimes appear to cause the decay of both protagonists or else reinforce the dominance of one color in a given region of space. A similar bias can affect transversal splitting. These peculiarities are not taken into account here: for simplicity, in all conflicting cases, we make the assumption that the result is non-empty and random with probability 1/2.

The model is now complete with parameters clearly related to empirical observations, plausible relative orders of magnitude and sense of variation: Probability *p*<sup>5</sup> is the main ingredient for the built-in propagation of the two families of LTBs (active states). In turn *p*<sup>1</sup> is obviously related to the behavior of the system close to decay at and slightly above Reg. The value given to probability *p*<sup>2</sup> will appear crucial to the 1D reduction of DP in a 2D medium as observed experimentally (Figure 4, right). Finally, we can anticipate that probability *p* <sup>4</sup> will control the one-sided/two-sided symmetry-restoring bifurcation, as it continuously grows from 0 beyond Event A at *R* ≈ 800.
