*1.2. Modeling Context: Directed Percolation, Probabilistic Cellular Automata, and Criticality Issues*

Various modeling approaches to transitional wall-bounded flows have received considerable attention recently, from low-order Galerkin expansions of the primitive equations [17,18], to phenomenological theories based on a deep physical analysis of the processes involved in a reaction-diffusion context [19], to analogical systems expressed in terms of deterministic coupled map lattices [6,10], and to more conceptual models implementing the dynamics of cellular automata with probabilistic evolution rules (PCA) [20–22]. The model developed below belongs to this last category, implementing rules that focus on the main qualitative features seen in experiments. Such models are based on the conventional modeling of DP [2] which is most appropriate to account for the absorbing versus active character of local states.

Let us briefly recall the PCA/DP framework. In the most general case, the activity at site *j* at time *t* + 1, call it *Sj* ∈ {0, 1}, depends on the activity at sites in a full *D*-dimensional neighbor V*<sup>j</sup>* of that site at time *t* and the status of the links, permitting or not the transfer of activity within the neighborhood. For convenience a (*D* + 1)-dimensional lattice is defined with one-way (directed) bonds in the direction corresponding to time so that *D*-dimensional directed percolation is often presented as a special (*D* + 1)-dimensional percolation problem. In the simplest case of one space dimension (*D* = 1), the neighborhood of a lattice site at *j* is the set of sites with *j*  ∈ [*j* − *r*1, *j* + *r*2], comprising *r*<sup>2</sup> + *r*<sup>1</sup> + 1 sites, and it is supposed that contamination of the state at *j* at time *t* + 1 depends on the status of full configuration, the sites' activity, and the bonds' transfer properties ("bond–site" percolation [23]). In some systems, the propagation rule is totalistic in the sense that the output only depends on the number of active sites in the neighborhood and not on their positions, i.e., *<sup>ς</sup><sup>j</sup>* = <sup>∑</sup>*j*∈V*<sup>j</sup> Sj* ; an interesting example is given in [24].

In view of future developments, let us discuss bond directed percolation in one dimension (*D* = 1) with two neighbors (*r*<sup>1</sup> = 0 or *r*<sup>2</sup> = 0), only depending on the probability *p* that bonds transfer activity. The evolution rule *S <sup>j</sup>* = R(*Sj*, *Sj*+1), where *S <sup>j</sup>* denotes the state at node *j* and time *t* + 1, is totalistic. With *ς<sup>j</sup>* = *Sj* + *Sj*+1, we have (a) R(*ς* = 0) = 0 with probability 1 (a site connected to two absorbing parents never gets active whatever the links) and (b) R(*ς* = 1) = 1 with probability *p* (closed link transmitting activity), so that (a') R(*ς* = 1) = 0 with probability 1 − *p* (open link preventing transmission), (c) R(*<sup>ς</sup>* = <sup>2</sup>) = 0 with probability (<sup>1</sup> − *<sup>p</sup>*)<sup>2</sup> (absorbing since the two links are open), and (d) R(*<sup>ς</sup>* = <sup>2</sup>) = 1 with probability 1 − (<sup>1</sup> − *<sup>p</sup>*)<sup>2</sup> = *<sup>p</sup>*(<sup>2</sup> − *<sup>p</sup>*), the complementary case.

The question is whether, depending on the value of *p*, once initiated, activity keeps continuing in the thermodynamic limit of infinite times in an infinitely wide system. An answer is readily obtained in the mean-field approximation where actual local states are replaced by their mean value, neglecting the effect of spatial correlations and stochastic fluctuations (we follow the presentation of [20]). The spatially-discrete Boolean variables *Sj* are, therefore, replaced by their spatial averages *S* = *Sj*(*t*) and this mean value is just the probability that any given site is active. It is then argued that the probability to get a future absorbing state, 1 − *S* , is given by activity not being transmitted (<sup>1</sup> − *pS*)2, which yields the mean-field equation:

$$1 - S' = (1 - pS)^2 = 1 - 2pS + p^2S^2, \quad \text{i.e.,} \quad S' = 2pS - p^2S^2. \tag{1}$$

Equilibrium states correspond to the fixed points of (1): *S* = *S* = *S*∗, which gives a nontrivial activity level *<sup>S</sup>*<sup>∗</sup> = (2*<sup>p</sup>* <sup>−</sup> <sup>1</sup>)/*p*<sup>2</sup> when *<sup>p</sup>* <sup>≥</sup> *<sup>p</sup>*<sup>c</sup> <sup>=</sup> 1/2. Close to threshold, defining *<sup>ε</sup>* = (*<sup>p</sup>* <sup>−</sup> *<sup>p</sup>*c)/*p*<sup>c</sup> <sup>=</sup> <sup>2</sup>*<sup>p</sup>* <sup>−</sup> <sup>1</sup> one gets *S*<sup>∗</sup> ≈ 4*ε*. In the mean-field (MF) approximation *S*<sup>∗</sup> is the order parameter of the transition supposed to vary as *εβ*, which defines the critical exponent *β*, here *β*MF = 1. Directed percolation is the prototype of non-equilibrium phase transitions and, as such, is associated with a set of critical exponents (see [2]). Both the critical probability *p*<sup>c</sup> and the mean activity *S*<sup>∗</sup> are affected by the effects of fluctuations, with *p*<sup>c</sup> ≈ 0.6445 > 1/2 expressing that a probability larger than the mean-field estimate is necessary to preserve activity, and *β*DP ≈ 0.276 when *D* = 1. The simple mean-field argument is not sensitive to the value of *D* in contrast with reality: *β*DP ≈ 0.584 when *D* = 2, ≈ 0.81 when *D* = 3, and trends upwards to 1 reached at *D* = 4 = *D*<sup>c</sup> = 4, called the upper critical dimension (see [2] for a review). Quite generally, mean-field arguments are valid for *D* > *D*c. We are interested in another critical exponent, *α*. When starting from a fully active system exactly poised at *pc*, the turbulent fraction is observed to decrease with time (the number of iteration steps) as *S* ∝ *t* <sup>−</sup>*<sup>α</sup>* with *<sup>α</sup>* <sup>≈</sup> 0.159 when *D* = 1 and 0.451 when *D* = 2, whereas the mean-field prediction, easily derived from (1), is *<sup>α</sup>*MF = 1. Scaling theory shows that *<sup>α</sup>* = *<sup>β</sup>*/*ν*, where *<sup>ν</sup>* is the exponent accounting for the decay of time correlations while *ν*<sup>⊥</sup> describes the decay of space correlations [2].

Universality is a key concept in the field of critical phenomena characterizing continuous phase transitions. It leads to the definition of universality classes expressing the insensitivity of critical properties to specific characteristics of the systems and retaining only properties linked to the symmetries of the order parameter and the dimension of space. For directed percolation, universality is conjectured to be ruled by a few conditions put forward by Grassberger and Janssen: that the transition is continuous into a unique absorbing state and characterized by a positive one-component order parameter, and that the processes involved are short-range and without weird properties such as quenched randomness; see [2]. Universality issues are discussed at length elsewhere in this special issue, in particular by Takeda et al. [25].

In this first approach, we shall examine how universality expectations hold for the ultimate decay stage of transitional channel flow at Reg, as described in Section 1.1, and limit the discussion to the consideration of exponents *β* and *α*. This will be done in Section 3, the next section being devoted to the derivation of the model and its mean-field study. Section 4 focuses on its ability to account for the symmetry-breaking bifurcation at Re2, and our conclusions are presented in Section 5.
