# Simple Rules of a Discrete Stochastic Process Leading to Catalan-like Recurrences
## 2 The System and Its Evolution Rule
### Evolution and Dynamic Parameters  
First, we define possible changes of state in individuals assigning relevant probability parameters. Next, the merging conditions of the clusters are defined. **Definition 3** (Transition). : _If an entity in the separative state is chosen, i.e., if \(v(k)=s\), then the_ **transition** _takes place with probability \(v\)--the \(k-th\) entity changes its state to \(a\). Otherwise, with probability \((1-v)\), it remains in state \(s\). If two integers \(k_{1}\) and \(k_{2}\) are chosen and \(k_{1}=k_{2}\), then the probability of transition is \(\vartheta\)._  
**Definition 4** (Separation, \(\gamma\)-activation, \(\delta^{+}\)-increase, \(\delta^{-}\)-decrease). : _If an entity in the aggregative state is chosen, i.e., if \(v(k)=a\), then it belongs to a cluster of some size \(i\), and one of the following changes can occur:_  
* **Separation:** _with probability_ \(\mu_{i}\)_, i.e., all entities belonging to the cluster containing the chosen_ \(k-th\) _entity change their states to_ \(s\)_;_
* \(\gamma\)**-activation:** _with probability_ \(\gamma_{i}\)_, the activation is valid in the current time step only;_ \(\gamma\)_-activated clusters can merge;_
* \(\delta^{+}\)**-increase:** _with probability_ \(\delta^{+}_{i}\)_, i.e., the cluster containing chosen_ \(k\)_-entity increases its size by_ \(1\) _via absorption of a separate unit with respective probability_ \(\delta^{+}_{i}\)_;_
* \(\delta^{-}\)**-decrease:** _with probability_ \(\delta^{-}_{i}\)_, i.e., the cluster containing chosen_ \(k\)_-entity decreases its size by_ \(1\) _via change of state of a one entity from the cluster to a separate state._  
_Otherwise, with probability_ \((1-\mu_{i}-\gamma_{i}-\delta^{+}_{i}-\delta^{-}_{i})\)_, nothing changes. If_ \(k_{1}\) _and_ \(k_{2}\) _belong to the same cluster, the respective probability parameters are denoted by tilde, that is,_ \(\beta_{i}\)_,_ \(\tilde{\gamma}_{i}\)_,_ \(\tilde{\delta}^{+}_{i}\)_, and_ \(\tilde{\delta}^{-}_{i}\)_. We also assume the notation convention_ \(\delta_{0}=0\)_._  
**Definition 5** (Merging of clusters). : _If there are two \(\gamma\)-activated clusters in the current time step, then they merge with probability \(\sigma^{+},\sigma^{-},\sigma^{0}\) depending on whether after merging the cluster additionally increases/does not change/decreases its size by \(1\). Otherwise, with probability \((1-(\sigma^{+}+\sigma^{0}+\sigma^{-}))\), nothing happens. Also, if there is only one \(\gamma\)-activated cluster, nothing happens._  
**Remark 1**. : _Note that the above merging process can be generalized in a simple way by introducing dependency of parameters \(\sigma\) on the sizes of the merged clusters, i.e., it is straightforward to define \(\sigma^{+}_{i,j},\sigma^{0}_{i,j},\sigma^{-}_{i,j}\) with possible dependencies on sizes \(i\) and \(j\) of the respective clusters._  
**Definition 6** (Evolution rule). : _In a given time step \(t\), pick up two integers \(k_{1}\in[N]\) and \(k_{2}\in[N]\). Then, for a given state of the system, depending on which of the configurations--(ss), (s), (sa), (a), (aa)--happens, the changes described in Definitions 3, 4, and 5 take place with their respective probabilities. Then, the same procedure is repeated in the next time step \((t+1)\)._  
All possible changes in the system during a single time step of the evolution are summarized in Table 1. The changes in a single entity according to Definitions 3 and 4 are illustrated in Figure 1. The evolution variants of the \((\mathbf{aa})\) configuration are presented in Figure 2.  
Figure 1: Possible changes in states of entities. On the left, a chosen \(k\)-entity in separative state \(s\), depicted as a black ball, changes its state to aggregative \(a\), depicted as a white ball, with probability \(v\). With probability \((1-v)\), it remains in a separative state.