Stochastic Model of Tree Growth

Abstract

Let T^N be a Caley tree, i.e. a partially ordered set with a unique minimal element $\bar{O}$such that for any element $a\ne \bar{O}$ there exists only one element â immediately preceding a and for any a the set C(a) of immediately following elements contains exactly N of them. We shall call the subsets of T^N configurations as in the statistical mechanics. To every element a∈T^N and any instant of discrete time t∈{O,l,...}we shall ascribe the logical variable X(a,t), which is “true” if a belongs to the configuration at the time t and “false” in the opposite case. Let us consider the stochastic model of tree growth, defined by the following Boolean equations

$$ \left\{ {\begin{array}{*{20}c} {X\left( {a,t + l} \right) = \mathop V\limits_{b \in C\left( a \right)} \,X\left( {b,t} \right)v\left( {\bar \eta \left( {a,t} \right) \wedge X\left( {a,t} \right)} \right)\gamma \left( {\xi \left( {a,t} \right) \wedge \bar X\left( {a,t} \right) \wedge X\left( {\hat a,t} \right)} \right)} & {} & {} & {} & {} \\ {\bar O \ne a \in T^N ,t \geqslant O,} & {} & {} & {} & {} \\ {X\left( {\bar O,t} \right) = true,t \geqslant O} & {} & {} & {} & {} \\ \end{array} } \right. $$

(1)

with some initial field of logical variables $X\left( a,o \right),a\in {{T}^{N}}\backslash \left\{ {\bar{O}} \right\}$. In these equations η and ξ are “external” random fields of logical variables such that ξ(a,t), as well as η(a,t) are jointly independent for all a and t and the dependence between ξ, and η can be given by the condition

$$\xi \left( a,t \right)\Rightarrow \bar{\eta }\left( \hat{a},t \right),a\in {{T}^{N}}\backslash \left\{ {\bar{o}} \right\}$$

(2)

.