Motzkin numbers out of Random Domino Automaton

Random Domino Automaton comes from very simplified view of earthquakes. The space -one dimensional lattice -corresponds to boundaries of two tectonic plates moving with relative constant velocity. Due to irregularities of surfaces, relative motion can be locked at some places producing stress accumulation. Beyond some threshold of the stress, a relaxation took place. The size of relaxation depends on the nearby accumulated stress. Energy in the automaton is represented by balls added to the randomly chosen cell (each one is equally possible) with a constant rate -one ball in one time step. If the chosen cell is empty, it becomes occupied with probability ν or the ball is scattered with probability (1 -ν) leaving the state of the automaton unchanged. If the chosen place is already occupied, there are also two possibilities: the ball is scattered with probability (1 -µ) or with probability µ the incoming ball triggers a relaxation -balls from the chosen cell and all adjacent occupied cells are removed. An example of relaxation of size five is presented in the diagram below.

The stationary state of the system may be described by the distribution of clusters. The number of clusters of the length i, for i = 1, 2, . . ., is denoted by n i ; the number of empty clusters of length 1 is denoted by n 0 1 . Then the number of all clusters n and and the density

(1)

for i ≥ 3, where

From the above set the balance equation for the total number of clusters n and for the density ρ can be derived

In the case which refers to equal probability of triggering an avalanche for each cluster, the parameters are fixed as follows: ν = const and µ = δ i , where δ = const. Equations ( 6) and ( 5) give ρ = (θ + 1) -1 , and n = Nθ[(θ + 1)(θ + 2)] -1 , where θ = δ ν ∈ (0, ∞). Together with simple form of n 0 1 = 2n/(3 + 2θ) it allows to reduce the set of equations ( 2)-( 4) to the following recurrence:

for i ≥ 2.

We define new variables c i for i = 0, 1, . . . , by

where

Then, the equation ( 9) can be rewritten in the form

which is valid for m ≥ 0 (m = i-2). Initial data c 0 and c 1 are easily obtained from equations ( 7)-( 8), when it is transformed according the rule of equation ( 10), namely

The above equation ( 13) has the form of Motzkin numbers recurrence [3][4][5][6]. It is similar to the ubiquitous Catalan numbers recurrence (however the order of ( 13) is not 1 but is 2) and can be solved using generating functions technique [3]. For the limit case, when θ = 0, we start with c 0 = c 1 = 1, and recurrence (13) produces Motzkin numbers:

1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, . . ., etc. Thus, it is explicitly shown, how to obtain them from the Random Domino Automaton.

Below we present solution of the set ( 7)-(9). The generating function

for the recurrence (13) is equal to

For c 0 = c 1 it reduces to

where k = (2c 0 -1 2 ). From expansion of the formula (16) one can read explicitly the form of coefficients c m being solutions of the recurrence (13). Using

where

for m ≥ 0 we have

Then it follows

For any m ≥ 0 it gives

Changing indices n + j = m + 2 for m ≥ 1 we have

Thus formula (10) gives explicit solution of equations (2)-(4) for the distribution n i s for any value of θ.

Using known assymptotics for Motzkin numbers (after Benoit Cloitre, see [5])

we can present explicit formula for asymptotic behaviour for the n i distribution. In the limit case θ -→ 0, while Nθ = const,

and using (23) one obtains

Thus we found one more example of inverse-power distribution with the power equal to 3 2 . Author would like to thank prof. Zbigniew Czechowski for fruitful discussions.

[1] M. Bia lecki and Z. Czechowski (2010), arXiv:1009.4609 [nlin.CG].