Random map: Cosmos — All that is, or was, or ever will be

Random map

cosmos 12th March 2017 at 11:55am

aka random map model, or random mapping

For each point in phase space, one chooses at random another point in phase space as being its successor in time, i.e. we have a random map, $T$ , from a finite Set of $M$ points, to itself. It can be shown to be a limiting case of a Kauffman Random Boolean network, with in-degree $K \rightarrow \infty$ .

To each attractor (labelled by $s$ ), we assign a weight $W_s$ corresponding to the fraction of points in its basin of attraction.

Statistics of attractors

Probability distribution of size of basin of attraction

Joint probability distribution of two attractor weights

Probability distribution of $Y=\sum\limits_s W_s^2$

Probability that a random map of $M$ points is indecomposable (i.e. map has a single attractor)

$Q_M = \frac{(M-1)!)}{M^M} \sum\limits_{n=0}^{M-1} \frac{M^n}{n!} \sim \sqrt{\frac{\pi}{2M}}$

$\sim$ is for large $M$

Probability that the map is indecomposable and the attractor is of period $l$ :

$Q_M(l) = \frac{M!}{(M-l)!M^{l+1}}$

Probability distribution of number of attractors

See Probability Distributions Related to Random Mappings , A Property of Randomness of an Arithmetical Functions

The average number of attractors is $\langle A \rangle = \frac{1}{2} \log{M} + O(1)$

Probabilities related to a point chosen at random

Probability that a randomly chosen point falls into an attractor of weight $W$ and period $l$

Probability that a randomly chosen point ends up on an attractor of weight $l$ :

$P(l) = \sum\limits_{n \geq l} \frac{\Gamma{(M)}}{\Gamma{(M-n+1)}} \frac{1}{M^n}$

For large $M$ , this gives

$P(l) = \frac{1}{\sqrt{M}} \int_x^\infty e^{-y^2/2} dy$

where $l = \sqrt{M} x$ . This gives the average $\langle l \rangle = \sqrt{M} \sqrt{\frac{\pi}{8}}$ , and variance $\langle l^2 \rangle - \langle l \rangle^2 = M \left ( \frac{2}{3} - \frac{\pi}{8} \right )$

Random mappings with constraints, and other extensions

In Probability Distributions Related to Random Mappings , some of the above results are extended to the case without self-1-loops, $T(i) \neq i$ , and where the function is one-to-one