On the pertinence to Physics of random walks induced by random dynamical systems: a survey
Résumé
Let $\mathbb{X}$ be an abstract space and $\mathbb{A}$ a denumerable (finite or infinite) alphabet. Suppose that $(p_a)_{a\in\mathbb{A}}$ is a family of functions $p_a:\mathbb{X}\to\mathbb{R}_+$ such that for all $x\in\mathbb{X}$ we have $\sum_{a\in\mathbb{A}} p_a(x)=1$ and $(S_a)_{a\in\mathbb{A}}$ a family of transformations $S_a:\mathbb{X}\to\mathbb{X}$.
The pair $ ((S_a)_a, (p_a)_a)$ is termed an \textit{iterated function system with place dependent probabilities}. Such systems can be thought as generalisations of random dynamical systems. As a matter of fact, suppose we start from a given $x\in\mathbb{X}$; we pick then randomly, with probability $p_a(x)$, \text{the} transformation $S_a$ and evolve to $S_a(x)$. We are interested in the behaviour of the system when the iteration continues indefinitely.
Random walks of the above type are omnipresent in both classical and quantum Physics. To give a small sample of occurrences we mention: random walks on the affine group, random walks on Penrose lattices, random walks on partially directed lattices, evolution of density matrices induced by repeated quantum measurements, quantum channels, quantum random walks, etc.
In this article, we review some basic properties of such systems and provide with a pathfinder in the extensive bibliography (both on mathematical and physical sides) where the main results have been originally published.