# Products of random matrices and generalised quantum point scatterers

### Alain Comtet 1, 2, Christophe Texier 2, 3, Yves Tourigny 4

#### Journal of Statistical Physics 140 (2010) 427-466

To every product of $2\times2$ matrices, there corresponds a one-dimensional Schr\'{o}dinger equation whose potential consists of generalised point scatterers. Products of {\em random} matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in $\text{SL}(2, {\mathbb R})$. We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.

• 1. Unite mixte de service de l'institut Henri Poincaré (UMSIHP),
CNRS : UMS839 – Université Paris VI - Pierre et Marie Curie
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Laboratoire de Physique des Solides (LPS),
CNRS : UMR8502 – Université Paris XI - Paris Sud
• 4. School of Mathematics,
University of Bristol