# Geometry of Gaussian signals

### Alberto Rosso 1, Raoul Santachiara 2, Werner Krauth 3

#### Journal of Statistical Mechanics: Theory and Experiment 1 (2005) L08001

We consider Gaussian signals, i.e. random functions $u(t)$ ($t/L \\in [0,1]$) with independent Gaussian Fourier modes of variance $\\sim 1/q^{\\alpha}$, and compute their statistical properties in small windows $[x, x+\\delta]$. We determine moments of the probability distribution of the mean square width of $u(t)$ in powers of the window size $\\delta$. We show that the moments, in the small-window limit $\\delta \\ll 1$, become universal, whereas they strongly depend on the boundary conditions of $u(t)$ for larger $\\delta$. For $\\alpha > 3$, the probability distribution is computed in the small-window limit and shown to be independent of $\\alpha$.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Instituut voor Theoretische Fysica,
Instituut voor Theoretische Fysica
• 3. Laboratoire de Physique Statistique de l'ENS (LPS),
CNRS : UMR8550 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot – Ecole Normale Supérieure de Paris - ENS Paris