Winding of planar gaussian processes

Abstract

We consider a smooth, rotationally invariant, centered gaussian process in the plane, with arbitrary correlation matrix $C_{t{t}^{\prime }}$. We study the winding angle ${\varphi }_t$ around its center. We obtain a closed formula for the variance of the winding angle as a function of the matrix $C_{t{t}^{\prime }}$. For most stationary processes $C_{t{t}^{\prime }}=C(t-t^{\prime })$ the winding angle exhibits diffusion at large time with diffusion coefficient $D={\int }_0^{\infty }ds{C}^{\prime }(s{)}^2/(C(0{)}^2-C(s{)}^2)$. Correlations of $\exp (in{\varphi }_t)$ with integer $n$, the distribution of the angular velocity ${\dot{\varphi }}_t$, and the variance of the algebraic area are also obtained. For smooth processes with stationary increments (random walks) the variance of the winding angle grows as $1/2(\ln t{)}^2$, with proper generalizations to the various classes of fractional Brownian motion. These results are tested numerically. Non integer $n$ is studied numerically.

12 pages, 6 figures

References in corpus (5)

• SLE for theoretical physicists
• Winding angle distribution of 2D random walks with traps
• Windings of the 2D free Rouse chain
• SLE on doubly-connected domains and the winding of loop-erased random walks
• Winding Number of Fractional Brownian Motion

Cited by in corpus (1)

• Interference in disordered systems: A particle in a complex random landscape