[cond-mat/0306716] Superconductors with broken time-reversal symmetry: Spontaneous magnetization and quantum Hall effects

Winding of planar Gaussian processes

Pierre Le Doussal¹, Yoav Etzioni² and Baruch Horovitz²

We consider a smooth, rotationally invariant, centered Gaussian process in the plane, with arbitrary correlation matrix C_tt'. We study the winding angle phi _t, around its center. We obtain a closed formula for the variance of the winding angle as a function of the matrix C_tt'. For most stationary processes C_tt' = C(t−t') the winding angle exhibits diffusion at large time with diffusion coefficient $D=\int _0^\infty \rmd s\, C'(s)^2/(C(0)^2-C(s)^2)$ . Correlations of exp(in phi _t) with integer n, the distribution of the angular velocity $\dot \phi_t$ , 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 $\frac {1}{2} (\ln t)^2$ , with proper generalizations to the various classes of fractional Brownian motion. These results are tested numerically. Non-integer n is studied numerically.

Received 9 April 2009 , accepted for publication 18 May 2009 Published 3 July 2009

Winding of planar Gaussian processes

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