TITLE: Twisted random walks
QUESTION [9 upvotes]: Suppose the points of two random walks in $\mathbb{R}^2$ are given the
step number (or time) as a third coordinate, so that they become paths in $\mathbb{R}^3$.
Here are several pairs of walks of $n=100$ steps, both starting at the origin,
with steps normally distributed with $\sigma=1$:
     
I would like to know the expected number of times that one path winds about the other,
as a function of $n$, the number of steps.
I believe for the three pairs illustrated, there is zero winding by $n=100$.
Experiments indicate winding becomes less likely as $n$ grows.
This is a bit counterintuitive to me.
The winding number of path $b(t)$ about path $a(t)$ up to $t=T$ could be defined by counting
the number of times (the normalization of) the vector $b(t)-a(t)$ turns around the origin for $t\in[0,T]$.

REPLY [4 votes]: Please find here three papers giving at least partial answers to your question:

M.A. Berger, "The random walk winding number problem: convergence to a diffusion process with excluded area." 1987:
http://iopscience.iop.org/0305-4470/20/17/028
Z. Shi, "Windings of Brownian motion and random walks in the plane." 1998:
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aop/1022855413
Y. Sun, "On the Expected Winding Number of a Random Walk on the Unit Lattice." 2005:
http://www.artofproblemsolving.com/Resources/Papers/YiSunIntel.pdf

The upshot is that the root mean square winding number grows logarithmically with the number of steps N for $N \to \infty$.