Let us consider a simple random walk on $\mathbb{Z}^2$ started at $(x,0)$ and killed upon hitting the origin. Define the total winding number $w_x$ around the origin to be the (signed) number of complete rotations around 0 (up to the last step before the random walk is killed). What is known about the distribution of $w_x$ as $x\to\infty$? More precisely, does $w/\log(x)$ converge in distribution? So far I have not been able to find a reference addressing this particular question. Indeed, most references on winding numbers of random walks seem to focus on the distribution after a (large) fixed number of steps.

One would expect the distribution to be related to the analogue for Brownian motion started at $(x,0)$ and killed upon hitting the unit circle. By going to radial coordinates it is easy to see that this total winding is given by the integer part of a Cauchy random variable with scale parameter $\log(x)/(2\pi)$.

  • $\begingroup$ See the earlier MO question, "Twisted random walks." Andreas Rüdinger provides three relevant references, including "On the Expected Winding Number of a Random Walk on the Unit Lattice." $\endgroup$ – Joseph O'Rourke Jun 23 '16 at 14:06
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    $\begingroup$ Thanks for the pointer. I checked those references, but did not see any results directly relevant to this problem. $\endgroup$ – Timothy Budd Jun 23 '16 at 14:29
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    $\begingroup$ Section 5 of Shi's paper is very relevant to your question.... $\endgroup$ – ofer zeitouni Jun 25 '16 at 7:35
  • $\begingroup$ Perhaps I was too quick to dismiss Shi's paper, it must be relevant in some way (disregarding the fact that the random walk is assumed to be spherically symmetric there). But I'll have to think about whether the result can be applied to say something about this particular problem. $\endgroup$ – Timothy Budd Jun 25 '16 at 10:57
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    $\begingroup$ I believe that what you need to consider is strong embedding together with an estimate that the contribution of "small windings" for random walk is asymptotic negligible. I have not checked the details though. $\endgroup$ – ofer zeitouni Jun 25 '16 at 17:11

Ofer Zeitouni's suggestions based on the paper of Shi can most likely be made precise to prove the asymptotic result using Brownian motion. At the risk of self-promotion and self-answering my 2-year-old question, let me advertise an alternative combinatorial solution that gives exact statistics for the total winding angle $\Theta_x~(\approx 2\pi w_x$) of a simple random walk started at $(x,x)$ and killed upon hitting $(0,0)$ for any finite $x\geq 1$.

Winding angle

Let $[\Theta_x]\in\pi(\mathbb{Z}+\tfrac{1}{2})$ be this angle rounded to the nearest half-integer multiple of $\pi$. Then it has characteristic function given explicitly by $$ \mathbb{E}[e^{i b[\Theta_x]}] = \frac{1}{2\cos(\tfrac{\pi b}{2})} [z^{2x}] \left(\frac{1-z}{1+z}\right)^{\!1-|b|}\qquad\text{for }b\in[-2,2], $$ where $[z^{2x}]\,\cdot$ means the coefficient of $z^{2x}$ in the series expansion around $z=0$. This is proved in Theorem 6 (together with Proposition 5) of

T. Budd, The peeling process on random planar maps coupled to an O(n) loop model (with an appendix by Linxiao Chen), arXiv:1809.02012,

where it arises as a byproduct of the study of a certain exploration process of random planar maps.

Straightforward singularity analysis then shows that $$ \mathbb{E}[\exp(ib \tfrac{[\Theta_x]}{\log x})] \xrightarrow{x\to\infty} e^{-|b|}\qquad\text{for }b\in\mathbb{R}, $$ which is precisely the characteristic function of the standard Cauchy random variable. Hence we deduce the convergence in distribution of $[\Theta_x]/\log x$ to Cauchy, and the same for $\Theta_x/\log x$ and $2\pi w_x/\log x$.


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