Winding number of a random walk on the square lattice before hitting the origin

Question

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 $\log(x)/(2\pi)$.

Answer 1

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 $x\geq 1$.

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 $2\pi w_x/\log x$.