Suppose we have a circle of radius $R$ centered in the origin of a $x,y$ cartesian reference frame. A particle starting from the center of the circle is moving with a speed given by:
$$\overrightarrow{v}=\frac{1}{\sqrt{k^2+h^2}}v\_0(k\underline i+h\underline j)$$ where $v\_0=const$, $\underline i$ and $\underline j$ are the unit vectors of $x$ and $y$ and $k\in\mathbb{N},h\in\mathbb{N}$ random integer numbers uniformly distribuited and uncorrelated between them, such that $-\alpha\le k\le \alpha$, $-\alpha\le h\le \alpha$, $k\neq 0,h\neq 0$ and $\alpha\in\mathbb{N}$. Every time step $\Delta t$, a random $k$ and a random $h$ are picked and so the particle runs through a space $s=|\overrightarrow v|\Delta t$. The question is: calling $P(t)$ the probability to find the particle at the point $(x(t),y(t))$ at time $t$, after how many time steps the probability to find the particle outside the circle $(x(t)^2+y(t)^2)\gt R$ is: $P(t)\ge P\_0?$
Thanks in advance.



Random walk in a circle
This is mainly for reference, I have not found a simple answer outside of the diffusion approximation. 


Assume $\alpha\gg 1$. The random walk has unit step size and the angle $\phi$ of a step with the $x$-axis is given by $\phi=\arctan(h/k)$ with $h,k$ two independent real numbers that are uniformly distributed in the range $(1,\alpha)\cup(-1,-α)$. The corresponding probability distribution $P(\phi)$ in the first quadrant $(0<\phi<\pi/2)$ is given by


$$P(\phi)=\frac{1}{2(\alpha-1)^2}\times\begin{cases}\alpha^2/\sin^2\phi-1/\cos^2\phi&{\rm if}\;\;1<\tan\phi<\alpha\\ \alpha^2/\cos^2\phi-1/\sin^2\phi&{\rm if}\;\;1/\alpha<\tan\phi<1\end{cases}$$


Here is a plot of $P(\phi)$ versus $\phi$ in the first quadrant for $\alpha=10$. (It is repeated identically in the other three quadrants.)


![](https://www.lorentz.leidenuniv.nl/beenakker/MO/randomwalk.png)


As expected the steps have the largest probability at a 45 degree angle with the $x$ and $y$ axes.


So to find the probability distribution $P(x,y,t)$ after $N=t/\Delta t$ steps we seek the probability distribution of the complex number $z=x+iy$ given by


$z=\sum\_{n=1}^{N}\exp(i\phi\_n)$ with $\phi\_n$, $n=1,2,\ldots N$ drawn independently from the distribution $P(\phi)$. Even for an isotropic distribution of $P(\phi)=$ constant, this has no simple answer (except for the large time limit, when the diffusion approximation holds and $P(x,y,t)$ is just a Gaussian).


