I apologize if this has been asked before. I tried searching all previous posts to look at different forms of this problem (such as random 2d walks on a lattice, relation to Isling's work, etc.). I posted this question to statistical physics section but I hope this is more relevant for this thread.
The problem is as follows (same as a random walk from the origin on a 2D lattice): there is an atom at the 2D cartesian origin (0,0) and hops in equal likelihood to all four directions. Understandably, the 2D joint distribution of the final coordinates $(x,y)$ after $N$ steps can be modeled through either a multinomial, or matrix convolution approach. I am trying to estimate the average number of hops it takes to reach a certain distance $d$.
I was able to decipher from numerous sources (including the original Whipple,McCrea paper that discussed the 2d random walk problem in detail) that the expected average number of hops is $ \approx d^2 $.
In order to show this, I attempted to compute the discrete probability mass function for the random variable $R_d$, defined as the number of steps it takes before it reaches distance $d$. For e.g., $P_{R_d}(R_d = r; d=10) = P_{x,y}(x_{(r)}^2 + y_{(r)}^2 \geq d^2 \text{ and } x_{(r-1)}^2 + y_{(r-1)}^2 < d^2 \text{ and } x_{(r-2)}^2 + y_{(r-2)}^2 < d^2 \cdots)$
where $x_{(r)},y_{(r)}$ are the coordinates at the $r$th step.
I tried simulating these with independent random samples and got expectations that go as $d^2 + \frac{d}{2}$, not $d^2$.
I am trying to estimate with certain precision $\tau$ and show that even when given the joint distribution on $x_{(i)},y_{(i)}$, we get $d^2$.
What would the best way be to estimate this expectation? Should I use MCMC methods, like Metropolis Hastings method? Would I set it up $R$ dependent on samples taken randomly from the distribution $p_{x,y}(x,y)$.