Distribution of all point to point distances on a square $L \times L$ lattice?

Question

Is it possible to derive a probability density function $P(d,L)$ which gives probability of having a point to point distance $d$ on a $2D$ square lattice of size $L$ or at least it's asymptotic behavior ?

Answer 1

Two lattice points which have a horizontal (or vertical) distance of $a$ and a vertical (or horizontal) distance of $b$, will have a point to point distance of $$d(a,b) = \sqrt{a^2 + b^2}$$

In an grid of $L \times L$ points, where $ a \lt L$ and $b \lt L$, there is a nice formula for the number of instances of the distance $d(a,b)$. For a given $a$ and $b$, where $a \ge b$, the number of instances of the distance $d(a,b)$, is $$ n = \begin{cases} 2(L-a)(L-b), & \text{if $b=0$ or $b=a$} \\ 4(L-a)(L-b), & \text{otherwise} \end{cases} $$ Since we know the total number of instances of all possible distances is $\binom {L^2}{2}$, it seems we can directly compute the probability of a given distance for a given lattice size.

However, as Ross Millikan pointed out, some distances can be generated in more than one way, i.e. for some distances $d(a,b) = d(e,f)$ where $(a,b) \ne (e,f)$. The distances where this occurs (or rather the square of those distances) can be seen here.

No formula for these distances appears available. Using the formula above, it is however fairly easy to make an algorithm which computes the probabilities. I did it by running through all possible values of $a$ and $b$ for a given grid size $L$, finding the distances which could be arrived at for different $a$ and $b$ and adding the instances for these together.

Below is a graph showing the probability distribution for $L=10$

And here is the distribution for $L=40$

There seems to be an asymptotic nature to this.

The above is for $L=70$. The extra large spikes occur where a distance can be generated in more than one way.