Given there is a 3D random walk starting at the origin $\vec{0}$. For simplicity, lets assume a lattice, where the walker randomly goes exactly one unit per step in one of the coordinate directions. This means after one step the walker is at one of the six positions $(0,0,1)$, $(0,0,-1)$,...,$(-1,0,0)$ with a probability of $1/6$.
The question is: For an arbitrary point on the lattice $\vec{p}$, what is the expected minimum distance of the walker to this specific point, given the random walk consists of $N$ steps.
More formally, let $\vec{r}(n)$ be the position of the walker at step n. The distance between walker and $p$ for each step is $d(n) = \sqrt{(\vec{r}(n)-\vec{p})^2}$.
What is the expected minimum distance $\left<d_{min}\right> = \left<\min\limits_{n = 0,\dots, N}d(n)\right>$?
Until now, I haven't found the solution for this in the literature. The problem is that I don't really know what to look for. Is there an established name for this problem? I assume that the solution is relevant when looking at the dynamics of diffusive processes. For example: I have a molecule with a certain diffusion coefficient. What is the probability that it is within 1 nm of my target site given a certain amount of time?