Collisions in random walk in $\mathbb{Z}^n$ Given a set $S$ of $r$ points in $\mathbb{Z}^n$, $S=(p_1,p_2,p_3.., p_r)$ , each a starting point for random walk with step size 1. What is the probability they will all eventually meet at the same point?
Ie. at each timestep add either -1 or 1 with 50% probability to a uniformly random coordinate of each point in $S$ separately.
Given S, what is the probability that eventually all points in S are the same?
 A: Please describe your problem more careful - you have some finite set $E\subset \mathbb{R}^n$ and what about random walks? Where does they start, how many of them, of what type?
Say, given $m$ stochastic processes in discrete time you can define 
$$
f(x_1,...,x_m) = \sum\limits_{i\neq j} (x_i - x_j)^2
$$
and look for the probability that $f(X_1,...,X_n)$ will reach the zero level. For any finite horizon you can solve it as a iterative procedure. For an infinite horizon it depends on the distribution of your random walks - please provide more details.
Edited: for a compact set the probability will be always $1$. On the other hand, the set $\mathbb{R}^n$ is not a compact. You can consider your process $X = (X_1,...,X_r)$ as a Markov chain - and hence this is a problem of reachability of the line $A = \{x_1=x_2=...=x_n\}$. Since your Markov chain has infinitely many states, I do not think there are developed algorithms to calculate this quantities.
A: Since a random walk in $n(r-1)$ dimensions is transient (it has a positive probability of not returning to the origin) for $n(r-1) \ge 3$, the probability that your random walk has all $n$ points coincide at some time is less than $1$. 
An upper bound for the probability that the walks will coincide at some point is the expected number of meetings $\sum_t P(\text{walks meet at time }t)$. This probability can be written as a multidimensional sum of binomial coefficients over a power of $2n$. 
A standard trick for random walks applies. Let $M(x)$ be the generating function for the probability of meeting after $t$ steps. Let $O(x)$ be the generating function for the probability that a random walk starting at the origin returns to the origin in $t$ steps. Let $F(x)$ be the generating function for the probability of first meeting after $t$ steps. Then $M(x) = O(x)F(x)$, so $F(x) = M(x)/O(x)$. To compute the probability of ever meeting $F(1),$ you can compute $M(1)/O(1)$, the expected number of meetings divided by the average time at the origin if the points start at the origin.
I don't anticipate a simple closed form for any of these, although it would not surprise me if there were a way to give the probabilities as values of hypergeometric functions, particularly for $n=1$.
