What is the probability of random walking ant to be at a position after some finite steps on an infinite grid?

Question

Is it even calculable? What if the grid is infinitely dimensional? Lets say that it is a simple random walk, and probability to move to any neighboring position is equal, but other types are also interesting.

Answer 1

On a finite-dimensional cubic grid, it's possible to calculate the probability that an ant starting at $P$ lands at point $Q$ after $n$ steps, since there are at any time only finitely many positions the ant can occupy, although I don't know a closed form for the probability.

On a cubic grid with uncountably many dimensions: Unless $P = Q$ and $n = 0$, the probability is $0$. For $n = 0$ and $P \neq Q$ the case is clear. So suppose $n \neq 0$. Suppose the ant makes a single step. Since it has uncountably many options, with probability $1$ the ant will not land on $Q$. Instead, it will land on a random point $X$ that differs from $Q$ in at least one coordinate. The chance that this coordinate will never change again is $1$, so the chance that it does change and the ant goes to $Q$ is $0$.

On a cubic grid with countably infinite dimensions, the ant will land on $Q$ with probability exactly $1/6$. Allow me to elaborate: Let $\Omega$ be the set of possible directions the ant can go. $\Omega$ is countably infinite. Consider a probability distribution $P$ on $\Omega$ such that $P(\{\omega\})$ is well-defined for all $\omega \in \Omega$ (i.e. $\{\omega\}$ is measurable for all $\omega \in \Omega$) and there exists a $p \in [0,1]$ such that $P(\{\omega\}) = p$ for all $\omega \in \Omega$.

By $\sigma$-additivity of probability distributions and by the law of total probability, we have

$$1 = P(\Omega) = P\left(\bigsqcup_{\omega \in \Omega}\{\omega\}\right) = \sum_{\omega \in \Omega}P(\omega) = \sum_{\omega \in \Omega}~p = \begin{cases}\infty & \text{if}~~p > 0 \\ 0 & \text{if}~~p = 0\end{cases}$$

which is clearly a contradiction. So there does not actually exist a probability distribution for the ant's behavior according to your specifications, therefore any such probability distribution will result in the ant landing on $Q$ with probability exactly $1/6$.

Answer 2

The probability wont be 0 for sure but the answer will be closer to 0 but it has some value > 0. The probability cant be determined as it an infinite grid and therefore we don't know the value in the denominator. In general probability of an event is ratio of number of favourable cases to the total number of cases(which here is infinity) and hence we cant determine that value which is for sure greater than 0.