Please consider a random walk on a finite N-dimensional lattice with vectors $(x_1, ..., x_N)$. We define the origin to be $(0, ..., 0)$ and the target to be at the point in the lattice furthest away from the origin - i.e. $(||x_1||, ..., ||x_N||)$ where $||x_k||$ is the integer length of the lattice in the $x_{k}$ dimension. Here, each step of the random walk is a uniformly distributed, strictly positive random integer in each of the N-dimensions with an upper-bound value defined by the requirement that one cannot exceed the dimensions of the lattice.

Is there a nice method, aside from explicit path-counting, to derive the probability density for hitting times provided an arbitrary lattice as defined above?

Some computational results: For the $N=1$ case I expected the target hitting time (defined as the number of steps to reach the target) to fit well with a logarithmic growth function of the form $A*ln(S)$ where A is a positive real number and $"S"$ is the number of integer steps one takes to reach the target from the origin. Running simulations (averaging 10,000 times) this yielded a decent fit with the value of $A$ ~ 1.146 for $||x|| = 100$, but $A$ decreases to ~1.095 for $||x|| = 1,000$ and decreased further ~1.069 for $||x||=10,000$.