What is the probability that a simple random walk(travelling only on the grid with step length 1 and equal probabilities for all direction parallel to the axis) , beginning at the origin, will return to the origin at time $t=2n$ for the first time in 2,3 dimensions(or higher)?
I've already understand how to derive the PDF in 1-d situation but not able to analyse the higher dimension situation. So I carried out a Monte Carlo simulation and get the following result for the asymptotic situation: for 2-d:$$P[S_{2n}^{(2)}=(0,0)]\sim N^{-1.244}*(0.3385)$$ and 3-d:$$P[S_{2n}^{(3)}=(0,0,0)]\sim N^{-1.517}*(0.3142)$$
The coefficients are not accurate as the simulation is only carried out for limited steps, but it's still interesting that the 3-d situation has the similar property as in 1-d situation. As for 1-d I got$$P[S_{2n}=0]\sim N^{-1.517}*(0.887)$$ which is in accord with the accurate result $$P[S_{2n}=0]=\frac{1}{2n-1}C_{2n}^n2^{-2n}\sim \frac{1}{n^{1.5}} $$