Hi,


consider the following random walk on the lattice $\{0,\dots,n\}^2$. It starts at $(0,0)$ and then move either up or right, with probability respectively $p$ and $1-p$. Once it reaches the right border (respectively the up border), it goes up (respectively it goes right) to $(n,n)$. What properties do we know about this random walk ? Especially can we say anything about the times when it crosses the main diagonal ?



Random walk on a two-dimensional uniform grid
Turn your square one fourth turn to the left and project it down.


To give some details: consider a simple random walk $Y$ on $\mathbb{Z}$ that is constrained to go left when yours goes up, and to go right when yours goes right. When your walk is on the boundary, then draw randomly independently the direction of $Y$. Since $Y$ is a simple random walk, all the results you can dream of are available. The only issue is that the two walks decorelate as soon as yours hit the boundary, but this is no problem for your question on the diagonal. Indeed, if $Z$ is the walk that is equal to $Y$ for steps $\leq n$, and equals the projection of $Y$ to $[-n+k,n-k]$ at steps $n+k$, then up to a $\sqrt{2}$ factor $Z$ is the projection on the second diagonal of your random walk, and it hits $0$ at time $t$ (meaning that your walk hits the diagonal at that time) if and only if $Y$ does.


