Say you have a building of $HJ$ rooms, where $H$ and $J$ are positive integers (a rectangular grid of rooms of size $H$ times $J$). You can label the rooms $(h,j)$ where $1 \le h \le H$ and $1 \le j \le J$. One person enters the building of rooms at $(1,j_0)$ and the other person enters at $(h_0,1)$. After each minute, they choose randomly another room which must be adjacent to it ; i.e. if you are at coordinate $(h,j)$, you can go to $(h \pm 1, j)$ or to $(h, j \pm 1)$ with probability $1/4$, but the probability goes to $1/3$ if you are on a "side" (i.e. at coordinate $(1,j)$ you go to $(2,j)$ or $(1, j \pm 1)$ with probability $1/3$, and at coordinate $(1,1)$ you go to $(2,1)$ or $(1,2)$ with probability $1/2$). When the person that entered at $(1,j_0)$ reaches the room $(h_0,1)$, she "exits" the building using the entrance the other person used (this is not the case if the person at $(1,j_0)$ goes back to $(1,j_0)$). The same applies to the other person ; if person 2 goes back to the entrance of person 1 she exits the building.
The question is : What is the probability that the two persons are in the same room at some point during their walk?
My attempts : for $H=J=1$, the probability is $1$. For $H=2$ and $J=1$ (or the opposite) and $h_0 = H$, the probability is zero, since the two persons will just switch rooms and leave, without meeting (the hallways don't count :P). So it seems that it depends a lot on $H$ and $J$. Actually, if $H = 1$ and $J \ge 1$ then the probability is $1$ if $j_0$ is odd and $0$ if $j_0$ is even, because then the number of rooms between the two is always even, so for this number to reach $0$ we must have $j_0$ odd. Conversely, if $j_0$ is odd then since the two persons are forced to cross the same rooms they must meet at some point. The same goes by symmetry if $J = 1$.
Obviously it cannot only depend on the parity of $H$ and $J$ ; when $H$ and $J$ are not $1$ there always exists paths where the two persons meet and don't meet. I can do by hand the computations for $H=J=h_0=j_0=2$ but in the general case I have no idea how to tackle this problem. This is actually a question in my friend's homework, and there is nothing in the course about stochastic processes.
Thanks in advance,