Consider the square grid $$\{-n, -n+1, ..., n-1, n\} \times \{-n, -n+1, ..., n-1, n\} \subset \mathbb{Z}^2$$ As usual, connect $(i,j)$ to $(i',j')$ with an edge if $|i-i'|+|j-j'|=1$.

We randomly specify the environment as follows: At each vertex we designate one of the 4 outgoing directions to have probability 1/2, and the remaining 3 outgoing directions to have probability 1/6. The choice of which outgoing edge gets probability 1/2 is done uniformly and independently at random for each vertex.

Then we consider the random walk starting at $(0,0)$ that moves from each vertex in each direction with the chosen probabilities $1/2$ or $1/6$.

I want to show that for n large, if I fix a random environment and start the random walk at $(0,0)$, then the probability that the random walk hits the top edge of the grid before any of the other edges is very likely to be close to 1/4.

existsa vertex (i,j) (which may depend on the random environment) such that the probability that the random walk starting at (i,j) hits the top edge of the grid before any of the other edges is very likely to be close to 1/4? $\endgroup$ – Alex Jan 10 '17 at 17:05