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Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$.

Let $T_{M}, T_{N}$ be the smallest $n$ such that $|X_{n}^{(1)}| = M$ and $|X_{n}^{(2)}| = N$ respectively, where $|\cdot|$ denotes absolute value and $M,N$ are positive integers.

What is the $P(\min\{T_{M},T_{N}\} = T_{M})$ ?

Of course, if $M = N$ the answer is just 1/2. But how does this extend to the general case?

Further, considering $[-M,M] ×[-1,1]$, I can calculate this probability exactly by noting that $X_{n}$ can hit either one of the vertical sides prior to the horizontal ones if and only if no vertical jump occurs for time equal (in distribution) to the exit time from $[−M,M]$ for a 1-D RW.

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  • $\begingroup$ When you say the answer is just $1/2$ for $M=N$, this means the probability that $T_M = T_N$ is $0$. Are you considering a Brownian motion instead of a random walk, then? For a Brownian motion, you can compute the probability by taking a Riemann map from the rectangle to the unit disk so that the origin is fixed, and then the probability of exiting through the horizontal sides is the proportion of the circle taken up by the image of the sides. See harmonic measure. $\endgroup$
    – Douglas Zare
    May 1, 2015 at 3:24
  • $\begingroup$ Is this a discrete walk on $\mathbb{Z}^2$? $\endgroup$
    – kodlu
    May 1, 2015 at 5:20
  • $\begingroup$ Arguments of this type (showing that this probability is bounded away from 0 if the ratio of $M$ and $N$ is bounded) are used by Bollobas and Riordan in their book on percolation theory. $\endgroup$
    – Anthony Quas
    May 1, 2015 at 6:20
  • $\begingroup$ @John Lotos: The change you made did not affect whether there are ties which make the probability greater than $1/2$. You can rule those out with a random walk whose steps are $(0,\pm 1),(\pm 1,0)$ rather than a product of random walks in each coordinate. Is that what you mean? Either way, the probability for Brownian motion (say, $0.9774$ for a $1 \times 3$ rectangle) is the limit of the probabilities for random walks with $M/N \to 3$ as $N \to \infty$. $\endgroup$
    – Douglas Zare
    May 1, 2015 at 10:32
  • $\begingroup$ @Douglas Zare: I consider simple symmetric 2 dimensional RW confined on a rectangle grid (or, equivalently, the set of sites of $Z^{2}$ within the rectangle $[-M,M] \times [-N,N]$). Simple symmetric means that jumps to every neighbor site occur with probability 1/4. (or, if you like to think of two independent 1-dimensional RW, simply toss a fair coin at every step to decide in which direction (vertical or horizontal) you move) $\endgroup$
    – user71202
    May 1, 2015 at 15:12

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