In internal diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is added to the aggregate; then the particle is started from infinity again, and so on.

This begs the question: What does it mean to let a particle do random walk from infinity? Or, in a more practical vein, how does one simulate such a walk?

The answer to the first question is, Random walk from infinity is just the limit as $v$ goes to infinity of random walk started at $v$. More precisely, for each $v$, we can look at random walk that starts from $v$ and stops when it hits (0,0) $T$ time-steps later (where the hitting-time $T$ is random), and we can re-index it so that it starts at $v$ at time $-T$ and hits (0,0) at time 0. This gives us a probability measure on paths that end at (0,0). Now we take the limit as $v$ gets farther and farther from (0,0), and we do some work and show that the law of the random path approaches a limit, and that this limit doesn't depend on how $v$ goes off to infinity. Actually I'm bluffing; I don't know how to prove this. I assume it's in the literature; can anyone point me to a relevant book or article?

Now we come to the problem of simulation. What we want is a black box that prints out the path in reverse, chugging away until we turn it off. Note that this is not the same as having a black box that, for any $n$, prints out the last $n$ steps of the path; for, after the box has printed out that path, if we decide we want to know "What happened before that?", we can't find out by increasing $n$ and consulting the box again, because then the black box will give us the last $n+1$ steps of a DIFFERENT random path. On the other hand, if we had a box that, given the last $n$ steps of the path, could sample from its (up to 4 different) continuations one step back into the past with the appropriate conditional probabilities, one could use this box iteratively to build the sort of black box I want. It might be very hard to compute these conditional probabilities in the case of the square grid, but I suppose that would be one way to do it. A more robust (and, to my way of thinking, prettier) solution would be something more combinatorial, using ideas like coupling and domination.

In diffusion-limited aggregation on the square lattice, one lets a particle do "random walk from infinity" until it hits the current aggregate, at which point the site occupied by the particle is added to the aggregate; then the particle is started from infinity again, and so on.

This begs the question: What does it mean to let a particle do random walk from infinity? Or, in a more practical vein, how does one simulate such a walk?

The answer to the first question is, Random walk from infinity is just the limit as $v$ goes to infinity of random walk started at $v$. More precisely, for each $v$, we can look at random walk that starts from $v$ and stops when it hits (0,0) $T$ time-steps later (where the hitting-time $T$ is random), and we can re-index it so that it starts at $v$ at time $-T$ and hits (0,0) at time 0. This gives us a probability measure on paths that end at (0,0). Now we take the limit as $v$ gets farther and farther from (0,0), and we do some work and show that the law of the random path approaches a limit, and that this limit doesn't depend on how $v$ goes off to infinity. Actually I'm bluffing; I don't know how to prove this. I assume it's in the literature; can anyone point me to a relevant book or article?

Now we come to the problem of simulation. What we want is a black box that prints out the path in reverse, chugging away until we turn it off. Note that this is not the same as having a black box that, for any $n$, prints out the last $n$ steps of the path; for, after the box has printed out that path, if we decide we want to know "What happened before that?", we can't find out by increasing $n$ and consulting the box again, because then the black box will give us the last $n+1$ steps of a DIFFERENT random path. On the other hand, if we had a box that, given the last $n$ steps of the path, could sample from its (up to 4 different) continuations one step back into the past with the appropriate conditional probabilities, one could use this box iteratively to build the sort of black box I want. It might be very hard to compute these conditional probabilities in the case of the square grid, but I suppose that would be one way to do it. A more robust (and, to my way of thinking, prettier) solution would be something more combinatorial, using ideas like coupling and domination.