Non-erasure probability in a loop-erased random walk in three dimensions

Question

Perform a simple random walk $S(0),S(1),S(2)...$ on $\mathbb{Z}^3$, that is $S:\mathbb{N}\to\mathbb{Z}^3$ with $||S(i)-S(i+1)||_1 = 1$ for all $i$. Now let $\Gamma_n$ be the loop-erasure of the first $n$ steps of $S$. Consider $n$ to be fixed. What are the best bounds (upper and lower) for the probability that $S$ never hits $\Gamma_n$ after time $n$, that is, the event $S(m)\notin\Gamma_n$ for $m>n$? Equivalently, perform a loop-erased random walk from the origin for $n$ (pre-erasure) steps. What is the probability that an independent simple random walk begun at the origin does not intersect the loop-erased random walk except at the origin?

In Loop-Erased Self-Avoiding Random Walk in Two and Three Dimensions, Lawler argues that for a fixed point in $\Gamma_n$ the probability that it is not hit by the proceeding simple random walk is about $n^{1-\alpha}$ where $\alpha$ is the growth exponent of loop-erased random walk in 3 dimensions (to my knowledge the best estimates on $\alpha$ are due to Kozma). Furthermore, he estimates the expected size of a loop when it is erased, but it's not clear that he looks at this specific intersection probability. However, it can be shown from Lawler's work that an infinite LERW and independent SRW both begun at the origin intersect infinitely often.

I apologize if this result is well known, but my search of the literature did not reveal the answer I was looking for.

Answer 1

You are restating the fundamental hard problem in this area. You are writing down the probability that the nth point on the simple random walk is not erased. Summing this from 0 to n gives the expected number of points that have not been erased from the first n steps of the random walk. This gives the growth exponent (If there are n^a points that have not been erased, then it takes n^a points of the loop-erased walk to reach distance n^{1/2}). As far as I know, the best bounds on this are still the ones that I mentioned in my paper. Basically, we know that "cut points" of the simple random walk are not erased which gives one bound and the "third moment" estimate gives a bound in the other direction. Neither of these bounds should be the correct answer.