Determine the exact three-dimensional LERW growth exponent

Determine the exact value of the growth exponent β for three-dimensional loop-erased random walk, whose currently known bounds are 1 < β ≤ 5/3.

Background

The paper defines β as the asymptotic exponent governing the length of the loop-erasure of a three-dimensional simple random walk stopped upon exiting a ball. Prior to the paper, the best rigorous information was β ∈ (1, 5/3], while simulations suggested β ≈ 1.624.

The paper proves the explicit lower bound β ≥ 1.0001 by establishing an upper bound below 1 for the three-dimensional Brownian intersection exponent ξ₃(1,1). It does not determine the exact value of β, so the exact exponent remains unresolved.