An explicit lower bound for the growth exponent of three-dimensional loop-erased random walk

Abstract: In this paper, we derive a new lower bound for the Hausdorff dimension of 3D Brownian cut points by proving an explicit upper bound ($<0.9999$) for $ξ_3(1,1)$, the intersection exponent for two independent Brownian motions in 3D. Consequently, the growth exponent of 3D loop-erased random walk is at least $1.0001$.