2026/07/20 by Gefei Cai
Mathematics · #math.PR #math-ph #math.MP
We introduce a family of random connected closed subsets of planar Brownian motion, called Brownian loop-catchers, which interpolate between the continuum loop-erased random-walk (LERW) and the Brownian trace. This provides the canonical continuation of Brownian loop soup clusters to central charges -2≤ c<0. For each such c, the corresponding loop-catcher satisfies the following recovery property: adding all loops from an independent Brownian loop soup of intensity -c/2 that intersect it recovers the Brownian trace. Furthermore, no such law exists for c<-2. We also show that its outer boundary is locally SLEκ with κ= (1)/(3)(13 - c - √((1-c)(25-c)))∈[2,\frac83), and the probability that it intersects an interior ball of radius ε is asymptotically proportional to |logε|-1+(c)/(2) when -2<c<0. Therefore, a planar Brownian trace contains an SLEκ-type curve for every κ∈[2,\frac83]. Our construction begins with a random-walk loop-catcher on any finite graph, whose law is determined by a finite linear system. We prove that its solution is nonnegative for -2≤ c<0, while nonnegativity can fail for c<-2. The key ingredient is a new entangled multipath LERW, which recovers a union of independent random-walk paths when decorated with a single common random-walk loop soup. We then prove that the random-walk loop-catcher converges to the Brownian loop-catcher under lattice approximations. To this end, we propose a novel Green function test which converts the recovery property of the Brownian loop-catcher into all mixed moments of Green functions in the remaining domain, based on the entangled multipath LERW. Consequently, the recovery property characterizes the full law of the Brownian loop-catcher, not only its filling. The Green function test also extends to the three-dimensional case.