Fluctuations of loop-erased random walk length and uniform spanning tree graph distance

Determine the scaling limit of the fluctuations of the length of loop-erased random walk up to time n and of graph distance in the uniform spanning tree, including the limiting process under the fluctuation scaling considered for graph distance and effective resistance of the random-walk trace.

Background

The paper identifies stable-law fluctuation limits for graph distance and effective resistance on the trace of a random walk in dimensions four and five. It suggests that the coupling between long-range intersections of the random-walk trace and long-range percolation could also address loop-erased random walk and uniform spanning tree observables.

The authors emphasize that loop-erased random walk depends strongly on long-range edges, while also requiring the temporal ordering of long-range intersections to be incorporated. The unresolved issue is to identify the appropriate fluctuation scaling limit and its limiting process for the loop-erased-walk length and the uniform-spanning-tree graph distance.