Scaling limits for minimal and random spanning trees in two dimensions

Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree

“…As already noted, this theorem extends the results of [2,48] to include scaling of the intrinsic metric and uniform measure. We further note that the tightness in [2,48] was essentially a finite-dimensional statement, since it described the shape in Euclidean space of the tree spanning a finite number of points, while the result above establishes tightness for the entire space.…”

Traveling Salesman Problem, Conformal Invariance, and Dense Polymers

“…Here the analog of TSPIII would involve so-called uniform spanning trees (USTs) in place of dense polymers. The equivalence of USTs and MSTs in d = 2 is not excluded by rigorous results [12], and in our view is supported by existing numerics [13]. A tree in two dimensions is equivalent to a nonintersecting loop (take the boundary of a "thickened" tree), and the universality classes of USTs and dense polymers are the same in d = 2 [14] (and also the same as stochastic Loewner evolution at parameter value κ = 8 [15]).…”

Theory of minimum spanning trees. I. Mean-field theory and strongly disordered spin-glass model

“…We note that for a different model of random trees on a lattice, that of uniform spanning trees (each spanning tree on a finite graph is given equal probability), similar behavior of the dimensions was proven, except that d c UST = 4 and D UST = 4 for d > 4 [39]. Hence the universal properties of MSTs and uniform spanning trees are distinct, at least in sufficiently high dimensions [40,41].…”