A general formulation for continuum scaling limits of stochastic spanning trees is presented. The paper is organized in eight sections. Section 1 is an introduction. In Section 2 the space of immersed trees is introduced. Section 3 contains a summary of some pertinent results as two criteria for systems of random curves, which permit one to deduce regularity and roughness statements (as in Theorem 1.1). The criteria require certain scale-invariant bounds on the probabilities of multiple traversals of annuli and of lengthwise traversals of rectangles by curves in the given random family. These criteria admit a conformally invariant formulation. In Sections 4 and 5 some auxiliary results are presented. Section 4 deals with a very useful free-wired bracketing principle, while in Section 5 preliminary results on the crossing probabilities for annuli with various boundary conditions are presented. In the next section the regularity criterion is verified, treating the three models separately. Then, in Section 7 the roughness criterion is verified by means of an argument that applies to all the models just discussed. Finally, in the last section the scale-invariant bounds derived in the previous two sections are used to prove the Theorems 1.1 and 1.2. Some further comments on the geometry of scaling limits are made, too. The discussion of crossing exponents is supplemented in the Appendix by deriving a quadratic lower bound \([\lambda(k)\geq\text{const}.(k-1)^2]\) for the rate of growth of the exponent associated with the probability of \(k\)-fold traversals.
The paper offers to the reader a varied, rigorous and very good study, which can be successfully continued.