Nearcritical spanning forests and renormalization
Abstract
We study random twodimensional spanning forests in the plane that can be viewed both in the discrete case and in their appropriately taken scaling limits as a uniformly chosen spanning tree with some Poissonian deletion of edges or points. We show how to relate these scaling limits to a stationary distribution of a natural coalescenttype Markov process on a statespace of abstract graphs with realvalued edgeweights. This Markov process can be interpreted as a renormalization flow. This provides a model for which one can rigorously implement the formalism proposed by the third author in order to relate the law of the scaling limit of a critical model to a stationary distribution of such a renormalization/Markov process: When starting from any twodimensional lattice with constant edgeweights, the Markov process does indeed converge in law to this stationary distribution that corresponds to a scaling limit of UST with Poissonian deletions. The results of this paper heavily build on the convergence in distribution of branches of the UST to SLE$_2$ (a result by Lawler, Schramm and Werner) as well as on the convergence of the suitably renormalized length of the looperased random walk to the "natural parametrization" of the SLE$_2$ (a recent result by Lawler and Viklund).
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 arXiv:
 arXiv:1503.08093
 Bibcode:
 2015arXiv150308093B
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 32 pages, 8 figures, to appear in Ann. Probab