A random walk that reflects off the boundary at infinity yields new algorithmic constructions of the free uniform spanning forest and a conjectural embedding framework for supercritical Liouville quantum gravity.
Interlacement percolation on transient weighted graphs
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abstract
In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value u_* for the percolation of the vacant set is finite. We also prove that, once G satisfies the isoperimetric inequality IS_6 (see (1.5)), u_* is positive for the product GxZ (where we endow Z with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value u_*.
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Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity
A random walk that reflects off the boundary at infinity yields new algorithmic constructions of the free uniform spanning forest and a conjectural embedding framework for supercritical Liouville quantum gravity.