Co-authors: Tom Hutchcraft (UBC), Asaf Nachmias (Tel Aviv), Gourab Ray (Cambridge)
We show that for a unimodular random planar map, many geometric and probabilistic properties are equivalent. These include local and global properties: Negative mean curvature, invariant non-amenability, gap between the critical and uniqueness parameters for percolation, distinction between free and wired uniform spanning forests, and more.
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