Abstract:Let $\mathcal G$ be an infinite graph -- not necessarily one-ended -- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and "reflects off of $\infty$" infinitely many times.
We show that the Aldous-Broder algorithm for the random walk reflected off of $\infty$ gives the free uniform spanning forest (FUSF) on $\mathcal G$. Furthermore, Wilson's algorithm for the random walk reflected off of $\infty$ gives the FUSF on $\mathcal G$ on the event that the FUSF is connected, but not in general.
We also apply the theory of random walk reflected off of $\infty$ to study random planar maps in the universality class of supercritical Liouville quantum gravity (LQG), equivalently LQG with central charge in $(1,25)$. Such random planar maps are infinite, with uncountably many ends. We define a version of the Tutte embedding for such maps under which they conjecturally converge to LQG. We also make several conjectures regarding the qualitative behavior of stochastic processes on such maps -- including the FUSF and critical percolation.
From: Jinwoo Sung [view email]
[v1]
Mon, 23 Jun 2025 16:41:47 UTC (164 KB)
[v2]
Tue, 30 Jun 2026 02:14:53 UTC (168 KB)
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