Abstract
A stationary random graph is a random rooted graph whose distribution is invariant under re-rooting along the simple random walk. We adapt the entropy technique developed for Cayley graphs and show in particular that stationary random graphs of subexponential growth are almost surely Liouville, that is, admit no non constant bounded harmonic functions. Applications include the uniform infinite planar quadrangulation and long-range percolation clusters.
Highlights
A stationary random graph (G, ρ) is a random rooted graph whose distribution is invariant under re-rooting along a simple random walk started at the root ρ
The entropy technique and characterization of the Liouville property for groups, homogeneous graphs or random walk in random environment [24, 25, 26, 27, 29, 30] are adapted to this context
We construct (Proposition 5.4) a stationary and reversible random graph of subexponential growth which is planar and transient. This indicates that the theory of local limits of random planar graphs of bounded degree developped in [10] can not be extended to the unbounded degree case in a straightforward manner
Summary
A stationary random graph (G, ρ) is a random rooted graph whose distribution is invariant under re-rooting along a simple random walk started at the root ρ (see Section 1.1 for a precise definition). In the case of graphs of bounded degree we show in Proposition 3.6 that stationary non-Liouville random graphs are ballistic. We reinterpret ideas from measured equivalence relations theory to prove (Theorem 4.4) that if a stationary random graph of bounded degree (G, ρ) is non reversible the simple random walk on G is ballistic, improving Theorem A of [37] and extending [38] in the case of transitive graphs. We construct (Proposition 5.4) a stationary and reversible random graph of subexponential growth which is planar and transient This indicates that the theory of local limits of random planar graphs of bounded degree developped in [10] can not be extended to the unbounded degree case in a straightforward manner. Added in proof: After the completion of this work, Gurel-Gurevich and Nachmias [21] proved that the UIPQ (and the UIPT) is recurrent which implies Corollary 1.2
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