5 problems

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Aldous–Lyons conjecture on soficity of unimodular random graphs
A unimodular random rooted graph $(\mathbb{G},o)$ is a random rooted graph satisfying the mass-transport principle, and it is sofic if it is a Benjamini–Schramm limit of finite gra…
Conjectureopensoficity unimodular-random-graphsarXiv posted 2 days ago
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Aldous–Lyons conjecture on ends of uniform spanning trees
Aldous–Lyons conjecture. The UST of $(G,o)$ is one-ended if and only if $G$ is one-ended.
Conjecturesolvedends-of-graphs uniform-spanning-trees unimodular-random-graphsarXiv posted 2 days ago
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Positive-speed conjecture for traces of branching random walks on unimodular random graphs
Positive-speed conjecture. The SRW on almost every trace of a transient BRW on a URG has positive speed.
Conjectureopenbranching-random-walks random-walks unimodular-random-graphsarXiv posted 2 days ago
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The soficity conjecture for minor-excluded unimodular random rooted graphs
Minor-excluded soficity conjecture. Every such unimodular random rooted graph $(G,\rho)$ is sofic.
Conjectureopengraph-minors sofic-groups unimodular-random-graphsarXiv posted 2 days ago
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Non-amenability conjecture for traces of branching random walks
Non-amenability conjecture. Let $\mu$ be the measure of the trace of a transient BRW on a URG. Then $\mu$ is non-amenable.
Conjectureopenbranching-random-walks non-amenable-graphs unimodular-random-graphsarXiv posted 2 days ago

5 problems

Aldous–Lyons conjecture on soficity of unimodular random graphs

Aldous–Lyons conjecture on ends of uniform spanning trees

Positive-speed conjecture for traces of branching random walks on unimodular random graphs

The soficity conjecture for minor-excluded unimodular random rooted graphs

Non-amenability conjecture for traces of branching random walks