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Floodings of metric graphs

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Abstract

We consider random labelings of finite graphs conditioned on a small fixed number of peaks. We introduce a continuum framework where a combinatorial graph is associated with a metric graph and edges are identified with intervals. Next we consider a sequence of partitions of the edges of the metric graph with the partition size going to zero. As the mesh of the subdivision goes to zero, the conditioned random labelings converge, in a suitable sense, to a deterministic function which evolves as an increasing process of subsets of the metric graph that grows at rate one while maximizing an appropriate notion of entropy. We call such functions floodings. We present a number of qualitative and quantitative properties of floodings and some explicit examples.

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Acknowledgements

We are grateful to Omer Angel, Jérémie Bettinelli, Ivan Corwin, Ted Cox, Nicolas Curien, Michael Damron, Dmitri Drusvyatskiy, Rick Durrett, Martin Hairer, Christopher Hoffman, Doug Rizzolo, Timo Seppalainen, Alexandre Stauffer, Wendelin Werner and Brent Werness for the most helpful advice. We thank the anonymous referee for many suggestions for improvement. The first author is grateful to the Isaac Newton Institute for Mathematical Sciences, where this research was partly carried out, for the hospitality and support.

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  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    Krzysztof Burdzy & Soumik Pal

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  1. Krzysztof Burdzy
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  2. Soumik Pal
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Correspondence to Krzysztof Burdzy.

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Research supported in part by NSF Grants DMS-1206276 and DMS-1612483, and by Simons Foundation Grant 506732.

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Burdzy, K., Pal, S. Floodings of metric graphs. Probab. Theory Relat. Fields 177, 577–620 (2020). https://doi.org/10.1007/s00440-020-00974-x

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  • DOI: https://doi.org/10.1007/s00440-020-00974-x

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