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The local limit of uniform spanning trees

Abstract

We show that the local limit of the uniform spanning tree on any finite, simple, connected, regular graph sequence with degree tending to \(\infty \) is the Poisson(1) branching process conditioned to survive forever. An extension to “almost” regular graphs and a quenched version are also given.

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Acknowledgements

This research is supported by ERC starting Grant 676970 RANDGEOM and by ISF Grants 1207/15 and 1294/19. Research of Y.P. was partially supported by NSF grant DMS-1900008. We wish to thank Jan Hladký for many useful conversations and for his permission to include his proof of Proposition 7.1. We also thank Matan Shalev for finding several errors in a previous version of this manuscript.

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Correspondence to Asaf Nachmias.

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Nachmias, A., Peres, Y. The local limit of uniform spanning trees. Probab. Theory Relat. Fields 182, 1133–1161 (2022). https://doi.org/10.1007/s00440-021-01072-2

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Mathematics Subject Classification

  • 05C81