Pete, Gábor and Timár, Ádám (2022) The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set. ANNALS OF PROBABILITY, 50 (6). pp. 2218-2243. ISSN 0091-1798
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Abstract
We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely.This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtu-ally free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (Probability on Trees and Networks (2016) Cambridge Univ. Press) in the negative.A version of our argument gives an example of a nonunimodular tran-sitive graph where WUSF not equal FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang (Electron. J. Probab. 26 (2021) Paper No. 141).
| Item Type: | Article |
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| Additional Information: | Export Date: 31 January 2023 Correspondence Address: Pete, G.; Alfréd Rényi Institute of MathematicsHungary; email: gabor.pete@renyi.hu Funding details: Budapesti Műszaki és Gazdaságtudományi Egyetem, BME Funding details: European Research Council, ERC, 772466 Funding details: Icelandic Centre for Research, RANNIS, 185233-051 Funding text 1: The first author is also at the Institute of Mathematics, Budapest University of Technology and Economics. The second author is also at the Alfréd Rényi Institute of Mathematics, Budapest. Our work was supported by the ERC Consolidator Grant 772466 “NOISE.” The second author was partially supported by Icelandic Research Fund Grant 185233-051. Funding text 2: Our work was supported by the ERC Consolidator Grant 772466 “NOISE.” The second author was partially supported by Icelandic Research Fund Grant 185233-051. |
| Uncontrolled Keywords: | Free Uniform Spanning Forest; Wilson's algorithm; virtually free groups; nonunimod-ular transitive graphs; nonuniversality at criticality; |
| Subjects: | Q Science / természettudomány > QA Mathematics / matematika |
| SWORD Depositor: | MTMT SWORD |
| Depositing User: | MTMT SWORD |
| Date Deposited: | 16 Mar 2023 15:38 |
| Last Modified: | 06 Apr 2023 14:30 |
| URI: | http://real.mtak.hu/id/eprint/162288 |
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