Abstract
We investigate the random loop model on the d-ary tree. For , we establish a (locally) sharp phase transition for the existence of infinite loops. Moreover, we derive rigorous bounds that in principle allow to determine the value of the critical parameter with arbitrary precision. Additionally, we prove the existence of an asymptotic expansion for the critical parameter in terms of . The corresponding coefficients can be determined in a schematic way and we calculate them up to order 6.
Funding Statement
The research of BL was supported by the Alexander von Humboldt Foundation.
Citation
Volker Betz. Johannes Ehlert. Benjamin Lees. Lukas Roth. "Sharp phase transition for random loop models on trees." Electron. J. Probab. 26 1 - 26, 2021. https://doi.org/10.1214/21-EJP677
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