PreprintCC BY 4.0arXiv
Infinite-mean Galton-Watson trees have zero random-interchange threshold
Abstract
We prove that, on a Galton-Watson tree with almost surely finite offspring and infinite offspring mean, the random interchange process has critical parameter zero for infinite cycles, conditionally on survival. Combined with the finite-mean strict inequality established in \emph{arXiv:2503.03319}, this gives a dichotomy for locally finite Galton-Watson trees with offspring mean in (1,∞]: the loop and link critical parameters coincide exactly in the infinite-mean case. The proof reduces the tree problem to a finite permutation estimate derived from peeling estimates for random surfaces.
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BibTeX
@misc{klippel2026infinitemean,
title = {{Infinite-mean Galton-Watson trees have zero random-interchange threshold}},
author = {Andreas Klippel and Benjamin Lees and Christian Mönch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08149},
eprint = {2610.08149},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08149},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinite-mean Galton-Watson trees have zero random-interchange threshold
AU - Andreas Klippel
AU - Benjamin Lees
AU - Christian Mönch
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08149
UR - https://arxiv.org/abs/2610.08149
AB - We prove that, on a Galton-Watson tree with almost surely finite offspring and infinite offspring mean, the random interchange process has critical parameter zero for infinite cycles, conditionally on survival. Combined with the finite-mean strict inequality established in \emph{arXiv:2503.03319}, this gives a dichotomy for locally finite Galton-Watson trees with offspring mean in (1,∞]: the loop and link critical parameters coincide exactly in the infinite-mean case. The proof reduces the tree problem to a finite permutation estimate derived from peeling estimates for random surfaces.
ER -