Skip to Main Content

Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Nonuniqueness and mean-field criticality for percolation on nonunimodular transitive graphs
HTML articles powered by AMS MathViewer

by Tom Hutchcroft HTML | PDF
J. Amer. Math. Soc. 33 (2020), 1101-1165 Request permission


We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a nonempty phase in which there are infinite light clusters, which implies the existence of a nonempty phase in which there are infinitely many infinite clusters. That is, we show that $p_c<p_h \leq p_u$ for any such graph. This answers a question of Häggström, Peres, and Schonmann (1999), and verifies the nonunimodular case of a well-known conjecture of Benjamini and Schramm (1996). We also prove that the triangle condition holds at criticality on any such graph, which implies that various critical exponents exist and take their mean-field values.

All our results apply, for example, to the product $T_k\times \mathbb {Z}^d$ of a $k$-regular tree with $\mathbb {Z}^d$ for $k\geq 3$ and $d \geq 1$, for which these results were previously known only for large $k$. Furthermore, our methods also enable us to establish the basic topological features of the phase diagram for anisotropic percolation on such products, in which tree edges and $\mathbb {Z}^d$ edges are given different retention probabilities. These features had only previously been established for $d=1$, $k$ large.

Similar Articles
  • Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 60B99, 60K35
  • Retrieve articles in all journals with MSC (2010): 60B99, 60K35
Additional Information
  • Tom Hutchcroft
  • Affiliation: Statslab, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambride, United Kingdom
  • MR Author ID: 1126713
  • Received by editor(s): November 29, 2017
  • Received by editor(s) in revised form: July 9, 2019, and February 23, 2020
  • Published electronically: September 23, 2020
  • Additional Notes: This work mainly took place while the author was a Ph.D. student at the University of British Columbia, during which time he was supported by a Microsoft Research Ph.D. Fellowship.
  • © Copyright 2020 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 33 (2020), 1101-1165
  • MSC (2010): Primary 60B99, 60K35
  • DOI:
  • MathSciNet review: 4155221