Electronic Communications in Probability

Non-amenable Cayley graphs of high girth have $p_c < p_u$ and mean-field exponents

Asaf Nachmias and Yuval Peres

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In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., $p_c< p_u$. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical exponents. In particular, the self-avoiding walk has positive speed.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 57, 8 pp.

Accepted: 3 December 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B43: Percolation [See also 60K35]

Percolation Self avoiding walk Non-amenable graphs

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Nachmias, Asaf; Peres, Yuval. Non-amenable Cayley graphs of high girth have $p_c &lt; p_u$ and mean-field exponents. Electron. Commun. Probab. 17 (2012), paper no. 57, 8 pp. doi:10.1214/ECP.v17-2139. https://projecteuclid.org/euclid.ecp/1465263190

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