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

   
@article{ECP2139,
	author = {Asaf Nachmias and Yuval Peres},
	title = {Non-amenable Cayley graphs of high girth have $p_c < p_u$ and mean-field exponents},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {17},
	year = {2012},
	keywords = {Percolation; Self avoiding walk; Non-amenable graphs},
	abstract = {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.},
	pages = {no. 57, 1-8},
	issn = {1083-589X},
	doi = {10.1214/ECP.v17-2139},    
        url = {http__/ecp.ejpecp.org/article/view/2139}}.html