2020/02/07 by Hutchcroft, Tom
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2002.02916
We study the distribution of finite clusters in slightly supercritical (p \downarrow pc) Bernoulli bond percolation on transitive nonamenable graphs, proving in particular that if G is a transitive nonamenable graph satisfying the L2 boundedness condition (pc0 such that Pp(n ≤ |K| lt; ∞) \asymp n-1/2 exp[ -Θ( |p-pc|2 n) ] and Pp(r ≤ Rad(K) lt; ∞) \asymp r-1 exp[ -Θ( |p-pc| r) ] for every p∈ (pc-δ,pc+δ) and n,r≥ 1, where all implicit constants depend only on G. We deduce in particular that the critical exponents γ' and Δ' describing the rate of growth of the moments of a finite cluster as p \downarrow pc take their mean-field values of 1 and 2 respectively. These results apply in particular to Cayley graphs of nonelementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius ρ<1/2. In particular, every finitely generated nonamenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on ℤd even for d very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.