For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation density P∞(p) is discontinuous at pc, then the critical exponent γ (defined by the divergence of expected cluster size, ∑nPn(p) ∼ (Pc-P)-γ as p ↑pc) must satisfy γ ≥ 2. (2)γ or γ′ (defined analogously to γ, but as p ↓pc) and δ [Pn(pc) ∼ (n-1-1/δ) as n → ∞ ] must satisfy γ, γ′ ≥ 2(1 - 1/δ). These inequalities for γ improve the previously known bound γ ≥ 1(Aizenman and Newman), since δ ≥ 2 (Aizenman and Barsky). Additionally, result 1 may be useful, in standard d-dimensional percolation, for proving rigorously (in d>2) that, as expected, Px has no discontinuity at pc.
- critical exponent inequalities
- rigorous results
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics