Abstract
We consider a general, translation invariant bond percolation model on ℤd with bonds characterized by couplings (Jx| X∈ℤ d ) and an inverse temperature parameter tf, with nontrivial critical value v c. We prove several inequalities including: (Da differential inequality for the infinite cluster density, P∞(v); and (2) an inequality relating the backbone density, Q∞(v), to p∞(v) and the expected size of finite clusters, χ’(v). If the above quantities exhibit critical scaling with exponents “defined” by P∞(v) ~ | v — v c |β, Q∞(v) ~ | v — v c |βQ, and χ’(v) ~ |v — v c|-γ’ as v↓v c these inequalities imply the mean field bounds: β ≤ 1 and 2β ≤ βQ ≤ β + γ’. Furthermore, a magnetic backbone exponent, δQ, is defined analogously to the standard magnetic backbone exponent, δ. Again assuming critical scaling, our inequalities also imply the mean field bounds δ ≥ 2δQ and δQ ≥ 1.
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© 1987 Springer-Verlag New York, Inc.
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Chayes, J.T., Chayes, L. (1987). The Mean Field Bound for the Order Parameter of Bernoulli Percolation. In: Kesten, H. (eds) Percolation Theory and Ergodic Theory of Infinite Particle Systems. The IMA Volumes in Mathematics and Its Applications, vol 8. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8734-3_5
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DOI: https://doi.org/10.1007/978-1-4613-8734-3_5
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