Some critical exponent inequalities for percolation

Journal Article · Sat Nov 01 04:00:00 EST 1986 · J. Stat. Phys.; (United States)

DOI:https://doi.org/10.1007/BF01021076· OSTI ID:6897481

Newman, C M

For a large class of independent (site or bond, short- or long-range) percolation models, the author shows the following: (1) If the percolation density P/sub infinity/(p) is discontinuous at p/sub c/, then the critical exponent ..gamma.. (defined by the divergence of expected cluster size, ..sigma.. n P/sub n/(p) approx. (p/sub c/-p)/sup -..gamma../ as p up arrow p/sub c/) must satisfy ..gamma.. greater than or equal to 2. (2) ..gamma.. or ..gamma..' (defined analogously to ..gamma.., but as p down arrow p/sub c/) and sigma (P/sub n/(p/sub c/)approx. n/sup -1-1/sigma/ as n ..-->.. infinity) must satisfy ..gamma.., ..gamma..' greater than or equal to 2(1-1/sigma). These inequalities for ..gamma.. improve the previously known bound ..gamma.. greater than or equal to 1 (Aizenman and Newman), since sigma greater than or equal to 2 (Aizenman and Barsky). Additionally, result 1 may be useful, in standard d-dimensional percolation, for proving rigorously (in d > 2) that, as expected, P/sub infinity/ has no discontinuity at p/sub c/.

• Research Organization: Univ. of Arizona, Tucson
• OSTI ID: 6897481
• Journal Information: J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 45:3/4; ISSN JSTPB
• Country of Publication: United States
• Language: English

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657002* -- Theoretical & Mathematical Physics-- Classical & Quantum Mechanics
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOUNDARY CONDITIONS
CHEMICAL BONDS
CONVERGENCE
CRYSTAL DEFECTS
CRYSTAL MODELS
CRYSTAL STRUCTURE
FERROMAGNETIC MATERIALS
FERROMAGNETISM
INVARIANCE PRINCIPLES
LATTICE PARAMETERS
MAGNETIC MATERIALS
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MATHEMATICAL MODELS
PHASE TRANSFORMATIONS
POINT DEFECTS
SPIN GLASS STATE
VACANCIES