On the upper critical dimension of Bernoulli percolation

Technical Report · Thu Jan 01 04:00:00 EST 1987

OSTI ID:6063790

Chayes, J T; Chayes, L

Derived is a set of inequalities for the d-dimensional independent percolation problem. Assuming the existence of critical exponents, these inequalities imply: f + nu greater than or equal to 1 + ..beta../sub Q/, ..mu.. + nu greater than or equal to 1 + ..beta../sub Q/, zeta greater than or equal to min (1, nu'/nu), where the above exponents are f: the flow constant exponent, nu (nu'): the correlation length exponent below (above) threshold, ..mu..: the surface tension exponent, ..beta../sub Q/: the backbone density exponent and zeta: the chemical distance exponent. Note that all of these inequalities are mean-field bounds, and that they relate the exponent nu defined from below the percolation threshold to exponents defined from above threshold. Furthermore, we combine the strategy of the proofs these inequalities with notions of finite-size scaling to derive: max (d nu, d nu') greater than or equal to 1 + ..beta../sub Q/, where d is the lattice dimension. Since ..beta../sub Q/ greater than or equal to 2..beta.., where ..beta.. is the percolation density exponent, the final bound implies that, below six dimensions, the standard order parameter and correlation length exponents cannot simultaneously assume their mean-field values; hence an implicit bound on the upper critical dimension: d/sub c/ greater than or equal to 6.

• Research Organization: Cornell Univ., Ithaca, NY (USA). Lab. of Atomic and Solid State Physics
• DOE Contract Number: AC02-83ER13044
• OSTI ID: 6063790
• Report Number(s): DOE/ER/13044-9; ON: DE87012412
• Country of Publication: United States
• Language: English

Related Subjects

640410* -- Fluid Physics-- General Fluid Dynamics
656002 -- Condensed Matter Physics-- General Techniques in Condensed Matter-- (1987-)
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
BERNOULLI LAW
FLOW MODELS
FLOW RATE
MANY-DIMENSIONAL CALCULATIONS
MATERIALS
MATHEMATICAL MODELS
POROUS MATERIALS
SURFACE PROPERTIES
SURFACE TENSION