David J. Bergman,
Physics and Astronomy
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, IL-69978 Tel Aviv, Israel
December 4, 2001
It has long been known that the critical exponent of the elastic stiffness of a -dimensional percolating network ( measures the closeness of the network to its percolation threshold ) satisfies the following inequalities , where is the critical exponent of the electrical conductivity of the same network and is the critical exponent of the percolation correlation length . Similarly, the critical exponents which characterize the divergences , of a percolating rigid/normal network (i.e., a random mixture of normal elastic bonds and totally rigid bonds) and a percolating superconducting/normal network (i.e., a random mixture of normal conducting bonds and perfectly conducting bonds; now measures the closeness of the rigid or superconducting constituent to its percolation threshold ) have long been known to satisfy . I now show that, when or , is in fact exactly equal to and is exactly equal to . This is achieved by a judicious use of some variational principles for electrical and elastic networks, and by a judicious treatment of constraints and short range correlations in those networks. An extension of these proofs to arbitrary (integer) values of the dimensionality should be possible.