David J. Bergman,
School of
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.