Statistical mechanics of conducting phase transitions

The critical behavior of the effective conductivity o* of the random resistor network in Zd, near its percolation threshold, is considered. The network has bonds assigned the conductivities 1 and E >_0 in the volume fractions p and 1 -p. Motivated by the statistical mechanics of an Ising ferromagnet at temperature T in a field H, we introduce a partition function and free-energy for the resistor network, which establishes a direct correspondence between the two problems. In particular, we show that the free energies for the resistor network and the Ising model both have the same type of integral representation, which has the interpretation of the complex potential due to a charge distribution on [0, l] in the s = l/( 1 -e) plane for the resistor network, and on the unit circle in the z=exp(-2BH) plane for the ferromagnet. Based on this correspondence, we develop a Yang-Lee picture of the onset of nonanalytic behavior of the effective conductivity o*, so that the percolation threshold p=pc is characterized as an accumulation point of zeros of the partition function in the complex p-plane as E -- 0. A scheme is developed to find the locations of a certain sequence of zeros in the p-plane, which is based on Pade approximation of a perturbation expansion of o*(p,e) around a homogeneous medium (E= 1). Furthermore, for E > 0, we construct a domain De containing [0, l] in the p-plane in which o*i(p,e) is analytic, and which collapses as E -- 0. The explicit construction of this domain allows us to obtain a lower bound on the size of the gap in zeros of the partition function around the percolation threshold p =pc , which leads to the gap exponent inequality A<_l. 0 1995 American Institute of Physics.