Resistance fluctuations in randomly diluted networks

Abstract

The resistance R(x,x') between two connected terminals in a randomly diluted resistor network is studied on a d-dimensional hypercubic lattice at the percolation threshold ${p}_{c}$. When each individual resistor has a small random component of resistance, R(x,x') becomes a random variable with an associated probability distribution, which contains information on the distribution of currents in the individual resistors. The noise measured between the terminals may be characterized by the cumulants ${M}_{q}$(x,x') of R(x,x'). When averaged over configurations of clusters, M\ifmmode\bar\else\textasciimacron\fi{}(x,x')\ensuremath{\sim}\ensuremath{\Vert}x-x'${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\psi}}}$${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}^{(\mathrm{q})}$. We construct low-concentration series for the generalized resistive susceptibility, ${\ensuremath{\chi}}^{(q)}$, associated with M${\ifmmode\bar\else\textasciimacron\fi{}}_{q}$, from which the critical exponents \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(q) are obtained. We prove that \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(q) is a convex monotonically decreasing function of q, which has the special values \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(0)=${D}_{B}$, \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(1)=\ensuremath{\zeta}${\ifmmode \tilde{}\else \~{}\fi{}}_{R}$, and \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\infty})=1/\ensuremath{\nu}. ($D sub B--- is the fractal dimension of the backbone, \ensuremath{\zeta}${\ifmmode \tilde{}\else \~{}\fi{}}_{R}$ is the usual scaling exponent for the average resistance, and \ensuremath{\nu} is the correlation-length exponent.) Using the convexity property and the accepted values of these three exponents, we construct two approximant functions for \ensuremath{\psi}(q)=\ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(q)\ensuremath{\nu}, both of which agree with the series results for all q>1 and with existing numerical simulations. These approximants enabled us to obtain explicit approximate forms for the multifractal functions \ensuremath{\alpha}(q) and f(q) which, for a given q, characterize the scaling with size of the dominant value of the current and the number of bonds having this current. This scaling description fails for sufficiently large negative q, when the dominant (small) current decreases exponentially with size. In this case ${\ensuremath{\chi}}^{(q)}$ diverges at a lower threshold ${p}^{\mathrm{*}}$(q), which vanishes as q\ensuremath{\rightarrow}-\ensuremath{\infty}.