Critical behavior of nonlinear random conductance networks: Real-space renormalization group approach

The real-space renormalization group theory is used to study the critical behavior of nonlinear random conductance networks in two and three dimensions, in which the components obey a current-voltage (I – V) relation of the form I = χμVγ with γ > 0 and μ = i,m where i,m represent the inclusion of volume fraction p and the host medium of volume fraction q (p + Q = 1), respectively. Two important limits are worth studying: (1) normal conductor-insulator (N/I) mixture (χm = 0) and (2) superconductor-normal conductor (S/N) mixture (χm = ∞). As the percolation threshold pc (or qc) is approached, the effective nonlinear response χe is found to behave as χe ≈ (p − pc)t in the N/I limit while χe ≈ (qc − q)−s in the S/N limit. We calculate the critical exponents t and s as a function of γ. A more general duality relation is found to obey in two dimensions. By a connection with the links-nodes-blobs picture, t and s can be related to critical exponents ζR and ζs, which describe the geometry of the percolating network. The results are compared with those of the series analysis and excellent agreements are found over a wide range of γ.