Protocol-Dependent Critical Exponents in Random Composites: Beyond Universality

Classical homogenization theory treats critical exponents as universal quantities depending only on spatial dimension, but recent evidence shows that this assumption fails for continuum composites once the mechanism of randomness generation is taken into account. We synthesize three complementary frameworks—structural approximation, structural sums, and self-similar renormalization—to develop a unified geometric theory of criticality in random composites. Dilute-regime expansions for the effective conductivity and shear modulus are expressed in terms of structural sums whose ensemble statistics depend sensitively on the randomness protocol. To bridge the dilute and critical regimes, we employ self-similar factor approximants, iterated-root approximants, additive approximants, and renormalization schemes based on minimal-difference and minimal-sensitivity conditions, combined with Borel summation. For maximally disordered protocols P(τ), the conductivity index s and the elasticity index S fall within comparable numerical ranges, indicating a shared geometric origin and spectral response to the continuous breaking of translational symmetry. A regular periodic arrangement of inclusions (τ=0) possesses full discrete translational symmetry; as a stochastic protocol P(τ) is applied (τ increases), this symmetry is gradually degraded until statistical chaos is reached. For instance, the parameter τ can be considered as a time of stirring. During this evolution, the system traverses a continuous spectrum of critical indices, s=sP(τ), which encodes the geometric and topological memory of the initial ordered state. It is established that the classical “universality” of percolation corresponds to a fixed point τ within a broader manifold of protocol-dependent critical behaviors. The framework developed here provides a coherent basis for inverse design, diagnostics, and classification of random composites by their disorder history, offering a geometric alternative to the universality paradigm.