INFINITE HIERARCHIES OF EXPONENTS IN A DILUTED FERROMAGNET AND THEIR INTERPRETATION

In analogy with other spatially inhomogeneous, scale invariant systems (strange attractors, diffusion limited aggregation, Anderson localization, random resistor networks,...) we propose a description of the scaling properties of local operators ψ(r) (e.g. local spin or energy density) at the critical point of a quenched diluted ferromagnet by universal convex functions Hψ(α) [the analog of f{hook}(α)], generalizing exponents. Using the random 2D q-state Potts model as a typical example of a random ferromagnet we illustrate that this is an appropriate description of random criticality, hence rather different from pure critical phenomena. The infinite set ηN = 2XN(σ) = 2N[xσ- 1 16(N-1)y + O(y2)] of independent exponents of the moments of the Potts spin (scaling dimension xσ without disorder) correlation function is obtained to 1 loop. [y is the RG eigenvalue of disorder causing the RG flow from the pure to the random Potts fixed point, nearby when y ∝ (q-2) is small.] From this the perturbative expression Hσ(α) = ( 1 2yB2){(α-α0)2-[B 3/(B2)2](α-α0) 3 + O((α-α0)4)} is inferred by Legendre transformation. Its minimum α0 = xσ+ 1 16y+O(y2) is a new exponent describing the asymptotic decay of the spin-spin correlation function in a fixed sample at criticality. Using the conformal map onto the cylinder it is shown (for q close to 2) to be universally related to the finite size scaling amplitude Aσ of the spin correlation length in the critical disorder d cylinder α0 = Aσ/2π. In general, multiplicatively renormalizable operators at random criticality should be classified by irreducible representations of the permutation group of the replicas. Moments of the Potts energy are shown to correspond to non-singlet representations under permutations. For the 2D random Ising model at the transition non-self-averaging logarithmic corrections to all moments of the spin correlation function are derived. The averaged spin correlation function is shown to be equal to the correlation function in the critical pure Ising Model at large distance. © 1990.