Exact relationship between the radius of gyration and the density-density correlation function: Application to diffusion limited aggregation

We derive an exact relationship between the density-density correlation function C(r) and the radius of gyration Rg. We show that if the correlation function scales as C(r) ≉ r-αƒ(rNγ) then from the exact relations between C(r) and Rg ≉ Nβ it follows that, unlike the commonly used expression α = d - 1/β, the dependence of α on β must include a γ dependence. The new relationship is 2β = (d + 2 − α)γ -1. Using the above expression on DLA clusters of up to 330 000 particles produces an excellent scaling collapse of C(r).