We rigorously establish that, in disordered three-dimensional isotropic solids, the stress autocorrelation function presents anisotropic terms that decay as 1/r3 at long-range, with r being the distance, as soon as local stress fluctuations are normal, by which we mean that the fluctuations of stress, as averaged over spherical domains, decay as the inverse domain volume. Since this property is required for macroscopic stress to be self-averaging, it is expected to hold generically in all glasses and we thus conclude that the presence of 1/r3 stress correlation tails is the rule in these systems. Our proof follows from the observation that, in an infinite medium, when both material isotropy and mechanical balance hold, (i) the stress autocorrelation matrix is completely fixed by just two radial functions: the pressure autocorrelation and the trace of the autocorrelation of stress deviators; furthermore, these two functions (ii) fix the decay of the fluctuations of sphere-averaged pressure and deviatoric stresses with the increasing sphere volume. Our conclusion is reached because, in view of (ii), the normal decay of stress fluctuations is only compatible with both the pressure autocorrelation and the trace of the autocorrelation of stress deviators being integrable; in turn, due to the precise analytic relation (i) fixed by isotropy and mechanical balance, this condition demands the spatially anisotropic stress correlation terms to decay as 1/r3 at long-range.
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