Scale invariant forces in one-dimensional shuffled lattices

We present a detailed and exact study of the probability density function P(F) of the total force F acting on a point particle belonging to a perturbed lattice of identical point sources of a power-law pair interaction. The main results concern the large-F tail of P(F) for which two cases are mainly distinguished: (i) Gaussian-like fast decreasing P(F) for lattice with perturbations forbidding any pair of particles to be found arbitrarily close to one each other and (ii) Levy-like power-law decreasing P(F) when this possibility is instead permitted. It is important to note that in the second case the exponent of the power-law tail of P(F) is the same for all perturbations (apart from very singular cases) and is in a one-to-one correspondence with the exponent characterizing the behavior of the pair interaction with the distance between the two particles.