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We show that information on the probability density of local fluctuations canbe obtained from a numerical renormalisation group calculation of a reduceddensity matrix. We apply this approach to the Anderson-Holstein impurity modelto calculate the ground state probability density $\rho(x)$ for thedisplacement $x$ of the local oscillator. From this density we can deduce aneffective local potential for the oscillator and compare its form with thatobtained from a semiclassical approximation as a function of the couplingstrength. The method is extended to infinite dimensional Holstein-Hubbard modelusing dynamical mean field theory. We use this approach to compare theprobability densities for the displacement of the local oscillator in thenormal, antiferromagnetic and charge ordered phases.

Numerical Renormalization Group Study of Probability Distributions for Local Fluctuations in the Anderson-Holstein and Holstein-Hubbard Models