We calculate high-temperature graph expansions for the Ising spin glass model with 4 symmetric random distribution functions for its nearest neighbor interaction constants J ij . Series for the Edwards-Anderson susceptibility χ EA are obtained to order 13 in the expansion variable (J/(k B T )) 2 for the general d-dimensional hyper-cubic lattice, where the parameter J determines the width of the distributions. We explain in detail how the expansions are calculated. The analysis, using the Dlog-Padé approximation and the techniques known as M1 and M2, leads to estimates for the critical threshold (J/(k B T c )) 2 and for the critical exponent γ in dimensions 4, 5, 7 and 8 for all the distribution functions. In each dimension the values for γ agree, within their uncertainty margins, with a common value for the different distributions, thus confirming universality.

We study the site and bond quantum percolation model on the two-dimensional square lattice using series expansion in the low concentration limit. We calculate series for the averages of P ij r k ij Tij(E), where Tij(E) is the transmission coefficient between sites i and j, for k = 0, 1, . . . , 5 and for several values of the energy E near the center of the band. In the bond case the series are of order p 14 in the concentration p (some of those have been formerly available to order p 10 ) and in the site case of order p 16 . The analysis, using the Dlog-Padé approximation and the techniques known as M1 and M2, shows clear evidence for a delocalization transition (from exponentially localized to extended or power-law-decaying states) at an energy-dependent threshold pq(E) in the range pc < pq(E) < 1, confirming previous results (e.g. pq(0.05) = 0.625 ± 0.025 and 0.740 ± 0.025 for bond and site percolation) but in contrast with the Anderson model. The divergence of the series for different k is characterized by a constant gap exponent, which is identified as the localization length exponent ν from a general scaling assumption. We obtain estimates of ν = 0.57 ± 0.10. These values violate the bound ν ≥ 2/d of Chayes et al.

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