Novel methods were used to generate and analyze new 15 term high temperature series for both the (connected) susceptibility \ensuremath{\chi} and the structure factor (disconnected susceptibility) ${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$ for the random field Ising model with dimensionless coupling K=J/kT, in general dimension d. For both the bimodal and the Gaussian field distributions, with mean square field
J2g, we find that (${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$-\ensuremath{\chi})/
K2g${\mathrm{\ensuremath{\chi}}}^{2}$=1 as T\ensuremath{\rightarrow}
Tc(g), for a range of [
h2]=
J2g and d=3,4,5. This confirms the exponent relation \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\gamma} (where ${\mathrm{\ensuremath{\chi}}}_{\mathit{d}}$\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}\mathrm{\ifmmode\bar\else\textasciimacron\fi{}}}$, \ensuremath{\chi}\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\gamma}}$, t=T-
Tc) providing that random field exponents are determined by two (and not three) independent exponents. We also present new accurate values for \ensuremath{\gamma}.
The first-order phase transition of the two-dimensional eight-state Potts model is shown to be rounded when long-range correlated disorder is coupled to energy density. Critical exponents are estimated by means of large-scale Monte Carlo simulations. In contrast to uncorrelated disorder, a violation of the hyperscaling relation γ / ν = d − 2 x σ is observed. Even though the system is not frustrated, disorder fluctuations are strong enough to cause this violation in the very same way as in the 3D random-field Ising model. In the thermal sector, too, evidence is given for such violation in the two hyperscaling relations α / ν = d − 2 x ε and 1/ ν = d − x ε . In contrast to the random field Ising model, at least two hyperscaling violation exponents are needed. The scaling dimension of energy is conjectured to be x ε = a /2, where a is the exponent of the algebraic decay of disorder correlations.