  We consider stochastic heat equations with fractional Laplacian on
$\mathbb{R}^d$. Here, the driving noise is generalized Gaussian which is white
in time but spatially homogenous and the spatial covariance is given by the
Riesz kernels. We study the large-scale structure of the tall peaks for (i) the
linear stochastic heat equation and (ii) the parabolic Anderson model. We
obtain the largest order of the tall peaks and compute the macroscopic
Hausdorff dimensions of the tall peaks for both (i) and (ii). These results
imply that both (i) and (ii) exhibit multi-fractal behavior in a macroscopic
scale even though (i) is not intermittent and (ii) is intermittent. This is an
extension of a recent result by Khoshnevisan et al to a wider class of
stochastic heat equations.
