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

On the large-scale structure of the tall peaks for stochastic heat equations with fractional Laplacian