Scale dependence of distributions of hotspots

Abstract

We consider a random field $\phi(\mathbf{r})$ in $d$ dimensions which is largely concentrated around small `hotspots', with `weights', $w_i$. These weights may have a very broad distribution, such that their mean does not exist, or else is not a useful estimate. In such cases, the median $\overline W$ of the total weight $W$ in a region of size $R$ is an informative characterisation of the weights. We define the function $F$ by $\ln \overline W=F(\ln R)$. If $F'(x)>d$, the distribution of hotspots is dominated by the largest weights. In the case where $F'(x)-d$ approaches a constant positive value when $R\to \infty$, the hotspots distribution has a type of scale-invariance which is different from that of fractal sets, and which we term \emph{ultradimensional}. The form of the function $F(x)$ is determined for a model of diffusion in a random potential.