Scale Dependence of Distributions of Hotspots

We consider a random field \(\phi ({\textbf{r}})\) 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 is dominated by unusually large values, thus not being 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 \(R\rightarrow \infty \), the hotspots distribution has a type of scale-invariance which is different from that of fractal sets, and which we term ultradimensional. The form of the function F(x) is determined for a model of diffusion in a random potential.