Scale Dependence of Distributions of Hotspots

We consider a random field ϕ (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 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 lnW=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→∞ , 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.