Scale Dependence of Distributions of Hotspots

We consider a random field φ (r) ϕ ( r ) in d dimensions which is largely concentrated around small ‘hotspots’, with ‘weights’, wᵢ 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 W̄ 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 W̄=F(ln R) ln W ¯ = F ( ln R ) . If $$F'(x)>d$$ F ′ ( x ) > d , the distribution of hotspots is dominated by the largest weights. In the case where $$F'(x)-d$$ F ′ ( x ) - d approaches a constant positive value when R→ ∞ 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.