Scale Dependence of Distributions of Hotspots

Wilkinson, Michael and Veytsman, Boris (2024). Scale Dependence of Distributions of Hotspots. Journal of Statistical Physics, 191(5), article no. 60.



We consider a random field ϕ(r) in d dimensions which is largely concentrated around small ‘hotspots’, with ‘weights’, wi. 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(lnR). 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.

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