The Statistics of the Number of Minima in a Random Energy Landscape

Abstract

We consider random energy landscapes constructed from d-dimensional lattices or trees. The distribution of the number of local minima in such landscapes follows a large deviation principle and we derive the associated law exactly for dimension 1. Also of interest is the probability of the maximum possible number of minima; this probability scales exponentially with the number of sites. We calculate analytically the corresponding exponent for the Cayley tree and the two-leg ladder; for 2 to 5 dimensional hypercubic lattices, we compute the exponent numerically and compare to the Cayley tree case.

arXiv comment18 pages, 8 figures, added background on landscapes and references

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• Universal distribution of the number of minima for random walks and Lévy flights
• The number of minima in random landscapes generated by constrained random walk and Lévy flights: universal properties