Complexity of random smooth functions of many variables

Details Complexity of random smooth functions of many variables Complexity of random smooth functions of many variables

2011-10-26

arxiv.org/abs/1110.5872v2

Mathematics

Probability

33 pages, 3 figures, references added

Scientific paper

We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom energy landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at energy levels below the limiting ground state energy the mean number of local minima is exponentially large. These two scenarios should correspond to the distinction between one-step replica symmetry breaking and full replica-symmetric breaking of the physics literature on spin glasses. In the former, we find a new way to derive the asymptotic complexity function as a function of the 1RSB Parisi functional.

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