Counting equilibria in non-gradient dynamics

The characterisation of equilibria of complex dynamical systems is a gen- eral problem arising in a variety of different fields ranging from economics to ecology.

A simple characteristic feature of these is the number of equilibria and their stability. In this talk, I will consider equilibria of N degrees of free- dom, each displaying a distinct relaxation rate, coupled by a random Gaussian field. This problem has seen a surge of interest in the recent years in physics and mathematics with many application in e.g. the study of spin-glasses, cosmology, inference problems or machine learning.

The average number of all equilibria and number of stable equilibria have re- cently been obtained for a gradient descent flow with inhomogeneous relaxation rates [2]. In particular, it was found that the system quite generally under- goes a ”topology trivialisation transition” form a ”trivial” phase where there is typically one (stable) equilibrium at low magnitude of the disordered field to a ”complex” phase where the number of equilibria become exponentially large in the system size. The behaviour at the transition has been shown to be universal for a large class of relaxation rates. In a somewhat orthogonal perspective, it was recently shown for a dynam- ics with homogeneous relaxation rates in a random field with both gradient and solenoidal components, that the total average number of equilibria displays the same transition [3]. An additional transition was exhibited from a phase with exponentially many stable equilibria to a phase with exponentially small probability to have any as the ratio of gradient versus solenoidal component is decreased.

I will show how to take into account both inhomogeneous relaxation rates and a random field with gradient and solenoidal components and how this impacts both these transitions [4].

References
[1] G. Ben Arous, P. Bourgade and B. McKenna. Landscape complexity beyond invariance and the elastic manifold, arXiv:2105.05051. 1
[2] G. Ben Arous, Y. V. Fyodorov, B. A. Khoruzhenko, Counting equilibria of large complex systems by instability index, PNAS 118(34) (2021).
[3] B. Lacroix-A-Chez-Toine, Y. V. Fyodorov, Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates, J Phys A, 55(14):144001, (2022).

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