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Probabilidade e Combinatória

How many stable equilibria will a large complex system have?
Boris Khoruzhenko

Queen Mary, University of London
Quarta-feira, 19 de fevereiro de 2020, 15:30
Sala 345

In 1972 Robert May argued that (generic) complex systems become unstable to small displacements from equilibria as the system complexity increases. May’s model was linear and his outlook was very much local. In search of a global signature of the May instability transition, I will analyse a minimal model for large nonlinear complex systems whereby $N$ degrees of freedom equipped with a stability feedback mechanism are coupled via a smooth homogeneous Gaussian vector field with longitudinal and transverse components. With the increase in complexity (as measured by the number of degrees of freedom and the strength of interaction relative to the relaxation strength), this model undergoes an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime of 'absolute instability' where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the interaction is purely longitudinal (purely gradient dynamics). When the complexity increases even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely transverse (purely solenoidal dynamics). The width of the instability transition region scales as $N^{-1/2}$ (same as in May’s model) and I will argue that in this region the unstable equilibria (saddles) on average have only a very a small proportion of unstable directions which scales as $N^{-1/4}$ with $N$. This talk is based on collaborative works with Yan Fyodorov (PNAS 2016), Gerard Ben Arous and Yan Fyodorov (manuscript in preparation) and Jacek Grela (manuscript in preparation) and our analysis uses Kac-Rice formula for counting zeros of random functions and theory of random matrices applied to the real elliptic ensemble.