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Abstract |
Understanding the statistical properties of complex and/or random landscapes in high-dimensional spaces is a central problem in a variety of different contexts, from glassy systems to computer science, ecology and biology. Among these properties, an important role is played by the statistics of the number of stationary points, which is expected to be relevant in determining the evolution of local dynamics within the landscape. In this talk, I will discuss the calculation of the typical number of local minima of the energy landscape of a simple model, which captures the competition between a deterministic ‘signal’ term and a noisy contribution. The model is obtained adding to the spherical p-spin Hamiltonian a term favoring configurations that are aligned to a given configuration on the sphere (the signal), and reproduces the spiked-tensor model for a specific choice of parameters. I will describe the phase transitions that occur in the structure of the landscape when changing the signal-to-noise ratio, and provide some details on the calculation of the quenched complexity, that is performed using a replicated version of the Kac-Rice formula. The talk is based on joint work with Gérard Ben Arous, Giulio Biroli and Chiara Cammarota. |
