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CRM-ISM Probability Seminar

Title: Number of saddles of the pure spherical p-spin model

Abstract:

The pure spherical p-spin model is a Gaussian random polynomial H of degree p on an N-dimensional sphere, with N large. The sphere is the state space of a physical system with many degrees of freedom, and the random function H is an energy landscape.

Using the Kac-Rice formula, Auffinger, Ben Arous and Cerny computed the average number of critical points of H with a given index, and with energy below a given value. This average is exponentially large in N for p > 2; the rate of growth, called the complexity, is a function of the index chosen and of the energy cutoff. The complexity describes the topology of H: below an energy threshold marking the bottom of the landscape, all critical points are local minima or saddles with an index not diverging in N, and the complexity reveals a nested structure in these finite-index saddles.

Subag鈥檚 remarkable 2017 paper used a second moment approach to show the total number of critical points concentrates around its mean. At the bottom of the landscape, the average number of critical points is dominated by minima at the exponential scale, demonstrating that typical behavior of the minima reflects their average behavior. We show the number of critical points of any finite, positive index concentrates around its mean. This is a step towards understanding dynamics in the landscape, thought to be related to protein folding and training neural nets. Joint work with Antonio Auffinger at Northwestern.

Link:

Meeting ID: 875 9669 4672

Passcode: problab

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