**Title:** (Remote) Complexity of barriers in the pure spherical p-spin model

**Speaker:** Julian Gold

**Speaker Info:** Northwestern

**Brief Description:** This is a remote talk. Link: https://northwestern.zoom.us/j/907400031 Meeting ID: 907 400 031

**Special Note**: **This is a remote talk. Link: https://northwestern.zoom.us/j/907400031 Meeting ID: 907 400 031**

**Abstract:**

I will discuss results and computations in a recent physics paper of Ros, Biroli and Cammarota (arXiv:1809.05440v1). The model studied is the pure spherical $p$-spin model, a Gaussian 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 a smooth assignment of energy to each state.In 2012, Auffinger, Ben Arous and Cerny used the Kac-Rice formula to count the average number of critical points of $H$ having a given index, and with energy below a given value. This number is exponentially large in $N$ for $p > 2$, and the rate of growth itself is a function of the index chosen and of the energy cutoff. This function, called the complexity, helps us understand the topology of $H$. In 2017, Subag counted pairs of critical points using the Kac-Rice formula, showing that the total number of critical points concentrates around its mean through a second moment computation. Subag's result implies the average behavior of the total number of critical points is also typical. The goal of the talk is to present how the replica trick is used on the Kac-Rice formula in Ros et al. to predict typical behavior of saddles close to a fixed minimum.

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