Fluctuations of the ground state of the spiked spherical Sherrington-Kirkpatrick model

TL;DR

This paper analyzes the fluctuations of the ground state of the spiked spherical Sherrington-Kirkpatrick model, providing leading order and fluctuation results for the maximum of the Hamiltonian with external or spike terms.

Contribution

It computes the leading order level and the first- and second-order fluctuations of the ground state in the spiked spherical SK model, extending understanding of its extremal properties.

Findings

Determined the leading order of the maximum Hamiltonian value.

Derived the first- and second-order fluctuation results.

Extended results to the TAP free energy maximum.

Abstract

The Sherrington-Kirkpatrick Hamiltonian is a random quadratic function on the high-dimensional sphere. This article studies the ground state (i.e. maximum) of this Hamiltonian with external field, or more generally with a non-linear "spike" term. We compute the level of the maximum to leading order, and under appropriate condition its first- and second-order fluctuations. The equivalent results are also derived for the maximum of the model's TAP free energy on the ball.