Concentration of the complexity of spherical pure $p$-spin models at arbitrary energies

Abstract

We consider critical points of the spherical pure $p$-spin spin glass model with Hamiltonian $H_N\left(\mathbf{\sigma }\right)=\frac{1}{N^{\left(p-1\right)/2}}{\sum }_{i_1,...,i_p=1}^N{J}_{i_1,...,i_p}{\sigma }_{i_1}\cdots {\sigma }_{i_p}$, where $\mathbf{\sigma }=\left({\sigma }_1,...,{\sigma }_N\right)\in {\mathbb{S}}^{N-1}:=\left\{\mathbf{\sigma }\in {\mathbb{R}}^N:{\Vert \mathbf{\sigma }\Vert }_2=\sqrt{N}\right\}$ and $J_{i_1,...,i_p}$ are i.i.d standard normal variables. Using a second moment analysis, we prove that for $p\ge 32$ and any $E>-E_{\infty }$, where $E_{\infty }$ is the (normalized) ground state, the ratio of the number of critical points $\mathbf{\sigma }$ with $H_N(\mathbf{\sigma })\le NE$ and its expectation asymptotically concentrates at $1$. This extends to arbitrary $E$ a similar conclusion of [Sub17a].

References in corpus (3)

• Free energy and complexity of spherical bipartite models
• Exponential growth of random determinants beyond invariance
• The number of saddles of the spherical $p$-spin model

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• Tight Lipschitz Hardness for Optimizing Mean Field Spin Glasses