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 $H_{N}\left(\boldsymbolσ\right)=\frac{1}{N^{\left(p-1\right)/2}}\sum_{i_{1},...,i_{p}=1}^{N}J_{i_{1},...,i_{p}}σ_{i_{1}}\cdotsσ_{i_{p}}$, where $\boldsymbolσ=\left(σ_{1},...,σ_{N}\right)\in \mathbb{S}^{N-1}:=\left\{ \boldsymbolσ\in\mathbb{R}^{N}:\,\left\Vert \boldsymbolσ\right\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 $p\geq 32$ and any $E>-E_\infty$, where $E_\infty$ is the (normalized) ground state, the ratio of the number of critical points $\boldsymbolσ$ with $H_N(\boldsymbolσ)\leq NE$ and its expectation asymptotically concentrates at $1$. This extends to arbitrary $E$ a similar conclusion of [Sub17a].

Theorems & Definitions (16)

• Theorem 1
• Corollary 2
• Lemma 3
• Lemma 4
• Corollary 5
• proof
• Lemma 6
• proof
• Lemma 7
• proof