The Hilbert-cube Overlap Gap Property conjecture for even p-spin tensors

Let $A$ and $\hat A$ be independent random tensors in $(\mathbb{R}^N)^{\otimes p}$ with independent entries distributed as $\mathcal{N}(0,N^{-(p-1)})$, and set

$$A_\tau=\sqrt{1-\tau}A+\sqrt{\tau}\hat A,\qquad 0\leq\tau\leq1,$$

with $\mathcal{A}=\{A_\tau:0\leq\tau\leq1\}$. For a domain $\mathcal{S}_N\subset\mathbb{R}^N$, say that $\mathcal{A}$ satisfies the Overlap Gap Property (OGP) with parameters $\mu>0$ and $0<\nu_1<\nu_2<1$ if every $A_j\in\mathcal{A}$ and every $u_j\in\mathcal{S}_N$ satisfying

$$\frac{1}{N}A_j(u_j)\leq\frac{1}{N}\inf_{w\in\mathcal{S}_N}A_j(w)+\mu$$

obey

$$\frac{|\langle u_1,u_2\rangle|}{\|u_1\|_2\|u_2\|_2}\in[0,\nu_1]\cup[\nu_2,1].$$

Hilbert-cube OGP conjecture. For every even $p\geq4$, there exist $\mu>0$ and $0<\nu_1<\nu_2<1$ such that $\mathcal{A}$ satisfies the OGP with domain $\mathcal{S}_N=\mathcal{H}_N$, with probability at least $1-\exp(-cN)$ for some $c>0$ and all sufficiently large $N$. Furthermore, for every $\delta>0$ and every $v_1,v_2\in\mathcal{H}_N$ satisfying

$$A(v_1)/N\leq(1-\delta)\mathbb{E}[\eta_N],\qquad \hat A(v_2)/N\leq(1-\delta)\mathbb{E}[\eta_N],$$

one has $|\langle v_1,v_2\rangle|\leq\delta N$ with probability at least $1-\exp(-cN)$ for some $c>0$ and all large $N$. The OGP has already been proved for the binary domain $\mathcal{B}_N$; the conjecture extends it to the Hilbert cube and is used as an assumption for the paper’s algorithmic barrier. The asserted chaos statement is included as part of the conjecture.