4 problems

Fix even $r$ and $\ell$ with $n\gg\ell\geq r/2$, and let $Z_S\sim\mathcal N(0,1)$ independently for each $r$-subset $S\subset[n]$. Define the Kikuchi matrix $M(\lambda)$, indexed b…

Let $p$ be odd and fix an integer $\ell\in[\lfloor p/2\rfloor,n-\lceil p/2\rceil]$. For $S\subseteq[n]$ with $|S|=\ell$ and $T\subseteq[n]$ with $|T|=\ell+1$, define the rectangula…

Fix an integer $p$ and let $\ell$ be constant independently of $n$. Log-free performance conjecture. There exists a constant $c_p(\ell)>0$, with $c_p(\ell)\to 0$ as $\ell\to\infty$…

Let $\mathcal{S}^N$ be the sphere, let $H_0$ be the pure-noise Hamiltonian, and for $\beta>0$ let $\pi_{0,\beta}$ be the Gibbs measure with density proportional to…