2 problems

Let $U:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $U(x)=U(-x)$ and … for some $\alpha>0$. For each $M>0$, define the interaction sequence $u^M=(u^M_n)$ by $u^M_n:=U(n/M)$. For…

Let $G(n,p)$ be an Erdős–Rényi random graph. Assume … for some constant $\epsilon>0$. Let $v$ be a random unit vector uniformly distributed on the $(n-1)$-dimensional unit sphere,…