Littlewood-Offord problems for the Curie-Weiss models

In this paper, we consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. Let $$Q_n^{+}:=\underset{x\in \mathbb{R}}{\sup }\underset{v_1,v_2,\dots ,v_n\ge 1}{\sup }P(\sum _{i=1}^n{v}_i{\epsilon }_i\in (x-1,x+1)),$$ $$Q_n=\underset{x\in \mathbb{R}}{\sup }\underset{\mathrm{\mid }{v}_1\mathrm{\mid },\mathrm{\mid }{v}_2\mathrm{\mid },\dots ,\mathrm{\mid }{v}_n\mathrm{\mid }\ge 1}{\sup }P(\sum _{i=1}^n{v}_i{\epsilon }_i\in (x-1,x+1))$$ where the random variables $({\epsilon }_i{)}_{1\le i\le n}$ are spins in Curie-Weiss models. We calculate the asymptotic properties of $Q_n^{+}$ and $Q_n$ as $n\to \mathrm{\infty }$ and observe the phenomena of phase transitions. Meanwhile, we also get that $Q_n^{+}$ is attained when $v_1=v_2=\cdots =v_n=1$. And $Q_n$ is attained when one half of $(v_i{)}_{1\le i\le n}$ equals to $1$ and the other half equals to $-1$ when $n$ is even.This is a generalization of classical Littlewood-Offord problems from Rademacher random variables to possibly dependent random variables. In particular, it includes the case of general independent and identically distributed Bernoulli random variables.