Littlewood-Offord problems for the Curie-Weiss models

Abstract

In this paper, we consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. Let \[Q_n^{+}:=\sup_{x\in\mathbb{R}}\sup_{v_1,v_2,\ldots,v_n\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1)),\] \[Q_n=\sup_{x\in\mathbb{R}}\sup_{|v_1|,|v_2|,\ldots,|v_n|\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1))\] 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 $n\to \infty$ 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.