Littlewood-Offord problems for Ising models

Abstract

We consider the one-dimensional Littlewood-Offord problem for general Ising models. More precisely, we consider the concentration function \[Q_n(x,v)=P\left(\sum_{i=1}^{n}\varepsilon_iv_i\in(x-1,x+1)\right),\] where $x\in \mathbb{R}$, $v_1,v_2,\dots ,v_n$ are real numbers such that $|v_1|\ge 1,|v_2|\ge 1,\dots ,|v_n|\ge 1$, and $({\epsilon }_i{)}_{i=1,2,\dots ,n}\in \{-1,1{\}}^n$ are random spins of some Ising model. Let $Q_n={\sup }_{x,v}{Q}_n(x,v)$. Under natural assumptions, we show that there exists a universal constant $C$ such that for all $n\ge 1$, \[\binom{n}{[n/2]}2^{-n}\leq Q_n\leq Cn^{-\frac{1}{2}}.\] As an application of the method, under the same assumption, we give a lower bound on the smallest eigenvalue of the truncated correlation matrix of the Ising model.