Littlewood-Offord problems for Ising models

We consider the one-dimensional Littlewood-Offord problem for general Ising models. More precisely, we consider the concentration function $$Q_n(x,v)=P(\sum _{i=1}^n{\epsilon }_i{v}_i\in (x-1,x+1)),$$ where $x\in \mathbb{R}$, $v_1,v_2,\dots ,v_n$ are real numbers such that $\mathrm{\mid }{v}_1\mathrm{\mid }\ge 1,\mathrm{\mid }{v}_2\mathrm{\mid }\ge 1,\dots ,\mathrm{\mid }{v}_n\mathrm{\mid }\ge 1$, 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 $n\ge 1$, $$\left(\genfrac{}{}{0px}{}{n}{[n\mathrm{/}2]}\right)2^{-n}\le Q_n\le C{n}^{-\frac{1}{2}}\mathrm{.}$$ 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.