Littlewood-Offord problems for Ising models

We consider the one dimensional Littlewood-Offord problem for general Ising models. More precisely, consider the concentration function \[Q_n(x,v)=P(∑ᵢ₌₁ⁿε_iv_i∈(x-1,x+1)),\] where x, v₁,v₂,…,vₙ are real numbers such that |v₁|≥ 1, |v₂|≥ 1,…, |vₙ|≥ 1, and (εᵢ)i=1,2,…,n are spins of some Ising model. Let Qₙ=x,vQₙ(x,v). Under natural assumptions, we show that there exists a universal constant C such that for all n≥ 1, \[{n}{[n/2]}2⁻ⁿ≤ Q_n≤ Cn-1/2.\]