Littlewood-Offord problems for the Curie-Weiss models

We consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. To be more precise, we are interested in \[Q_n⁺:=ₓv_1,v_2,…,v_n≥ 1P(∑ᵢ₌₁ⁿv_iε_i∈(x-1,x+1)),\] \[Q_n=ₓ|v_1|,|v_2|,…,|v_n|≥ 1P(∑ᵢ₌₁ⁿv_iε_i∈(x-1,x+1))\] where the random variables (εᵢ)i=1,2,…,n are spins in Curie-Weiss models. 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 i.i.d. Bernoulli random variables. We calculate the asymptotics of Qₙ⁺ and Qₙ as n→∞ and observe the phenomena of phase transitions. Besides, we prove that the supremum in the definition of Qₙ⁺ is attained when v₁=v₂=⋯=vₙ=1. When n is even, the supremum in the definition of Qₙ is attained when one half of (vᵢ)ᵢ equals to $1$ and the other half equals to $-1$.