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Littlewood-Offord problems for Ising models

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  • Chang, Yinshan

Abstract

We consider the one-dimensional Littlewood-Offord problem for general Ising models. More precisely, we consider the concentration functionQn(x,v)=P(∑i=1nεivi∈(x−1,x+1)),where x∈R, v1,v2,…,vn are real numbers such that |v1|≥1,|v2|≥1,…,|vn|≥1, and (εi)i=1,2,…,n∈{−1,1}n are random spins of some Ising model. Let Qn=supx,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≤Qn≤Cn−12.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.

Suggested Citation

  • Chang, Yinshan, 2026. "Littlewood-Offord problems for Ising models," Stochastic Processes and their Applications, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:spapps:v:195:y:2026:i:c:s0304414926000049
    DOI: 10.1016/j.spa.2026.104872
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