Littlewood-Offord problems for the Curie-Weiss models

• Published in: arXiv.org
• Main Authors: Chang, Yinshan, Xue Peng
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 31.07.2024
• Subjects: Dependent variables; Phase transitions; Random variables
• ISSN: 2331-8422

We consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. To be more precise, we are interested in \[Q_n^{+}:=\sup_{x\in\mathbb{R}}\sup_{v_1,v_2,\ldots,v_n\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1)),\] \[Q_n=\sup_{x\in\mathbb{R}}\sup_{|v_1|,|v_2|,\ldots,|v_n|\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1))\] where the random variables \((\varepsilon_i)_{i=1,2,\ldots,n}\) 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 \(Q_n^{+}\) and \(Q_n\) as \(n\to\infty\) and observe the phenomena of phase transitions. Besides, we prove that the supremum in the definition of \(Q_n^{+}\) is attained when \(v_1=v_2=\cdots=v_n=1\). When \(n\) is even, the supremum in the definition of \(Q_n\) is attained when one half of \((v_i)_i\) equals to \(1\) and the other half equals to \(-1\).