Littlewood-Offord problems for the Curie-Weiss models
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915021370425344 |
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| author | Chang, Yinshan Peng, Xue |
| author_facet | Chang, Yinshan Peng, Xue |
| contents | In this paper, we consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. Let \[Q_n^{+}:=\sup_{x\in\mathbb{R}}\sup_{v_1,v_2,\ldots,v_n\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1)),\] \[Q_n=\sup_{x\in\mathbb{R}}\sup_{|v_1|,|v_2|,\ldots,|v_n|\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1))\] where the random variables $(\varepsilon_i)_{1\leq i\leq n}$ are spins in Curie-Weiss models. We calculate the asymptotic properties of $Q_n^{+}$ and $Q_n$ as $n\to\infty$ and observe the phenomena of phase transitions. Meanwhile, we also get that $Q_n^{+}$ is attained when $v_1=v_2=\cdots=v_n=1$. And $Q_n$ is attained when one half of $(v_i)_{1\leq i\leq n}$ equals to $1$ and the other half equals to $-1$ when $n$ is even.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 independent and identically distributed Bernoulli random variables. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_00127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Littlewood-Offord problems for the Curie-Weiss models Chang, Yinshan Peng, Xue Probability 60C05, 82B20 In this paper, we consider the Littlewood-Offord problems in one dimension for the Curie-Weiss models. Let \[Q_n^{+}:=\sup_{x\in\mathbb{R}}\sup_{v_1,v_2,\ldots,v_n\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1)),\] \[Q_n=\sup_{x\in\mathbb{R}}\sup_{|v_1|,|v_2|,\ldots,|v_n|\geq 1}P(\sum_{i=1}^{n}v_i\varepsilon_i\in(x-1,x+1))\] where the random variables $(\varepsilon_i)_{1\leq i\leq n}$ are spins in Curie-Weiss models. We calculate the asymptotic properties of $Q_n^{+}$ and $Q_n$ as $n\to\infty$ and observe the phenomena of phase transitions. Meanwhile, we also get that $Q_n^{+}$ is attained when $v_1=v_2=\cdots=v_n=1$. And $Q_n$ is attained when one half of $(v_i)_{1\leq i\leq n}$ equals to $1$ and the other half equals to $-1$ when $n$ is even.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 independent and identically distributed Bernoulli random variables. |
| title | Littlewood-Offord problems for the Curie-Weiss models |
| topic | Probability 60C05, 82B20 |
| url | https://arxiv.org/abs/2408.00127 |