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A point process on the unit circle with antipodal interactions

Year: 2025
Type: article
Abstract: We introduce the point process 1 Z n ∏ 1 ≤ j < k ≤ n | e i θ j + e i θ k | β ∏ j = 1 n d θ j , θ 1 , … , θ n ∈ ( − π , π ] , β > 0 , where Z n is the normalization constant. This point process is attractive : it involves n dependent, uniformly distributed random variables on the unit circle that attract each other. (For comparison, the well-studied C β E involves n uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form ∑ j = 1 n g ( θ j ) as n → ∞ , where g ∈ C 1 , q and 2 π -periodic. We prove that the leading order fluctuations around the mean are of order n and given by ( g ( U ) − ∫ − π π g ( θ ) d θ 2 π ) n , where U ∼ Uniform ( − π , π ] . We also prove that the subleading fluctuations around the mean are of order n and of the form N R ( 0 , 4 g ′ ( U ) 2 / β ) n , i.e. that the subleading fluctuations are given by a Gaussian random variable that itself has a random variance. Our proof uses techniques developed by McKay and Isaev (McKay, 1990; Isaev and McKay, 2018) to obtain asymptotics of related n -fold integrals.
DOI: 10.1016/j.jat.2025.106161
Institution: Lund University
Language: English
Open Access status: hybrid
Estimated APC paid: $2,910
OpenAlex ID: W4408240926
FWCI:
Cites: 4
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Related to: 10