A point process on the unit circle with antipodal interactions

We introduce the point process [Formula presented] where Zn 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 CβE involves n uniformly distributed random variables on the unit circle that repel each other.) We consider linear statistics of the form ∑j=1ng(θj) as n→∞, where g∈C1,q and 2π-periodic. We prove that the leading order fluctuations around the mean are of order n and given by [Formula presented], where U∼Uniform(−π,π]. We also prove that the subleading fluctuations around the mean are of order n and of the form NR(0,4g′(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.