On Gegenbauer Point Processes on the unit interval

TL;DR

This paper analyzes Gegenbauer determinantal point processes on the interval [-1,1], computing their logarithmic energy, comparing asymptotic behaviors, and deriving exact and minimal energy values, revealing limitations in matching the minimal energy.

Contribution

It provides the first detailed computation of logarithmic energy for Gegenbauer DPPs, including exact values for Chebyshev polynomials and a closed form for the minimal energy.

Findings

All Gegenbauer families share the same third-order asymptotic energy.

Chebyshev polynomials achieve exact logarithmic energy values.

DPPs cannot surpass the minimal energy beyond the third asymptotic term.

Abstract

In this note we compute the logarithmic energy of points in the unit interval $[-1,1]$ chosen from different Gegenbauer Determinantal Point Processes. We check that all the different families of Gegenbauer polynomials yield the same asymptotic result to third order, we compute exactly the value for Chebyshev polynomials and we give a closed expresion for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.