On Gegenbauer Point Processes on the Unit Interval

Abstract

In this paper 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 expression for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.

Expected Energy of Zeros of Elliptic Polynomials

“…The asymptotic expression above is indeed very close to the minimal logarithmic energy of N points on the sphere, see Sect. 4. Working in a more general setting in [43][44][45] and [25] for the geodesic distance, the same expression (1) was obtained but with a o(N ) remainder.…”