Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry
Volume 123 / 2019
Abstract
This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kähler–Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in $\mathbb C ^{n}.$ A fair amount of background material is included. Along the way the results are generalized to the non-compact setting of $\mathbb C ^{n}.$ This yields a probabilistic construction of Kähler solutions to Einstein’s equations in $\mathbb C ^{n}$, with cosmological constant $-\beta $, from a gas of interpolation nodes in equilibrium at positive inverse temperature $\beta .$ In the infinite temperature limit, $\beta \rightarrow 0$, solutions to the Calabi–Yau equation are obtained. In the opposite zero temperature case the results may be interpreted as “transcendental” analogs of classical asymptotics for orthogonal polynomials, with the inverse temperature $\beta $ playing the role of the degree of a polynomial.
