Statistical Mechanics of Interpolation Nodes, Pluripotential theory and Complex Geometry

TL;DR

This survey explores how probabilistic methods from statistical mechanics can construct Kahler-Einstein metrics and solutions to Einstein's equations on complex manifolds, linking classical approximation problems to modern geometric analysis.

Contribution

It introduces a novel probabilistic framework connecting interpolation nodes with Kahler-Einstein metrics and extends these ideas to non-compact complex spaces, including solutions to the Calabi-Yau equation.

Findings

Probabilistic construction of Kahler-Einstein metrics from interpolation nodes.

Extension of methods to non-compact complex spaces.

Connection between temperature limits and classical polynomial asymptotics.

Abstract

This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in complex n-space. A fair amount of background material is included. Along the way, the results are generalized to the non-compact setting of complex n-space. This yields a probabilistic construction of Kahler solutions to Einstein's equations in complex n-space, with cosmological constant -beta, from a gas of interpolation nodes in equilibrium at positive inverse temperature -beta. In the infinite temperature limit, solutions to the Calabi-Yau equation are obtained. In the opposite zero-temperature the results may be interpreted as "transcendental" analogs of classical asymptotics for orthogonal polynomials, with the inverse temperature playing the role of the degree of a polynomial.