2018/10/16 by Berman, Robert J.
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1810.06939
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.