Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry

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 C n . A fair amount of background material is included. Along the way, the results are generalized to the non-compact setting of C n . This yields a probabilistic construction of Kähler solutions to Einstein's equations in C n , with cosmological constant −β, from a gas of interpolation nodes in equilibrium at positive inverse temperature β. In the infinite temperature limit, β → 0, 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 β playing the role of the degree of a polynomial.

Priors leading to well-behaved Coulomb and Riesz gases versus zeroth-order phase transitions – a potential-theoretic characterization

“…Complex-geometric analogs of the results, where the role of the Laplacian is played by the complex Monge-Ampère operator, are described in [9]. Our results yield new probabilistic constructions of capacities, equilibrium measures etc, using random point processes, in contrast to the usual probabilistic approach based on Markov processes (and their hitting probabilities [33,22]).…”

“…The "only if" direction in Theorem 1.3 was first shown in the complex-geometric setting of compact Kähler manifolds X in [9] and generalized to the non-compact setting pluripotential setting of C n in [9], using a compactification argument. See also [35] for a far-reaching generalization of [8, Theorem 2.1] to general measures not charging pluripolar subsets and bounded weights.…”

“…In particular, the case when X is the Riemann sphere implies (by a compactification argument, as in [9]) that such a rate holds for the Coulomb gas in R [2] (see also [1, formula 18] for an explicit integral formula for µ T,φ in the case of the Coulomb gas on R with φ(x) = x 2 and µ 0 the Gaussian measure, so that µ T,φ converges, as T → 0, to Wigner's semi-circle law). However, it should be stressed that in the general setting of determining measures µ 0 , studied in the present paper, one can not hope for quantitative rates, unless regularity assumptions on µ 0 are imposed.…”

“…This means that the corresponding Coulomb gas in C defines a determinantal point process with correlation kernel K k (z, w), where K k is the integral kernel of the orthogonal projection from L 2 (C, µ 0 ) onto the space P k (C) of all polynomials p k (z) on C of degree at most k := N − 1 : for an orthonormal base p j in P k (C) (known as the Christoffel-Darboux kernel in the literature on orthogonal polynomials and the Bergman kernel in the complex analysis literature). In fact, this is the case for any measure µ 0 on C not charging polar subsets (see, for example, [9]). Accordingly, it follows from general properties of determinantal point processes that…”

“…Remark 2.4. The notation is inspired by the corresponding complex-geometric setup, using the notation in [5,9], applied to the special case when d = α = 2. In this case, when φ is expressed as the restriction to S of a function with logarithmic growth in C, the symbol ω φ denotes the signed measure ω φ := ∆φ (extended by zero from C to the Riemann sphere) and E ω φ is the corresponding pluricomplex energy.…”