Discrete and continuous green energy on compact manifolds

Beltrán, Carlos; Corral, Núria; Rey, Juan G. Criado del

doi:10.1016/j.jat.2018.09.004

Cited by 31 publications

(35 citation statements)

References 39 publications

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“…This cost is positive definite and has the same homogeneity near zero as the Coulomb potential |x − y| 2−d where d = dim(M ). This type of minimization has been studied in [BCC17]. For the case of embedded submanifolds M ⊂ R D this fits in the same framework as the celebrated Smale's 7 th problem [Sm98].…”

Section: Resultsmentioning

confidence: 99%

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Cotar

Petrache

2019

Advances in Mathematics

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Motivated by a problem arising from Density Functional Theory, we provide the sharp next-order asymptotics for a class of multimarginal optimal transport problems with cost given by singular, long-range pairwise interaction potentials. More precisely, we consider an N -marginal optimal transport problem with N equal marginals supported on R d and with cost of the form i =j |x i − x j | −s . In this setting we determine the second-order term in the N → ∞ asymptotic expansion of the minimum energy, for the long-range interactions corresponding to all exponents 0 < s < d. We also prove a small oscillations property for this second-order energy term. Our results can be extended to a larger class of models than power-law-type radial costs, such as non-rotationally-invariant costs. The key ingredient and main novelty in our proofs is a robust extension and simplification of the Fefferman-Gregg decomposition [F85], [Ge89], extended here to our class of kernels, and which provides a unified method valid across our full range of exponents. Our first result generalizes a recent work of Lewin, Lieb and Seiringer [LLS17], who dealt with the second-order term for the Coulomb case s = 1, d = 3 .1 and we will prove the following asymptotic expansion for F OT N,c (µ). For all 0 < s < d, if µ has density ρ ∈ L 1+s/d (R d ), then we have (see Theorem 1.1 below)Moreover we will prove that the strictly positive constant C(s, d) is independent of the choice of the marginal dµ(x) = ρ(x)dx and therefore can be interpreted as arising in an independent model problem. Furthermore, we will derive a small oscillations principle (with respect to N ) forwhich could also be interpreted as a rough third order asymptotic bound for F OT N,c (µ) with c as in (1.3). Our methods are extendable to more general costs, as non-rotationally-invariant ones. See Remark 1.3 below.For d = 3, s = 1 , the F OT N,c (µ) was introduced in the physics literature in the context of Density Functional Theory (DFT) by Seidl, Perdew, Levy, Gori-Giorgi, and Savin [Sd99, SPL99, SGGS07], without them being aware of its meaning in optimal transport. Namely, for s = 1, d = 3 , (1.1) is a natural semiclassical limit to the famous Hohenberg-Kohn (HK) functional from quantum mechanics, originally introduced by Hohenberg-Kohn in [HK64], and rigorously proved by Levy and Lieb in [Ly79], [Li83]. The connection to optimal transport was mathematically established later by [CFK11] and [BdPGG12] for N = 2 , later further extended to N = 3 in [BdP17], and recently independently proved for all N ≥ 2 by [CFK17] and [Lw17].In the process of establishing our two main results, we were required to prove, as key new tools for them, a set of additional secondary results of independent interest and of possible use to other settings, leading to generalized versions of the Fefferman-Gregg decomposition and positive definiteness criteria, as described in Section 4.1. What we use is a decomposition of positive definite kernels following the strategy established for s = 1, d = 3 by Fefferman [F85],...

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Section: Resultsmentioning

confidence: 99%

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Cotar

Petrache

2019

Advances in Mathematics

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“…For example, one can study the natural analogues of Riesz's energy as in Theorem 3.3 below. A very natural measure is given by the value of Green's energy of [3]:…”

Section: Introductionmentioning

confidence: 99%

“…Minimizers of Green's energy are assymptotically well-distributed (see [3,Main Theorem]). Our second main result will follow from the computation of the expected value of Green's energy for the projective ensemble.…”

Section: Introductionmentioning

confidence: 99%

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The Projective Ensemble and Distribution of Points in Odd-Dimensional Spheres

Beltrán

Etayo

2018

Constr Approx

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We define a determinantal point process on the complex projective space that reduces to the so-called spherical ensemble for complex dimension 1 under identification of the 2-sphere with the Riemann sphere. Through this determinantal point process we propose a point processs in odddimensional spheres that produces fairly well-distributed points, in the sense that the expected value of the Riesz 2-energy for these collections of points is smaller than all previously known bounds.

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“…There are many studies related to the extremal problems for different kinds of energy of a discrete charge (see, e. g., the papers [2][3][4], [9], [10], [12], and the references therein). In contrast to the previous research, we consider the Green energy (see also the recent articles [1] and [5]). In addition, an extremal problem with an alternating charge is studied.…”

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confidence: 99%

Green energy and extremal decompositions

Дубинин¹

2019

Issues of Analysis

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We give two precise estimates for the Green energy of a discrete charge, concentrated in an even number of points on the circle, with respect to the concentric ring. The lower estimate for the Green energy is attained for the points with a nonstandard symmetry. The well-known Pólya-Schur inequality for the logarithmic energy is a special case of this estimate. The proof is based on the application of dissymmetrization and an asymptotic formula for the conformal capacity of a generalized condenser in the case when some of its plates contract to given points. The upper bound is established for a charge that takes values of opposite signs. Its proof reduces to solving a problem on the so-called extremal decomposition of a circular ring with free poles on a circle.

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Discrete and continuous green energy on compact manifolds

Cited by 31 publications

References 39 publications

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

Next-order asymptotic expansion for N-marginal optimal transport with Coulomb and Riesz costs

The Projective Ensemble and Distribution of Points in Odd-Dimensional Spheres

Green energy and extremal decompositions

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