Minima of Epstein’s Zeta function and heights of flat tori

Sarnak, Peter; Strömbergsson, Andreas

doi:10.1007/s00222-005-0488-2

Cited by 110 publications

(170 citation statements)

References 23 publications

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“…In dimension d ≥ 3, some authors have studied the critical points and the (local or global) minima of the Jacobi theta function and the Epstein zeta function. In a famous article [218], Sarnak and Strömbergsson determined special local minima in dimensions 4, 8 and 24 (see also [62,58,63]). In dimension 3, Ennola has proved that the face centered cubic (FCC) lattice is a non-degenerate local minimum of ζ 3 (S, s) for all s > 0 [80].…”

Section: Optimal Lattices and Special Functionsmentioning

confidence: 99%

“…This excludes the HCP (Hexagonal Close Packed) lattice in dimension 3, since it is not a Bravais lattice. We refer to [167] for an explicit link between zeta functions and quantum field theory, to [226] for the link with optimal quadrature point repartition, and to [186,217,218] for the link with the optimization of the determinant of the Laplace operator: det(−∆) = e −ζ ′ d (S,0) . As a conclusion, determining the optimal periodic lattice can, for some simple potentials, be related to the study of special functions.…”

Section: Optimal Lattices and Special Functionsmentioning

confidence: 99%

See 1 more Smart Citation

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

142

191

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In this article we describe the crystallization conjecture. It states that, in appropriate physical conditions, interacting particles always place themselves into periodic configurations, breaking thereby the natural translation-invariance of the system. This famous problem is still largely open. Mathematically, it amounts to studying the minima of a real-valued function defined on R 3N where N is the number of particles, which tends to infinity. We review the existing literature and mention several related open problems, of which many have not been thoroughly studied.

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Section: Optimal Lattices and Special Functionsmentioning

confidence: 99%

Section: Optimal Lattices and Special Functionsmentioning

confidence: 99%

The crystallization conjecture: a review

Blanc¹,

Lewin²

2015

EMS Surv. Math. Sci.

142

191

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“…Therefore, the Fourier transform of α(r; R) is the following nonnegative function of k: It has been shown that finding the point process that minimizes the number variance σ 2 (R) is equivalent to finding the ground state of a certain repulsive pair potential with compact support [18]. This problem is directly related to an outstanding problem in number theory involving generalized zeta functions and lattices [25]. Understanding such ground states can be facilitated by utilizing "duality" relations that link ground states in real space to those in Fourier space [26].…”

Section: Number Variance and Hyperuniformitymentioning

confidence: 99%

Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory

2008

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Abstract.It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line R. Here we analytically provide exact generalizations of such a point process in d-dimensional Euclidean space R d for any d, which are special cases of determinantal processes. In particular, we obtain the n-particle correlation functions for any n, which completely specify the point processes in R d . We also demonstrate that spinpolarized fermionic systems in R d have these same n-particle correlation functions in each dimension. The point processes for any d are shown to be hyperuniform, i.e., infinite wavelength density fluctuations vanish, and the structure factor (or power spectrum) S(k) has a nonanalytic behavior at the origin given by S(k) ∼ |k| (k → 0). The latter result implies that the pair correlation function g 2 (r) tends to unity for large pair distances with a decay rate that is controlled by the power law 1/r d+1 , which is a well-known property of bosonic ground states and more recently has been shown to characterize maximally random jammed sphere packings. We graphically display one-and two-dimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be characterized by an effective "hard-core" diameter that grows like the square root of d. The nearest-neighbor distribution functions for these point processes are also evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius r in dimension d behaves like a Poisson point process but in dimension d + 1, i.e., this probability is given by expPoint processes in arbitrary dimension 2 for large r and finite d, where κ(d) is a positive d-dependent constant. We also show that as d increases, the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than 1/2 d . This coverage fraction has a special significance in the study of sphere packings in high-dimensional Euclidean spaces.

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“…However, using more sophisticated tools, Sarnak and Strömbergsson proved in [16] Their proof relies essentially on a certain property of the automorphism group of those lattices which is shown to imply the desired property, at least for s > n 2 (the proof for s in the "critical strip" 0 < s < n 2 is more involved and requires some extra arguments). Inspired by Delone and Ryshkov's theorem, one may ask for an explanation of this result in terms of spherical designs.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, it should apply to a wider class of lattices, for which the group theoretic tools used in [16] are not available, but for which one can prove however that the 4-design properties hold for all layers. In particular we prove that essentially all the extremal modular lattices (up to certain restrictions on both the dimension and the level) share with D 4 , E 8 and Λ 24 the property of being ζ-extreme at s for any s > The paper is organized as follows : in Section 1 we collect some preliminary results about lattices, spherical designs and Epstein ζ function.…”

Section: Introductionmentioning

confidence: 99%

Spherical designs and zeta functions of lattices

Coulangeon¹

2006

International Mathematics Research Notices

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ABSTRACT. We set up a connection between the theory of spherical designs and the question of minima of Epstein's zeta function. More precisely, we prove that a Euclidean lattice, all layers of which hold a 4-design, achieves a local minimum of the Epstein's zeta function, at least at any real s > n 2 . We deduce from this a new proof of Sarnak and Strömbergsson's theorem asserting that the root lattices D 4 and E 8 , as well as the Leech lattice Λ 24 , achieve a strict local minimum of the Epstein's zeta function at any s > 0. Furthermore, our criterion enables us to extend their theorem to all the so-called extremal modular lattices (up to certain restrictions) using a theorem of Bachoc and Venkov, and to other classical families of lattices (e.g. the Barnes-Wall lattices).INTRODUCTION.

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Minima of Epstein’s Zeta function and heights of flat tori

Cited by 110 publications

References 23 publications

The crystallization conjecture: a review

The crystallization conjecture: a review

Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory

Spherical designs and zeta functions of lattices

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