A variational principle for Gaussian lattice sums

Bétermin, Laurent; Faulhuber, Markus; Steinerberger, Stefan

doi:10.48550/arxiv.2110.06008

Cited by 3 publications

(3 citation statements)

References 53 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This phenomenon has appeared and conjectured in two component Bose-Einstein condensates (see in Mueller-Ho [17] and Luo-Wei [15]). When κ = 1 the optimality of hexagonal lattice for sum of Gaussian lattice has been investigated recently in [8].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

Luo,

Wei,

Zou

2021

Preprint

View full text Add to dashboard Cite

We consider the minimizing problem for energy functionals with two types of competing particles and completely monotone potential on a lattice. We prove that the minima of sum of two completely monotone functions among lattices is located exactly on a special curve which is part of the boundary of the fundamental region. We also establish a universal result for square lattice being the optimal in certain interval, which is surprising. Our result establishes the hexagonal-rhombic-square-rectangular transition lattice shapes in many physical and biological system (such as Bose-Einstein condensates and two-component Ginzburg-Landau systems). It turns out, our results also apply to locating the minimizers of sum of two Eisenstein series, which is new in number theory.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

Luo,

Wei,

Zou

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Whereas different rigorous techniques exist for studying lattice energies in dimension 𝑑 = 2 (see, e.g., Refs. 22,54,59,66,67), where dim  2 = 3 and dim  2 (𝑉) = 2 for all 𝑉 > 0, the situation in dimension 𝑑 = 3 is much more complicated. Indeed the so-called fundamental domain containing only one copy of each lattice (up to dilation, isometry, and symmetry) is five-dimensional for  3 (1) and has a complicated shape (see, e.g., Terras 69 ).…”

Section: Methodsmentioning

confidence: 99%

Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies

2022

View full text Add to dashboard Cite

The Riesz potential fs(r)=r−s$f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard‐Jones–type potentials fn,mLJ(r):=ar−n−br−m$f_{n,m}^{\rm {LJ}}(r):=a r^{-n}-b r^{-m}$, n>m$n>m$ that are widely used in molecular simulations. In this paper, we investigate analytically and numerically the minimizers among three‐dimensional lattices of Riesz and Lennard‐Jones energies. We discuss the minimality of the body‐centered‐cubic (BCC) lattice, face‐centered‐cubic (FCC) lattice, simple hexagonal (SH) lattices, and hexagonal close‐packing (HCP) structure, globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for s<0$s<0$ and the HCP lattice is computed to have higher energy than the FCC (for s>3/2$s>3/2$) and BCC (for s<3/2$s<3/2$) lattices. In the Lennard‐Jones case with exponents 3

show abstract

“…An important conjecture by Strohmer and Beaver [58] expects the condition number of (deterministic) Gabor frames with Gaussian windows to be optimized by a hexagonal lattice. Significant progress towards a proof has been made by Faulhuber and collaborators [26], and a preprint with a full solution of the problem has recently been posted in [18].…”

Section: Theorem 1 All Local Minima Of H (Z) Are Zeros Of F(z) and No...mentioning

confidence: 99%

Local Maxima of White Noise Spectrograms and Gaussian Entire Functions

Abreu

2022

J Fourier Anal Appl

View full text Add to dashboard Cite

We confirm Flandrin’s prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions), a consequence of the conjectured double honeycomb mean model for the network of zeros and local maxima, where the area of local maxima centered hexagons is three times larger than the area of zero centered hexagons. More precisely, we show that Gaussian spectrograms, normalized such that their expected density of zeros is 1, have an expected density of 5/3 critical points, among those 1/3 are local maxima, and 4/3 saddle points, and compute the distributions of ordinate values (heights) for spectrogram local extrema. This is done by first writing the spectrograms in terms of Gaussian Entire Functions (GEFs). The extrema are considered under the translation invariant derivative of the Fock space (which in this case coincides with the Chern connection from complex differential geometry). We also observe that the critical points of a GEF are precisely the zeros of a Gaussian random function in the first higher Landau level. We discuss natural extensions of these Gaussian random functions: Gaussian Weyl–Heisenberg functions and Gaussian bi-entire functions. The paper also reviews recent results on the applications of white noise spectrograms, connections between several developments, and is partially intended as a pedestrian introduction to the topic.

show abstract

A variational principle for Gaussian lattice sums

Cited by 3 publications

References 53 publications

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

On universally optimal lattice phase transitions and energy minimizers of completely monotone potentials

Three‐dimensional lattice ground states for Riesz and Lennard‐Jones–type energies

Local Maxima of White Noise Spectrograms and Gaussian Entire Functions

Contact Info

Product

Resources

About