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11-13 Dec 2024 PARIS (France)
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Abstracts
(in order received)
Marco Salvalaglio
Title: Local structures in hyperuniform patterns: characterisation via persistent homology and perspectives
Abstract:
Hyperuniformity refers to the suppression of density fluctuations at large scales. Its classical definition utilizes information at the largest accessible scales, such as the scaling of the number variance for large sampling window size or the structure factor at small wave numbers. At the same time, hyperuniform (HU) configurations have distinctive local characteristics: strong HU characters can even be guessed by the naked eye of an expert. More importantly, such characteristics are expected to affect properties like response to deformation and filtration characteristics, and it is thus of general interest to tailor them for various applications. This presentation illustrates how local features in HU configurations can be characterized via their topological properties. In particular, we focus on point clouds and characterize their persistent homology as well as the statistics of local graph neighborhoods. We then discuss implications related to this novel information. We find that varying the structure factor results in configurations with systematically different topological properties. Moreover, these topological properties are almost conserved for subsets of HU point clouds, establishing a connection between finite-sized systems and idealized reference arrangements. As with classical measures, HU arrangements result in a distinct class of patterns realizing an order-to-disorder transition also when looking at local topological neighborhoods. Finally, concepts for inverse design and detection of HU patterns leveraging persistent homology are discussed together with future perspectives and potential applications.
David Dereudre
Title: Hyperuniformity of dependent perturbed lattices.
Abstract: We ask whether a stationary lattice in dimension $d$ whose points are shifted by identically distributed but possibly dependent perturbations remains hyperuniform. When $d = 1$ or $2$, we show that it is the case when the perturbations have a finite $d$-moment, and that this condition is sharp. When $d \geq 3$, we construct arbitrarily small perturbations such that the resulting point process is not hyperuniform.
Joint work with: Daniela Flimel, Martin Huesmann and Thomas Leblé
Daniela Flimmel
Title: Fitting regular point patterns with hyperuniform perturbed lattices
Abstract: When detecting regularity within a dataset, Gibbsian models became a common choice. It has been shown that a large class of repulsive potentials lead to non-hyperuniform Gibbs point processes. On the other hand, perturbed lattices could preserve their natural repulsivity and therefore, should be considered a valid candidate for the model as well. It is nowadays understood that perturbed lattices form a wide inventory of point processes from class I hyperuniform to above-Poisson fluctuating. Depending on the variability in the dataset, one can choose a suitable perturbation and expect straightforward intuition in the model as well as fast computation times. Selected results will be presented on the grain centers of nitinol alloy in 3D.
Martin Huesmann
Title: Hyperuniformity, Coulomb Energy and Wasserstein distance
Abstract: We show an interplay between the three properties of two-dimensional point processes: A) HU, B) Finite regularized Coulomb energy, and C) Finite 2-Wasserstein distance to Lebesgue. In particular, we will show that B) implies C) implies A) and there are counterexamples for the converse directions.
This is based on joint work with Thomas Leblé.
Mathias Kasiulis
Title: Fast generation of large hyperuniform point patterns
Abstract: In physics, recent work has highlighted the potential of correlated disordered materials (aperiodic non-Poissonian structures) to achieve functions. For instance, in photonics, such structures may display properties usually associated with periodic media (transparency, birefringence, structural coloration), and even give rise to unique phenomena like Anderson localization. The majority of existing results on correlated disordered media address only a few classes of disorder, e.g. stealthy hyperuniform systems. Furthermore, they are largely limited to small systems, consisting of only a few hundreds or thousands of particles. Both constraints stem from limitations in the numerical algorithms used to generate correlated disordered structures.
In this talk, I will discuss how the Fast Reciprocal-Space Correlator (FReSCo) addresses the issue by generating large point patterns using an optimization strategy that relies on non-uniform Fast Fourier Transforms. I will show that this strategy lets me generate truly arbitrary structures, at lightning speed.
In the particular example of hyperuniform structures, I will show that we obtained the largest ever disordered patterns with robust hyperuniform scalings in both structure factor and density fluctuations. Motivated by physical realizations of power-law hyperuniformity, I will show that the same strategy can be generalized to include short-range repulsion, thus leading to more realistic hyperuniform large particle packings akin to critical points of absorbing-state models.
Thomas Leblé
Title: Hyperuniformity, rigidity and translation-invariance for the two-dimensional one-component plasma
Abstract: I will present some recent results about the two-dimensional one-component plasma (or Coulomb gas): at all temperatures, its infinite-volume “limit” is hyperuniform, number-rigid and translation-invariant. This generalizes what was known for the Ginibre ensemble, which corresponds to a particular value of the temperature.
I will also mention some open problems, among which the Jancovici-Lebowitz-Manificat law, which is a precise and strong form of hyperuniformity.
Günther Koliander
Title: Hyperuniformity and Non-Hyperuniformity of Zeros of Twisted Stationary Gaussian Random Functions
Abstract: We present our recent results on the (non-)hyperuniformity of the zero set of a class of twisted stationary Gaussian random functions on the complex plane. We show that if zeros are weighted with their positive or negative winding number (charged zero set), the charged zero distribution is always hyperuniform. However, considering the (uncharged) zero set without taking the winding number into account, the distribution is not hyperuniform in simple scenarios. Only for the special case of a (scaled) Gaussian entire function we could show that the uncharged zero distribution is hyperuniform but also coincides with the charged zero distribution as the winding number of all zeros is one. Whether this is the only case where the uncharged zero distribution is hyperuniform is still an open problem.
Examples of the class of twisted stationary Gaussian random functions appear naturally as the short-time Fourier transform of signals in white noise. We will show the results of simulations in this setting and see that a hyperuniform-like behavior, i.e., a linear growth of the variance with respect to the radius, is already observed in finite domains.
Examples of the class of twisted stationary Gaussian random functions appear naturally as the short-time Fourier transform of signals in white noise. We will show the results of simulations in this setting and see that a hyperuniform-like behavior, i.e., a linear growth of the variance with respect to the radius, is already observed in finite domains.
Bartek Blaszczyszyn
Title: Ergodic Learning of Point Processes: From Theory to Practice
Abstract: Ergodicity provides a fundamental bridge between probability theory and statistics, particularly in spatial statistics, where it links spatial averages to their corresponding mathematical expectations. A noteworthy, yet often overlooked, consequence of ergodicity is that a single complete realization of a stationary ergodic model theoretically allows for an almost sure estimation of the underlying distribution.
In practice, however, we only observe partial realizations within finite spatial windows. Given a sufficiently large window (i.e., a high number of observed points), can we approximate the unknown underlying distribution and use it to generate new realizations?
Motivated by recent advancements in gradient descent methods for maximum entropy models, we propose, in joint work with Brochard et al. (2022), a novel approach to generating similar point patterns. Our method begins with an initial Poisson process realization and iteratively moves its particles toward a target counting measure.
We assess the performance of our model on various point processes exhibiting distinct geometric structures, using spectral and topological data analyses, and compare it to the statistical reconstruction method of Tscheschel and Stoyan (2006). Notably, this approach successfully captures the spectral characteristics of the input process and has inspired subsequent work by Mastrilli et al. (2024) on estimating the hyperuniformity exponent of point processes.
References:
Brochard, A., Błaszczyszyn, B., Mallat, S., & Zhang, S. (2022). Particle gradient descent model for point process generation. Statistics and Computing, 32(1), 1-25.
Mastrilli, G., Błaszczyszyn, B., & Lavancier, F. (2024). Estimating the hyperuniformity exponent of point processes. arXiv preprint, arXiv:2407.16797.
Tscheschel, A., & Stoyan, D. (2006). Statistical reconstruction of random point patterns. Computational Statistics & Data Analysis, 51(2), 859-871.
In practice, however, we only observe partial realizations within finite spatial windows. Given a sufficiently large window (i.e., a high number of observed points), can we approximate the unknown underlying distribution and use it to generate new realizations?
Motivated by recent advancements in gradient descent methods for maximum entropy models, we propose, in joint work with Brochard et al. (2022), a novel approach to generating similar point patterns. Our method begins with an initial Poisson process realization and iteratively moves its particles toward a target counting measure.
We assess the performance of our model on various point processes exhibiting distinct geometric structures, using spectral and topological data analyses, and compare it to the statistical reconstruction method of Tscheschel and Stoyan (2006). Notably, this approach successfully captures the spectral characteristics of the input process and has inspired subsequent work by Mastrilli et al. (2024) on estimating the hyperuniformity exponent of point processes.
References:
Brochard, A., Błaszczyszyn, B., Mallat, S., & Zhang, S. (2022). Particle gradient descent model for point process generation. Statistics and Computing, 32(1), 1-25.
Mastrilli, G., Błaszczyszyn, B., & Lavancier, F. (2024). Estimating the hyperuniformity exponent of point processes. arXiv preprint, arXiv:2407.16797.
Tscheschel, A., & Stoyan, D. (2006). Statistical reconstruction of random point patterns. Computational Statistics & Data Analysis, 51(2), 859-871.
Guenter Last
Title: Persistence of hyperuniformity under invariant allocations
Abstract: We consider invariant allocations transporting a stationary random measure. Under a suitable mixing assumption we prove that the resulting random measure has the same asymptotic variance. In particular we obtain a mixing criterion for the persistence of hyperuniformity under allocations. In examples the mixing assumption can be checked with a factorial moment expansion combined with stopping set techniques.
The talk is based on joint work with M. Klatt (Ulm), L. Lotz (Cologne)
and D. Yogeshwaran (Bangalore).
Raphaël Lachièze-Rey
Title: Rigidity of stationary point processes
Abstract: Rigidity is an intriguing phenomenon affecting some point processes. A point process X is rigid if some features of X on a subset A of R^d can be inferred from the knowledge of X on other parts of the space, and this phenomenon seems strongly related to the hyperuniformity of X. Number rigidity, which attracted a lot of attention, means the number of points on A can be recovered, k-rigidity that the moment of order k can be recovered, and maximal rigidity that all of X on A can be recovered. We show in all generality that k-rigidity occurs if X's structure factor has a zero of order k, and that this condition is sufficient if S has a nice structure, such as isotropy, separability, or a finite number of zeros. It allows for instance to characterise the class of rigid DPPs. We also discuss some interesting phenomena involving maximal rigidity for more general random measures.
Raphaël Butez
Title: On the Wasserstein distance between a hyperuniform point process and its mean
Abstract: In this talk I will present a recent result obtained in collaboration with S.Dallaporta and D.Garcia-Zelada. We study the p-Wasserstein distance between the counting measure of a point process and its intensity and we show that, under a condition on the p-th centered moments of counting statistics which amount to hyperuniformity when p=2, the average transport cost is bounded by a multiple of the number of points.We also show that hyperuniform point processes satisfy the condition for any value of p.
Gabriel Mastrilli
Title: Estimating the hyperuniformity exponent of spatial point processes.
Abstract: Hyperuniform point processes are characterized by reduced fluctuations compared to Poisson processes, with the variance of points in large spatial domains growing more slowly than the volume of the domain. Originally studied in statistical physics, hyperuniformity has found applications in both theoretical and applied fields. A key feature of hyperuniformity is the structure factor S, related to Bartlett’s spectral measure, which often follows a power-law behavior near zero. The exponent of this power law, known as the hyperuniformity exponent, quantifies the degree of hyperuniformity. In this presentation, we introduce an estimator for the hyperuniformity exponent, analyze its asymptotic properties, and discuss strategies for reliable estimation using a single realization of the point process.
Jonas Jalowy
Title: ox-Covariances of Hyperuniform Point Processes
Abstract: When Hyperuniform point processes are characterized by specific variance asymptotics of growing boxes, a natural question arises: What is the (relative) covariance between two boxes? In this talk, we review recent related results, discuss a curious universal and a non-universal covariance structure of Hyperuniform point processes depending on their class/exponent and present the proofs relying solely on symmetry assumptions of the truncated pair correlation measure.
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