Gaussian Fluctuations Research Articles

Beyond Gaussian fluctuations of quantum anharmonic nuclei

Beyond Gaussian fluctuations of quantum anharmonic nuclei

A martingale approach to Gaussian fluctuations and laws of iterated logarithm for Ewens–Pitman model

The Ewens–Pitman model refers to a distribution for random partitions of [n]={1,…,n}, which is indexed by a pair of parameters α∈[0,1) and θ>−α, with α=0 corresponding to the Ewens model in population genetics. The large n asymptotic properties of the Ewens–Pitman model have been the subject of numerous studies, with the focus being on the number Kn of partition sets and the number Kr,n of partition subsets of size r, for r=1,…,n. While for α=0 asymptotic results have been obtained in terms of almost-sure convergence and Gaussian fluctuations, for α∈(0,1) only almost-sure convergences are available, with the proof for Kr,n being given only as a sketch. In this paper, we make use of martingales to develop a unified and comprehensive treatment of the large n asymptotic behaviours of Kn and Kr,n for α∈(0,1), providing alternative, and rigorous, proofs of the almost-sure convergences of Kn and Kr,n, and covering the gap of Gaussian fluctuations. We also obtain new laws of the iterated logarithm for Kn and Kr,n.

Particle production and hadronization temperature in the massive Schwinger model

We study the pair production, string breaking, and hadronization of a receding electron-positron pair using the bosonized version of the massive Schwinger model in quantum electrodynamics in 1+1 space-time dimensions. Specifically, we study the dynamics of the electric field in Bjorken coordinates by splitting it into a coherent field and its Gaussian fluctuations. We find that the electric field shows damped oscillations, reflecting pair production. Interestingly, the computation of the asymptotic total particle density per rapidity interval for large masses can be fitted using a Boltzmann factor, where the temperature can be related to the hadronization temperature in QCD. Lastly, we discuss the possibility of an analog quantum simulation of the massive Schwinger model using ultracold atoms, explicitly matching the potential of the Schwinger model to the effective potential for the relative phase of two linearly coupled Bose-Einstein condensates. Published by the American Physical Society 2024

Instantons, Fluctuations, and Singularities in the Supercritical Stochastic Nonlinear Schrödinger Equation.

Recently, Josserand etal. proposed a stochastic nonlinear Schrödinger model for finite-time singularity-mediated turbulence [Phys. Rev. Fluids 5, 054607 (2020)PLFHBR2469-990X10.1103/PhysRevFluids.5.054607]. Here, we use instanton calculus to quantify the effect of extreme fluctuations on the statistics of the energy dissipation rate. While the contribution of the instanton alone is insufficient, we obtain excellent agreement with direct simulations when including Gaussian fluctuations and the corresponding zero mode. Fluctuations are crucial to obtain the correct scaling when quasisingular events govern the turbulence statistics.

KPZ on torus: Gaussian fluctuations

KPZ on torus: Gaussian fluctuations

The stability of deep learning for 21cm foreground removal across various sky models and frequency-dependent systematics

ABSTRACT Deep learning (DL) has recently been proposed as a novel approach for 21cm foreground removal. Before applying DL to real observations, it is essential to assess its consistency with established methods, its performance across various simulation models, and its robustness against instrumental systematics. This study develops a commonly used U-Net and evaluates its performance for post-reionization foreground removal across three distinct sky simulation models based on pure Gaussian realizations, the Lagrangian perturbation theory, and the Planck sky model. Consistent outcomes across the models are achieved provided that training and testing data align with the same model. On average, the residual foreground in the U-Net reconstructed data is $\sim 10~{{\ \rm per\ cent}}$ of the signal across angular scales at the considered redshift range. Comparable results are found with traditional approaches. However, blindly using a network trained on one model for data from another model yields inaccurate reconstructions, emphasizing the need for consistent training data. The study then introduces frequency-dependent Gaussian beams and bandpass fluctuations to the test data. The network struggles to denoise data affected by ‘unexpected’ systematics without prior information. However, after re-training consistently with systematics-contaminated data, the network effectively restores its reconstruction accuracy. Our results highlight the importance of incorporating prior knowledge during network training compared with established blind methods. Our work provides critical guidelines for using DL for 21cm foreground removal, tailored to specific data attributes. Notably, it is the first time that DL has been applied to the Planck sky model being most realistic foregrounds at present.

Spectral distortions from acoustic dissipation with non-Gaussian (or not) perturbations

A well-known route to form primordial black holes in the early universe relies on the existence of unusually large primordial curvature fluctuations, confined to a narrow range of wavelengths that would be too small to be constrained by Cosmic Microwave Background (CMB) anisotropies. This scenario would however boost the generation of μ-type spectral distortions in the CMB due to an enhanced dissipation of acoustic waves. Previous studies of μ-distortion bounds on the primordial spectrum were based on the assumptions of Gaussian primordial fluctuations. In this work, we push the calculation of μ-distortions to one higher order in photon anisotropies. We discuss how to derive bounds on primordial spectrum peaks obeying non-Gaussian statistics under the assumption of local (perturbative or not) non-Gaussianity. We find that, depending on the value of the peak scale, the bounds may either remain stable or get tighter by several orders of magnitude, but only when the departure from Gaussian statistics is very strong. Our results are translated in terms of bounds on primordial supermassive black hole mass in a companion paper.

Random sequential adsorption with correlated defects : A series expansion approach.

The random sequential adsorption (RSA) problem holds crucial theoretical and practical significance, serving as a pivotal framework for understanding and optimizing particle packing in various scientific and technological applications. Here the problem of the one-dimensional RSA of k-mers onto a substrate with correlated defects controlled by uniform and power-law distributions is theoretically investigated: the coverage fraction is obtained as a function of the density of defects and several scaling laws are examined. The results are compared with extensive Monte Carlo simulations and more traditional methods based on master equations. Emphasis is given in elucidating the scaling behavior of the fluctuations of the coverage fraction. The phenomenon of universality breaking and the issues of conventional Gaussian fluctuations and the Lévy type fluctuations from a simple perspective, relying on the central limit theorem, are also addressed.

Cosmological Background Interpretation of Pulsar Timing Array Data.

We discuss the interpretation of the detected signal by pulsar timing array (PTA) observations as a gravitational wave background of cosmological origin. We combine NANOGrav 15-years and EPTA-DR2new datasets and confront them against backgrounds from supermassive black hole binaries (SMBHBs), and cosmological signals from inflation, cosmic (super)strings, first-order phase transitions, Gaussian and non-Gaussian large scalar fluctuations, and audible axions. We find that scalar-induced, and to a lesser extent audible axion and cosmic superstring signals, provide a better fit than SMBHBs. These results depend, however, on modeling assumptions, so further data and analysis are needed to reach robust conclusions. Independently of the signal origin, the data strongly constrain the parameter space of cosmological signals, for example, setting an upper bound on primordial non-Gaussianity at PTA scales as |f_{nl}|≲2.34 at 95%C.L.

Tunable Superdiffusion in Integrable Spin Chains Using Correlated Initial States.

Although integrable spin chains host only ballistically propagating particles, they can still feature diffusive charge transfer. This diffusive charge transfer originates from quasiparticle charge fluctuations inherited from the initial state's magnetization Gaussian fluctuations. We show that ensembles of initial states with quasi-long-range correlations lead to superdiffusive charge transfer with a tunable dynamical exponent. We substantiate our prediction with numerical simulations and discuss how finite time and finite size effects might cause deviations.

Rotating SU(2) gluon matter and deconfinement at finite temperature

In this work a non-abelian gauge theory is reformulated in a rotating frame. With this new formalism, the influence of the background rotation on the color deconfinement transition for a pure SU(2) gluon system has been studied. The KvBLL caloron, which is a color neutral and topologically nontrivial solution of Yang-Mills equation at finite temperature, is reexamined under rotation and adopted as the confined vacuum of such a system. With rotation-modified solutions of the caloron's constituent solitons, i.e. dyons, the non-perturbative part of effective potential of the rotating system has been obtained. Combining with the perturbative potential by Gaussian fluctuations, the critical temperature of confinement-deconfinement phase transition is investigated. It is found that neither the rotational semi-classical potential nor Gaussian fluctuations can confine color charges more tightly when the rotation becomes faster. While only a stronger coupling constant is able to make the critical temperature increase with angular velocity, as what indicated in lattice simulations. And it is found a non-monotonic dependence of the critical temperature on the angular velocity is established because of the competition between these two contrary contributions.

Entropy and heat capacity of the transverse momentum distribution for pp collisions at RHIC and LHC energies

We investigate the transverse momentum distribution (TMD) statistics from three different theoretical approaches. In particular, we explore the framework used for string models, wherein the particle production is given by the Schwinger mechanism. The thermal distribution arises from the Gaussian fluctuations of the string tension. The hard part of the TMD can be reproduced by considering heavy tailed string tension fluctuations, for instance, the Tsallis q-Gaussian function, giving rise to a confluent hypergeometric function that fits the entire experimental TMD data. We also discuss the QCD-based Hagerdon function, another family of fitting functions frequently used to describe the spectrum. We analyze the experimental data of minimum bias pp collisions reported by the BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider (LHC) experiments (from s=0.2 TeV to s=13 TeV). We extracted the corresponding temperature by studying the behavior of the spectra at low transverse momentum values. For the three approaches, we compute all moments, highlighting the average, variance, and kurtosis. Finally, we compute the Shannon entropy and the heat capacity through the entropy derivative with respect to the temperature. We found that the q-Gaussian string tension fluctuations lead to a monotonically increasing heat capacity as a function of the center-of-mass energy, which is also observed for the Hagedorn fitting function. This behavior is consistent with the experimental observation that the temperature slowly rises with increments of the collision energy. Published by the American Physical Society 2024

Gaussian Fluctuations for the Stochastic Burgers Equation in Dimension d≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 2$$\\end{document}

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a d-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. https://doi.org/10.1103/PhysRevLett.54.2026). In both the critical d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} and super-critical d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 3$$\\end{document} cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 3$$\\end{document} the scaling adopted is the classical diffusive one, while in d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way.

Gaussian fluctuations of the elephant random walk with gradually increasing memory

The elephant random walk (ERW) is a discrete-time random walk introduced by Schütz and Trimper (2004) in order to investigate how long-range memory affects the behavior of the random walk. Its particularity is that the next step of the walker depends on its whole past through a parameter p∈[0,1] . In this work, we investigate the validity of the central limit theorem of the ERW when the walker has only a gradually increasing memory. Our contribution provides a positive answer to a conjecture raised in a recent work by Gut and Stadtmüller (2022 Stat. Probab. Lett. 189 109598).

Asymptotic Gaussian Fluctuations of Eigenvectors in Spectral Clustering

Asymptotic Gaussian Fluctuations of Eigenvectors in Spectral Clustering

Limit Theorems for Default Contagion and Systemic Risk

We consider a general tractable model for default contagion and systemic risk in a heterogeneous financial network subjected to an exogenous macroeconomic shock. We show that under certain regularity assumptions, the default cascade model can be transformed into a death process problem represented by a balls-and-bins model. We state various limit theorems regarding the final size of default cascades. Under appropriate assumptions on the degree and threshold distributions, we prove that the final sizes of default cascades have asymptotically Gaussian fluctuations. We next state limit theorems for different system-wide wealth aggregation functions, which enable us to provide systemic risk measures in relation to the structure and heterogeneity of the financial network. Lastly, we demonstrate how these results can be utilized by a social planner to optimally target interventions during a financial crisis given a budget constraint and under partial information of the financial network.

A basic study on sound absorption characteristics of disordered hyperuniform periodic structures

Disordered hyperuniform systems are exotic states of matter that cover the intermediate regime between random and periodic structures. Despite appearing disordered and isotropic, they possess a hidden long-range order. Our study aims to develop disordered hyperuniform microperiodic structures that combine short-scale randomness and long-scale periodicity, making them versatile and useful for sound-absorbing materials, such as wall panels. To construct these structures, we generate a large number of three-dimensionally random waves within a specific frequency range, while applying periodic boundary conditions. Using the finite element method, we analyze the sound absorption of the hyperuniform structures. Compared to simple Gaussian random fluctuations, we observe an increase in surface area, resulting in higher airflow resistivity, and a rise in the sound absorption coefficient within the low to mid frequency range.

Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-dimensional sphere (d≥2). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for diverging sequences of Laplace eigenvalues. Our approach takes advantage of a recent result by Bally, Caramellino and Poly (2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (2015) for Hermite-rank 2 functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish second order Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough. Our argument requires some novel estimates on moments of products of Gegenbauer polynomials that may be of independent interest, which we prove via the link between graph theory and diagram formulas.

Critical assessment of the recent report on the gigaparsec-scale correlation of the orientations of large quasar groups

ABSTRACT Recently, it was reported that large quasar groups (LQGs) identified from the Sloan Digital Sky Survey (SDSS) data release seven catalogue are not randomly oriented but preferentially aligned or orthogonal over scales 1–2 Gpc. To confirm this claim, I reproduced the same LQG sample and performed Sobolev tests of uniformity on the LQG orientation axes in the redshift space. Contrary to the original report based on the bimodal distribution of the LQG position angles in the sky, I found no departure from uniformity in the distribution of the LQG orientation axes. I also examined whether the LQGs are physical structures using a statistically more reliable data set constructed from the SDSS data release 16 (DR16) large-scale structure (LSS) quasar catalogue. Considering the Gaussian primordial density fluctuations and shot noise, I estimated the mass density contrasts of the LQGs from the number counts of the DR16 LSS quasars and found that most of the LQGs do not trace statistically significant high-density regions. I conclude that the LQG sample is a collection of unphysical chance associations and should not be used for any cosmological studies.

Assessment of the partial saddle point approximation in field-theoretic polymer simulations.

Field-theoretic simulations are numerical treatments of polymer field theory models that go beyond the mean-field self-consistent field theory level and have successfully captured a range of mesoscopic phenomena. Inherent in molecularly-based field theories is a "signproblem" associated with complex-valued Hamiltonian functionals. One route to field-theoretic simulations utilizes the complex Langevin (CL) method to importance sample complex-valued field configurations to bypass the signproblem. Although CL is exact in principle, it can be difficult to stabilize in strongly fluctuating systems. An alternate approach for blends or block copolymers with two segment species is to make a "partial saddle point approximation" (PSPA) in which the stiff pressure-like field is constrained to its mean-field value, eliminating the signproblem in the remaining field theory, allowing for traditional (real) sampling methods. The consequences of the PSPA are relatively unknown, and direct comparisons between the two methods are limited. Here, we quantitatively compare thermodynamic observables, order-disorder transitions, and periodic domain sizes predicted by the two approaches for a weakly compressible model of AB diblock copolymers. Using Gaussian fluctuation analysis, we validate our simulation observations, finding that the PSPA incorrectly captures trends in fluctuation corrections to certain thermodynamic observables, microdomain spacing, and location of order-disorder transitions. For incompressible models with contact interactions, we find similar discrepancies between the predictions of CL and PSPA, but these can be minimized by regularization procedures such as Morse calibration. These findings mandate caution in applying the PSPA to broader classes of soft-matter models and systems.