Gaussian Fluctuations Research Articles

Gaussian fluctuations in the equipartition principle for Wigner matrices

Abstract The total energy of an eigenstate in a composite quantum system tends to be distributed equally among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the bulk eigenvectors of Wigner matrices with an arbitrary deformation.

Kinetics of the Chiral Phase Transition in a Quark–Meson \(\sigma \) Model

Using the two-particle irreducible (2PI) $\Phi$-functional formalism for self-consistent approximations of a linear-$\sigma$ model for quarks and mesons in and out of equilibrium, the build-up of fluctuations of net-baryon number during the time evolution of an expanding fireball is studied within a kinetic theory for the order parameter ($\sigma$ field) and quark distribution functions. Initializing the system with purely Gaussian fluctuations a fourth-order cumulant is temporarily built up due to the evolution of the $\sigma$-field. This is counterbalanced, however, by the dissipative evolution due to collisions between quarks, anti-quarks, mesons, and the mean field, depending on the speed of the fireball expansion.

Insights into the initial conditions and evolution of hadronic collisions with flow observables

Studies of anisotropic flow in heavy-ion collisions are one of the critical elements in understanding the production and evolution of the Quark Gluon Plasma. These proceedings review the most recent results on anisotropic flow measurements at the LHC energies. In particular, recent measurements of pT-differential v2{4} in Pb–Pb collisions for a plethora of light-flavour particles are discussed and compared to models with and without partonic coalescence. The measurements of v2{2} and v2{4} allow for the calculation of relative flow fluctuations, which for the first time are also reported for (multi)strange baryons. The study is further extended to higher moments of the underlying charged-particle v2 PDF, where a transition from non-Gaussian to Gaussian fluctuations is observed. Finally, the correlation between v2 and mean transverse momentum [pT] is studied in terms of the Pearson Correlation Coefficient. This opens new venues to study nuclear deformation in ultra-central heavy-ion collisions and could provide a tool to identify processes in the pre-QGP phase.

Gaussian fluctuation for spatial average of the stochastic pseudo-partial differential equation with fractional noise

Gaussian fluctuation for spatial average of the stochastic pseudo-partial differential equation with fractional noise

1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency

On–off intermittency occurs in nonequilibrium physical systems close to bifurcation points, and is characterised by an aperiodic switching between a large-amplitude ‘on’ state and a small-amplitude ‘off’ state. Lévy on–off intermittency is a recently introduced generalisation of on–off intermittency to multiplicative Lévy noise, which depends on a stability parameter α and a skewness parameter β. Here, we derive two novel results on Lévy on–off intermittency by leveraging known exact results on the first-passage time statistics of Lévy flights. First, we compute anomalous critical exponents explicitly as a function of arbitrary Lévy noise parameters for the first time, by a heuristic method, extending previous results. The predictions are verified using numerical solutions of the fractional Fokker–Planck equation. Second, we derive the power spectrum S(f) of Lévy on–off intermittency, and show that it displays a power law at low frequencies f, where depends on the noise parameters . An explicit expression for κ is obtained in terms of . The predictions are verified using long time series realisations of Lévy on–off intermittency. Our findings help shed light on instabilities subject to non-equilibrium, power-law-distributed fluctuations, emphasizing that their properties can differ starkly from the case of Gaussian fluctuations.

Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale \(\varepsilon \), and tuned logarithmically by a factor \(\frac{1}{\sqrt{\log \varepsilon ^{-1}}}\) in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as \(\varepsilon \downarrow 0\). The tool we used here is the Malliavin-Stein’s method. We also give a functional version of this result.

Lyapunov Exponent, Universality and Phase Transition for Products of Random Matrices

Products of M i.i.d. random matrices of size \(N \times N\) are related to classical limit theorems in probability theory (\(N=1\) and large M), to Lyapunov exponents in dynamical systems (finite N and large M), and to universality in random matrix theory (finite M and large N). Under the two different limits of \(M\rightarrow \infty \) and \(N\rightarrow \infty \), the local singular value statistics display Gaussian and random matrix theory universality, respectively. However, it is unclear what happens if both M and N go to infinity. This problem, proposed by Akemann et al. (J Phys A 47(39):395202, 2014) and Deift (SIGMA Symmetry Integr Geom Methods Appl 13, 2017), lies at the heart of understanding both kinds of universal limits. In the case of complex Gaussian random matrices, we prove that there exists a crossover phenomenon as the relative ratio of M and N changes from 0 to \(\infty \): sine and Airy kernels from the Gaussian Unitary Ensemble (GUE) when \(M/N \rightarrow 0\), Gaussian fluctuation when \(M/N \rightarrow \infty \), and new critical phenomena when \(M/N \rightarrow \gamma \in (0,\infty )\). Accordingly, we further prove that the largest singular value undergoes a phase transition between the Gaussian and GUE Tracy–Widom distributions.

Fluctuations of subgraph counts in graphon based random graphs

AbstractGiven a graphon $W$ and a finite simple graph $H$ , with vertex set $V(H)$ , denote by $X_n(H, W)$ the number of copies of $H$ in a $W$ -random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$ . Towards this we introduce a notion of $H$ -regularity of graphons and show that if the graphon $W$ is not $H$ -regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$ . On the other hand, if $W$ is $H$ -regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$ . Our proofs use the asymptotic theory of generalised $U$ -statistics developed by Janson and Nowicki [22]. We also investigate the structure of $H$ -regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$ -regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$ . We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.

Two-dimensional exciton-polariton interactions beyond the Born approximation

We present a many-body theory for the interactions of two-dimensional exciton-polaritons beyond the Born approximation. Taking into account Gaussian quantum fluctuations via the Bogoliubov theory, we find that the exciton-exciton scattering is strongly renormalized by the light-matter coupling and acquires an inverse logarithmic dependence on the lower polariton energy. As a result, the polariton interaction strength may be enhanced at red photon-exciton detuning when the exciton-exciton scattering length is large. We also derive the exact Tan relations for exciton-polaritons, calculate Tan's contact coefficient and find an even more significant enhancement. Our predictions may provide a qualitatively correct guide for studies of exciton and polariton nonlinearities and suggest a route to achieving strongly nonlinear polariton gases.

The Kraichnan Model and Non-equilibrium Statistical Physics of Diffusive Mixing

We discuss application of methods from the Kraichnan model of turbulent advection to the study of non-equilibrium concentration fluctuations arising during diffusion in liquid mixtures at high Schmidt numbers. This approach treats nonlinear advection of concentration fluctuations exactly, without linearization. Remarkably, we find that static and dynamic structure functions obtained by this method reproduce precisely the predictions of linearized fluctuating hydrodynamics. It is argued that this agreement is an analogue of anomaly non-renormalization which does not, however, protect higher-order multi-point correlations. The latter should thus yield non-vanishing cumulants, unlike those for the Gaussian concentration fluctuations predicted by linearized theory.

Gaussian Fluctuation for Smoothed Local Correlations in CUE

Motivated by the Rudnick-Sarnak theorem we study limiting distribution of smoothed local correlations of the form ∑j1,j2,…,jnf(N(θj2-θj1),N(θj3-θj1),…,N(θjn-θj1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{j_1, j_2, \\ldots , j_n} f(N(\ heta _{j_2}-\ heta _{j_1}), N(\ heta _{j_3}-\ heta _{j_1}), \\ldots , N(\ heta _{j_n}-\ heta _{j_1})) \\end{aligned}$$\\end{document}for the Circular United Ensemble of random matrices for sufficiently smooth test functions.

Signal of cosmic strings in cross-correlation of 21-cm redshift and CMB polarization maps

We study the signal of cosmic string wakes present before the time of reionization in the cross-correlation signal of 21-cm redshift and B-mode CMB polarization maps. The specific non-Gaussian signal of strings in the position space cross-correlation maps can be extracted by means of a matched filtering analysis. Signals of strings with tension somewhat lower than those corresponding to the current upper bound can be identified when embedded in a background of Gaussian fluctuations from a Planck best-fit LCDM model.

Central Limit Theorem for Kernel Estimator of Invariant Density in Bifurcating Markov Chains Models

Bifurcating Markov chains are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. Motivated by the functional estimation of the density of the invariant probability measure which appears as the asymptotic distribution of the trait, we prove the consistency and the Gaussian fluctuations for a kernel estimator of this density based on late generations. In this setting, it is interesting to note that the distinction of the three regimes on the ergodic rate identified in a previous work (for fluctuations of average over large generations) disappears. This result is a first step to go beyond the threshold condition on the ergodic rate given in previous statistical papers on functional estimation.

Path Integral Derivation and Numerical Computation of Large Deviation Prefactors for Non-equilibrium Dynamics Through Matrix Riccati Equations

For many non-equilibrium dynamics driven by small noise, in physics, chemistry, biology, or economy, rare events do matter. Large deviation theory then explains that the leading order term of the main statistical quantities have an exponential behavior. The exponential rate is often obtained as the infimum of an action, which is minimized along an instanton. In this paper, we consider the computation of the next order sub-exponential prefactors, which are crucial for a large number of applications. Following a path integral approach, we derive the dynamics of the Gaussian fluctuations around the instanton and compute from it the sub-exponential prefactors. As might be expected, the formalism leads to the computation of functional determinants and matrix Riccati equations. By contrast with the cases of equilibrium dynamics with detailed balance or generalized detailed balance, we stress the specific non locality of the solutions of the Riccati equation: the prefactors depend on fluctuations all along the instanton and not just at its starting and ending points. We explain how to numerically compute the prefactors. The case of statistically stationary quantities requires considerations of non trivial initial conditions for the matrix Riccati equation.

Equilibrium fluctuations in mean-field disordered models.

Mean-field models of glasses that present a random first order transition exhibit highly nontrivial fluctuations. Building on previous studies that focused on the critical scaling regime, we here obtain a fully quantitative framework for all equilibrium conditions. By means of the replica method we evaluate Gaussian fluctuations of the overlaps around the thermodynamic limit, decomposing them in thermal fluctuations inside each state and heterogeneous fluctuations between different states. We first test and compare our analytical results with numerical simulation results for the p-spin spherical model and the random orthogonal model, and then analyze the random Lorentz gas. In all cases, a strong quantitative agreement is obtained. Our analysis thus provides a robust scheme for identifying the key finite-size (or finite-dimensional) corrections to the mean-field treatment of these paradigmatic glass models.

Quantitative two-scale stabilization on the Poisson space

We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one—that we evaluate and compare at two different scales—and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization—see Penrose and Yukich (Ann. Appl. Probab. 11 (2001) 1005–1041) and Penrose (Ann. Probab. 33 (2005) 1945–1991). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (Ann. Appl. Probab. 27 (2017) 1588–1645) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbour graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.

Three-dimensional Gaussian fluctuations of spectra of overlapping stochastic Wishart matrices

In [I. Dumitriu and E. Paquette, Spectra of overlapping Wishart matrices and the gaussian free field, Random Matrices: Theory Appl. 07(2) (2018) 1850003], the authors consider eigenvalues of overlapping Wishart matrices and prove that its fluctuations asymptotically convergence to the Gaussian free field. In this brief note, their result is extended to show that when the matrix entries undergo stochastic evolution, the fluctuations asymptotically converge to a three-dimensional Gaussian field, which has an explicit contour integral formula. This is analogous to the result of [A. Borodin, CLT for spectra of submatrices of Wigner random matrices, Moscow Math. J. 14(1) (2014) 29–38] for stochastic Wigner matrices.

Theory and simulation of electrokinetic fluctuations in electrolyte solutions at the mesoscale

Electrolyte solutions play an important role in energy storage devices, whose performance relies heavily on the electrokinetic processes at sub-micron scales. Although fluctuations and stochastic features become more critical at small scales, the long-range Coulomb interactions pose a particular challenge for both theoretical analysis and simulation of fluid systems with fluctuating hydrodynamic and electrostatic interactions. Here, we present a theoretical framework based on the Landau–Lifshitz theory to derive closed-form expressions for fluctuation correlations in electrolyte solutions, indicating significantly different decorrelation processes of ionic concentration fluctuations from hydrodynamic fluctuations, which provides insights for understanding transport phenomena of coupled fluctuating hydrodynamics and electrokinetics. Furthermore, we simulate fluctuating electrokinetic systems using both molecular dynamics (MD) with explicit ions and mesoscopic charged dissipative particle dynamics (cDPD) with semi-implicit ions, from which we identify that the spatial probability density functions of local charge density follow a gamma distribution at sub-nanometre scale (i.e. $0.3\,{\rm nm}$ ) and converge to a Gaussian distribution above nanometre scales (i.e. $1.55\,{\rm nm}$ ), indicating the existence of a lower limit of length scale for mesoscale models using Gaussian fluctuations. The temporal correlation functions of both hydrodynamic and electrokinetic fluctuations are computed from all-atom MD and mesoscale cDPD simulations, showing good agreement with the theoretical predictions based on the linearized fluctuating hydrodynamics theory.

Gaussian fluctuation for spatial average of super-Brownian motion

Let be the density of one-dimensional super-Brownian motion starting from Lebesgue measure. Using the Laplace functional of super-Brownian motion, we prove that as the normalized spatial integral converges jointly in (t, x) to Brownian sheet in distribution.

Dimensional crossovers in the Gaussian critical fluctuations above Tc of two-layer and three-layer superconductors

By using a Ginzburg–Landau functional in the Gaussian approximation, we calculate the energy of superconducting fluctuations above the transition, at zero external magnetic field, of a system composed by a small number N of parallel two-dimensional superconducting planes, each of them Josephson coupled to its first neighbour, with special focus in the N = 2 and 3 cases. This allows us to obtain expressions for the critical contributions to various observables (fluctuation specific heat and magnetic susceptibility and Aslamazov–Larkin paraconductivity). Our results suggest that these systems may display deviations from pure 2D behaviour and interesting crossover effects, with both similitudes and differences to those known to occur in infinite-layers superconductors. Some challenges for future related research are also outlined.