Global fluctuations in physical systems: a subtle interplay between sum and extreme value statistics

Abstract

Fluctuations of global additive quantities, like total energy or magnetization for instance, can in principle be described by statistics of sums of (possibly correlated) random variables. Yet, it turns out that extreme values (the largest value among a set of random variables) may also play a role in the statistics of global quantities, in a direct or indirect way. This review discusses different connections that may appear between problems of sums and of extreme values of random variables, and emphasizes physical situations in which such connections are relevant. Along this line of thought, standard convergence theorems for sums and extreme values of independent and identically distributed random variables are recalled, and some rigorous results as well as more heuristic reasonings are presented for correlated or non-identically distributed random variables. More specifically, the role of extreme values within sums of broadly distributed variables is addressed, and a general mapping between extreme values and sums is presented, allowing us to identify a class of correlated random variables whose sum follows (generalized) extreme value distributions. Possible applications of this specific class of random variables are illustrated on the example of two simple physical models. A few extensions to other related classes of random variables sharing similar qualitative properties are also briefly discussed, in connection with the so-called BHP distribution.

58 pages, final version, typo corrected in Theorem 6

References in corpus (71)

• Shape Fluctuations and Random Matrices
• Models of traps and glass phenomenology
• Statistical Aging and Non Ergodicity in the Fluorescence of Single Nanocrystals
• Universality Classes for Extreme Value Statistics
• Non-Gaussian equilibrium in a long-range Hamiltonian system
• Expansions for nearly Gaussian distributions
• Environmental effects in the third moment of voltage fluctuations in a tunnel junction
• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Aging on Parisi's tree
• Rank-Ordering Statistics of Extreme Events: Application to the Distribution of Large Earthquakes
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Similarity of fluctuations in correlated systems: The case of seismicity
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Stochastic Path Integral Formulation of Full Counting Statistics
• 1/f Noise and Extreme Value Statistics
• Magnetic fluctuations in the classical XY model: the origin of an exponential tail in a complex system
• Injected Power Fluctuations in Langevin Equation
• Time Resolved Correlation measurements of temporally heterogeneous dynamics
• Generalised extreme value statistics and sum of correlated variables
• Width distribution of contact lines on a disordered substrate
• Sub-diffusion and localization in the one dimensional trap model
• Width Distributions and the Upper Critical Dimension of KPZ Interfaces
• Extreme Value Statistics of Hierarchically Correlated Variables: Deviation from Gumbel Statistics and Anomalous Persistence
• Renormalization Group and Probability Theory
• Multiple scaling regimes in simple aging models
• Broad distribution effects in sums of lognormal random variables
• Global fluctuations and Gumbel statistics
• Roughness distributions for 1/f^alpha signals
• Out-of-equilibrium dynamical fluctuations in glassy systems
• Intermittency exponents and energy spectrum of the Burgers and KPZ equations with correlated noise
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Maximal height statistics for 1/f^alpha signals
• Extremal Paths on a Random Cayley Tree
• Statistics of extremal intensities for Gaussian interfaces
• Universal interface width distributions at the depinning threshold
• Lévy Statistics for Random Single--Molecule Line Shapes in a Glass
• Intermittency of velocity time increments in turbulence
• Extreme statistics for time series: Distribution of the maximum relative to the initial value
• Critical Behavior of Light
• Anomalous diffusion, Localization, Aging and Sub-aging effects in trap models at very low temperature
• Nonextensive Thermodynamics and Glassy behaviour in Hamiltonian systems
• Universal Fluctuations of the Danube Water Level: a Link with Turbulence, Criticality and Company Growth
• Level Density of a Bose Gas and Extreme Value Statistics
• Universal fluctuations and extreme value statistics
• Punctuated evolution for the quasispecies model
• Dynamical ultrametricity in the critical trap model
• Universal Extremal Statistics in a Freely Expanding Jepsen Gas
• Criterion for universality class independent critical fluctuations: example of the 2D Ising model
• Linear and non linear response in the aging regime of the 1D trap model
• Universal Magnetic Fluctuations with a Field Induced Length Scale
• Scaling and universality of critical fluctuations in granular gases
• Central limit theorem for anomalous scaling due to correlations
• Comment on "Universal Fluctuations in Correlated Systems"
• Distribution of injected power fluctuations in electroconvection
• Velocity Correlations, Diffusion and Stochasticity in a One-Dimensional System
• Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces
• From laser cooling to aging: a unified Levy flight description
• Energy fluctuations in vibrated and driven granular gases
• On the first Sonine correction for granular gases
• Comment on Bramwell et al, "Universal Fluctuations in Correlated Systems"
• Non-Gaussianity of resistance fluctuations near electrical breakdown
• Relevance of soft modes for order parameter fluctuations in the Two-Dimensional XY model
• Scaling functions for nonequilibrium fluctuations: A picture gallery
• Temperature dependent fluctuations in the two-dimensional XY model
• Chemical fracture and distribution of extreme values
• Scaling and Commonality in Anomalous Fluctuation Statistics in Models for Turbulence and Ferromagnetism
• Universal energy distribution for interfaces in a random field environment
• Origin of the approximate universality of distributions in equilibrium correlated systems
• An exactly solvable dissipative transport model
• Corrections to Universal Fluctuations in Correlated Systems: the 2D XY-model
• Fluctuating Fronts as Correlated Extreme Value Problems: An Example of Gaussian Statistics

Cited by in corpus (31)

• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• The Single Big Jump Principle in Physical Modelling
• Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution
• Universal fluctuations and extreme statistics of avalanches near the depinning transtition
• Generalized Gibbs ensemble and work statistics of a quenched Luttinger liquid
• Reinterpreting the development of extensive air showers initiated by nuclei and photons
• Renormalization flow in extreme value statistics
• Relaxation dynamics of the Ising $p$-spin disordered model with finite number of variables
• The Hatano-Sasa equality: transitions between steady states in a granular gas
• Fluctuation relation and heterogeneous superdiffusion in glassy transport
• Real-Space Renormalization for disordered systems at the level of Large Deviations
• The derivative of the Kardar-Parisi-Zhang equation is not in the KPZ universality class
• Anomalous scaling due to correlations: Limit theorems and self-similar processes
• Phase transition in the Jarzynski estimator of free energy differences
• Area distribution and the average shape of a Lévy bridge
• Central limit theorem for a class of globally correlated random variables
• On the parametrization of the distributions of depth of shower maximum of ultra-high energy extensive air showers
• Statistics of sums of correlated variables described by a matrix product ansatz
• Entropic aging and extreme value statistics
• On leaders and condensates in a growing network
• Renormalization flow for extreme value statistics of random variables raised to a varying power
• Exact and limit distributions of the largest fitness on correlated fitness landscapes
• Extreme value distributions for weakly correlated fitnesses in block model
• Shot noise multifractal model for turbulent pseudo-dissipation
• Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction (shortened version)
• Modeling Flocks and Prices: Jumping Particles with an Attractive Interaction
• Dissipation induced non-Gaussian energy fluctuations
• Nature of the global fluctuations in the spherical model at criticality
• Journey from the Wilson exact RG towards the Wegner-Morris Fokker-Planck RG and the Carosso field-coarsening via Langevin stochastic processes
• Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value
• Late-time universal distribution functions of observables in one-dimensional many-body quantum systems