Extreme value statistics of correlated random variables: a pedagogical review

Abstract

Extreme value statistics (EVS) concerns the study of the statistics of the maximum or the minimum of a set of random variables. This is an important problem for any time-series and has applications in climate, finance, sports, all the way to physics of disordered systems where one is interested in the statistics of the ground state energy. While the EVS of `uncorrelated' variables are well understood, little is known for strongly correlated random variables. Only recently this subject has gained much importance both in statistical physics and in probability theory. In this review, we will first recall the classical EVS for uncorrelated variables and discuss the three universality classes of extreme value limiting distribution, known as the Gumbel, Fréchet and Weibull distribution. We then show that, for weakly correlated random variables with a finite correlation length/time, the limiting extreme value distribution can still be inferred from that of the uncorrelated variables using a renormalisation group-like argument. Finally, we consider the most interesting examples of strongly correlated variables for which there are very few exact results for the EVS. We discuss few examples of such strongly correlated systems (such as the Brownian motion and the eigenvalues of a random matrix) where some analytical progress can be made. We also discuss other observables related to extremes, such as the density of near-extreme events, time at which an extreme value occurs, order and record statistics, etc.

arXiv comment37 pages, 5 Figures. Review article to appear in Physics Reports. This is a considerably expanded version of the lecture notes in arXiv:1406.6768

References in corpus (192)

• Level-Spacing Distributions and the Airy Kernel
• Shape Fluctuations and Random Matrices
• Diffusion with Stochastic Resetting
• On Orthogonal and Symplectic Matrix Ensembles
• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Persistence and First-Passage Properties in Non-equilibrium Systems
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Matrix Models for Beta Ensembles
• The one-dimensional KPZ equation: an exact solution and its universality
• Statistical physics of inference: Thresholds and algorithms
• Diffusion with Optimal Resetting
• First Passage Under Restart
• Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals
• Universality Classes for Extreme Value Statistics
• First order transition for the optimal search time of Lévy flights with resetting
• Free-energy distribution of the directed polymer at high temperature
• Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts
• Growing interfaces uncover universal fluctuations behind scale invariance
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Run and tumble particle under resetting: a renewal approach
• A KPZ Cocktail- Shaken, not stirred: Toasting 30 years of kinetically roughened surfaces
• Beta ensembles, stochastic Airy spectrum, and a diffusion
• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Stochastic resetting in backtrack recovery by RNA polymerases
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
• Edge Electron Gas
• Optimal first arrival times in Lévy flights with resetting
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• 1/f Noise and Extreme Value Statistics
• Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function
• The statistics of critical points of Gaussian fields on large-dimensional spaces
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• Records in stochastic processes -- Theory and applications
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Universal Record Statistics of Random Walks and Lévy Flights
• From Random Matrices to Stochastic Operators
• Generalised extreme value statistics and sum of correlated variables
• Topological and Dynamical Complexity of Random Neural Networks
• Statistical distribution of quantum entanglement for a random bipartite state
• Random Convex Hulls and Extreme Value Statistics
• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Critical points and supersymmetric vacua
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• A branching random walk seen from the tip
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• On the top eigenvalue of heavy-tailed random matrices
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Maximal Height Scaling of Kinetically Growing Surfaces
• Maximum of the characteristic polynomial of random unitary matrices
• Exact distribution of the maximal height of p vicious walkers
• The Single Big Jump Principle in Physical Modelling
• Statistical Mechanics of Logarithmic REM: Duality, Freezing and Extreme Value Statistics of $1/f$ Noises generated by Gaussian Free Fields
• Extreme Value Statistics of Hierarchically Correlated Variables: Deviation from Gumbel Statistics and Anomalous Persistence
• Convex Hull of N Planar Brownian Motions: Exact Results and an Application to Ecology
• Statistics at the tip of a branching random walk and the delay of traveling waves
• First passage under restart with branching
• Localization transition induced by learning in random searches
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Universal gap fluctuations in the superconductor proximity effect
• Density of States below the Thouless Gap in a Mesoscopic SNS Junction
• Phase transitions and edge scaling of number variance in Gaussian random matrices
• Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time
• Cosmology From Random Multifield Potentials
• Universal scaling of the order-parameter distribution in strongly disordered superconductors
• Exact solutions for the statistics of extrema of some random 1D landscapes, Application to the equilibrium and the dynamics of the toy model
• Spatial extent of an outbreak in animal epidemics
• Noninteracting fermions in a trap and random matrix theory
• Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution
• On the time to reach maximum for a variety of constrained Brownian motions
• Non-intersecting Brownian Interfaces and Wishart Random Matrices
• Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
• Extreme Value Statistics and Traveling Fronts: An Application to Computer Science
• Probability distributions of Linear Statistics in Chaotic Cavities and associated phase transitions
• On the area under a continuous time Brownian motion till its first-passage time
• Extreme Value Statistics and Traveling Fronts: Various Applications
• Generalized arcsine laws for fractional Brownian motion
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Density of near-extreme events
• Hitting probability for anomalous diffusion processes
• Maximal height statistics for 1/f^alpha signals
• Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem
• Exact Asymptotic Results for a Model of Sequence Alignment
• Large deviations of the maximal eigenvalue of random matrices
• Time to reach the maximum for a random acceleration process
• Finite-size scaling in extreme statistics
• An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution
• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Distribution of the time at which N vicious walkers reach their maximal height
• Extremal Paths on a Random Cayley Tree
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• Universal Order Statistics of Random Walks
• Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian
• Statistics of extremal intensities for Gaussian interfaces
• Extreme-Value Statistics of Fractional Brownian Motion Bridges
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• Statistical mechanics of a single particle in a multiscale random potential: Parisi landscapes in finite dimensional Euclidean spaces
• Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries
• A note on the second order universality at the edge of Coulomb gases on the plane
• Extreme statistics for time series: Distribution of the maximum relative to the initial value
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Extremal properties of random trees
• The longest excursion of stochastic processes in nonequilibrium systems
• Renormalization group theory for finite-size scaling in extreme statistics
• On the joint distribution of the maximum and its position of the Airy2 process minus a parabola
• Mixed order transition and condensation in exactly soluble one dimensional spin model
• Multifractality and Freezing Phenomena in Random Energy Landscapes: an Introduction
• A record-driven growth process
• Endpoint distribution of directed polymers in 1+1 dimensions
• Nonintersecting Brownian motions on the half-line and discrete Gaussian orthogonal polynomials
• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• Extremes of $2d$ Coulomb gas: universal intermediate deviation regime
• Near-extreme statistics of Brownian motion
• How Xenopus laevis replicates DNA reliably even though its origins of replication are located and initiated stochastically
• Adaptive walks and extreme value theory
• Universal Order and Gap Statistics of Critical Branching Brownian Motion
• Maximum distributions of bridges of noncolliding Brownian paths
• Perturbative Expansion for the Maximum of Fractional Brownian Motion
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
• Traveling Front Solutions to Directed Diffusion Limited Aggregation, Digital Search Trees and the Lempel-Ziv Data Compression Algorithm
• Universal Extremal Statistics in a Freely Expanding Jepsen Gas
• Extremal Properties of Random Systems
• Temporal correlations of the running maximum of a Brownian trajectory
• Spatial Extent of Branching Brownian Motion
• Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line
• Large time zero temperature dynamics of the spherical p=2-spin glass model of finite size
• Universal statistics of longest lasting records of random walks and Lévy flights
• Exact extreme value statistics at mixed order transitions
• Large deviations of the maximum of independent and identically distributed random variables
• The right tail exponent of the Tracy-Widom-beta distribution
• Right tail expansion of Tracy-Widom beta laws
• Intermediate deviation regime for the full eigenvalue statistics in the complex Ginibre ensemble
• The convex hull for a random acceleration process in two dimensions
• Branching Brownian Motion Conditioned on Particle Numbers
• Asymptotics for the Expected Maximum of Random Walks and Lévy Flights with a Constant Drift
• Free fermions at the edge of interacting systems
• Large deviations of radial statistics in the two-dimensional one-component plasma
• Subgap states in dirty superconductors and their effect on dephasing in Josephson qubits
• On the Gap and Time Interval between the First Two Maxima of Long Random Walks
• Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks
• Painlevé representation of Tracy-Widom${}_{\beta }$ distribution for $\beta =6$
• Phase transitions in the distribution of the Andreev conductance of superconductor-metal junctions with multiple transverse modes
• Spectral density asymptotics for Gaussian and Laguerre $\beta$-ensembles in the exponentially small region
• Statistics of the longest interval in renewal processes
• Maximum of N Independent Brownian Walkers till the First Exit From the Half Space
• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
• Near-extreme eigenvalues and the first gap of Hermitian random matrices
• Mean perimeter and mean area of the convex hull over planar random walks
• Universality of the weak pushed-to-pulled transition in systems with repulsive interactions
• Fluctuation-dominated phase ordering at a mixed order transition
• Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal $1/f$ noise
• Ground state energy of noninteracting fermions with a random energy spectrum
• On certain functionals of the maximum of Brownian motion and their applications
• Pickands' constant at first order in an expansion around Brownian motion
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
• Large deviations for the branching Brownian motion in presence of selection or coalescence
• Maximum relative height of elastic interfaces in random media
• Statistics of fermions in a $d$-dimensional box near a hard wall
• On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces
• Statistics of the first passage time of Brownian motion conditioned by maximum value or area
• Slower deviations of the branching Brownian motion and of branching random walks
• Gap statistics close to the quantile of a random walk
• Non-interacting fermions in hard-edge potentials
• Probability distribution of the maximum of a smooth temporal signal
• Large-displacement statistics of the rightmost particle of the one-dimensional branching Brownian motion
• Convex hull of a Brownian motion in confinement
• Joint distributions of partial and global maxima of a Brownian Bridge
• Number of Common Sites Visited by N Random Walkers
• Understanding Search Trees via Statistical Physics
• The Statistics of the Number of Minima in a Random Energy Landscape
• The density of eigenvalues seen from the soft edge of random matrices in the Gaussian beta-ensembles
• Exact distributions of cover times for $N$ independent random walkers in one dimension
• Order statistics of 1/f^α signals
• Extreme statistics of non-intersecting Brownian paths
• Extreme value distributions and Renormalization Group
• $\delta$-exceedance records and random adaptive walks
• The statistical mechanics of combinatorial optimization problems with site disorder
• Maximal distance travelled by N vicious walkers till their survival
• Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice
• Renormalization flow for extreme value statistics of random variables raised to a varying power
• Phase Transition in a Random Minima Model: Mean Field Theory and Exact Solution on the Bethe Lattice
• Distribution of Maximal Luminosity of Galaxies in the Sloan Digital Sky Survey
• First Gap Statistics of Long Random Walks with Bounded Jumps
• On the Gap and Time Interval between the First Two Maxima of Long Continuous Time Random Walks

Cited by in corpus (109)

• Stochastic Resetting and Applications
• Extreme events in dynamical systems and random walkers: A review
• The inspection paradox in stochastic resetting
• Stochastic resetting with stochastic returns using external trap
• First passage under restart for discrete space and time: application to one dimensional confined lattice random walks
• Geometric Brownian Motion under Stochastic Resetting: A Stationary yet Non-ergodic Process
• Chain breaking and Kosterlitz-Thouless scaling at the many-body localization transition in the random field Heisenberg spin chain
• Extreme statistics and spacing distribution in a Brownian gas correlated by resetting
• Extremal statistics for stochastic resetting systems
• Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Random acceleration process under stochastic resetting
• Record statistics for random walks and Lévy flights with resetting
• Resetting with stochastic return through linear confining potential
• Distribution of the time of the maximum for stationary processes
• Distribution of the Time Between Maximum and Minimum of Random Walks
• First-passage Brownian functionals with stochastic resetting
• Number of distinct sites visited by a resetting random walker
• Survival probability of a run-and-tumble particle in the presence of a drift
• Extreme value statistics and arcsine laws for heterogeneous diffusion processes
• Martingales for physicists: A treatise on stochastic thermodynamics and beyond
• Mitigating long transient time in deterministic systems by resetting
• Time to reach the maximum for a stationary stochastic process
• Universal framework for record ages under restart
• Exact extreme, order and sum statistics in a class of strongly correlated system
• Erratic non-Hermitian skin localization
• Discrete Sampling of Extreme Events Modifies Their Statistics
• Edge fluctuations and third-order phase transition in harmonically confined long-range systems
• Arrhenius law for interacting diffusive systems
• Depletion of Resources by a Population of Diffusing Species
• Dynamical phase transition in the first-passage probability of a Brownian motion
• Search efficiency in the Adam-Delbrück reduction-of-dimensionality scenario versus direct diffusive search
• Dynamically emergent correlations between particles in a switching harmonic trap
• Generating discrete-time constrained random walks and Lévy flights
• Geometrical optics of large deviations of fractional Brownian motion
• First-passage times of multiple diffusing particles with reversible target-binding kinetics
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• Extreme value statistics of edge currents in Markov jump processes and their use for entropy production estimation
• Extreme value theory for constrained physical systems
• Harmonically confined long-ranged interacting gas in the presence of a hard wall
• Reversible Target-Binding Kinetics of Multiple Impatient Particles
• Analytical Survival Analysis of the Ornstein-Uhlenbeck Process
• Nonequilibrium steady state of trapped active particles
• Statistical properties of avalanches via the c-record process
• Dynamically emergent correlations in bosons via quantum resetting
• Fluctuations and first-passage properties of systems of Brownian particles with reset
• First-passage functionals of Brownian motion in logarithmic potentials and heterogeneous diffusion
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier
• Last-passage time for linear diffusions and application to the emptying time of a box
• Noninteracting particles in a harmonic trap with a stochastically driven center
• Extremal statistics for a resetting Brownian motion before its first-passage time
• Rogue waves in discrete-time quantum walks
• Accurately approximating extreme value statistics
• A sluggish random walk with subdiffusive spread
• Breaking the chains: extreme value statistics and localization in random spin chains
• Record Ages of Scale Invariant non-Markovian Random Walks
• Rogue waves in quantum lattices with correlated disorder
• Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble
• Optimal conditions for first passage of jump processes with resetting
• Target Search Optimization by Threshold Resetting
• Extreme value statistics of positive recurrent centrally biased random walks
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
• Extreme statistics of the excitations in the random transverse Ising chain
• A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles
• Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
• Ground-state energy distribution of disordered many-body quantum systems
• Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols
• Rare Events in Extreme Value Statistics of Jump Processes with Power Tails
• From Maximum of Intervisit Times to Starving Random Walks
• Nonlinear-Cost Random Walk: exact statistics of the distance covered for fixed budget
• Exact finite-size scaling for the random-matrix representation of bond percolation on square lattice
• Dynamically emergent correlations in a Brownian gas with diffusing diffusivity
• Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous and discrete time stochastic processes
• Drift-diffusive resetting search process with stochastic returns: speed-up beyond optimal instantaneous return
• Coarsening, condensates and extremes in aggregation-fragmentation models
• Continuous Gated First-Passage Processes
• Wavefunction extreme value statistics in Anderson localization
• Maxima of log-correlated fields: some recent developments
• Decorrelation of a leader by the increasing number of followers
• Competition in a system of Brownian particles: Encouraging achievers
• Asymptotics for the fastest among n stochastics particles: role of an extended initial distribution and an additional drift component
• Fluctuation dominated phase ordering in coarse-grained depth models: Domain wall structures, extreme values and coarsening
• Extreme matrices or how an exponential map links classical and free extreme laws
• Arrival time for the fastest among $N$ switching stochastic particles
• Field theory of survival probabilities, extreme values, first passage times, and mean span of non-Markovian stochastic processes
• Autocorrelation functions and ergodicity in diffusion with stochastic resetting
• Exact joint distributions of three global characteristic times for Brownian motion
• Interaction-correlated random matrices
• Eigenvalue spectral tails and localization properties of asymmetric networks
• First-passage and extreme value statistics for overdamped Brownian motion in a linear potential
• On the weak scaling of the contact distance between two fluctuating interfaces with system size
• A 2D Levy-flight model for the complex dynamics of real-life financial markets
• Extreme-Value Statistics of Stochastic Transport Processes: Applications to Molecular Motors and Sports
• Estimation of heavy tails in optical non-linear processes
• Counting of level crossings for inertial random processes: Generalization of the Rice formula
• Eigenstate plateau transition and equilibration in 1D quantum lattice models
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles
• The distribution of the maximum of independent resetting Brownian motions
• A causal interactions indicator between two time series using extreme variations in the first eigenvalue of lagged correlation matrices
• Probabilities of moderately atypical fluctuations of the size of a swarm of Brownian Bees
• Finite-size relaxational dynamics of a spike random matrix spherical model
• Universality of Top Rank Statistics for Brownian Reshuffling
• How large should be the redundant numbers of copy to make a rare event probable
• A matrix model of a non-Hermitian $\beta$-ensemble
• Exact finite-size scaling of the maximum likelihood spectra in the quenched and annealed Sherrington-Kirkpatrick spin glass
• First-order condensation transition in the position distribution of a run-and-tumble particle in one dimension
• Big jump principle for heavy-tailed random walks with correlated increments
• Extreme value statistics in a continuous time branching process: a pedagogical primer