Maxima of log-correlated fields: some recent developments

Abstract

We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other $L$-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov \& Keating concerning the extreme value statistics, moments of moments, connections to Gaussian Multiplicative Chaos, and explicit formulae derived from the theory of symmetric functions.

arXiv comment69 pages, to appear in J. Phys. A: Math. Theor

References in corpus (60)

• Shape Fluctuations and Random Matrices
• Matrix Models for Beta Ensembles
• Glass transition of a particle in a random potential, front selection in nonlinear RG and entropic phenomena in Liouville and SinhGordon models
• Extreme value statistics of correlated random variables: a pedagogical review
• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
• Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function
• Critical Gaussian multiplicative chaos: Convergence of the derivative martingale
• Freezing Transitions and Extreme Values: Random Matrix Theory, $\zeta (1/2+it)$, and Disordered Landscapes
• Maximum of the characteristic polynomial of random unitary matrices
• Extreme events on complex networks
• Extreme local extrema of two-dimensional discrete Gaussian free field
• On the averages of characteristic polynomials from classical groups
• Convergence of the centered maximum of log-correlated Gaussian fields
• Secular determinants of random unitary matrices
• On the maximum of the C$\beta$E field
• Autocorrelation of Random Matrix Polynomials
• Toeplitz determinants with merging singularities
• Extreme value problems in Random Matrix Theory and other disordered systems
• Thick points of the Gaussian free field
• Continuous lower bounds for moments of zeta and L-functions
• Sharp conditional bounds for moments of the Riemann zeta function
• Finite N Fluctuation Formulas for Random Matrices
• Large GCD sums and extreme values of the Riemann zeta function
• Extreme statistics of complex random and quantum chaotic states
• Extreme values of the Riemann zeta function and its argument
• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• More than 41% of the zeros of the zeta function are on the critical line
• Howe pairs, supersymmetry, and ratios of random characteristic polynomials for the unitary groups U(N)
• Moments of zeta and correlations of divisor-sums: I
• The Fyodorov-Bouchaud formula and Liouville conformal field theory
• On the distribution of maximum value of the characteristic polynomial of GUE random matrices
• Entanglement between two subsystems, the Wigner semicircle and extreme value statistics
• Optimal local law and central limit theorem for $\beta$-ensembles
• Critical Liouville measure as a limit of subcritical measures
• Sums of divisor functions in $F_q[t]$ and matrix integrals
• Subcritical multiplicative chaos for regularized counting statistics from random matrix theory
• Critical Gaussian chaos: convergence and uniqueness in the derivative normalisation
• On the partition function of the Riemann zeta function, and the Fyodorov--Hiary--Keating conjecture
• Moments of zeta and correlations of divisor-sums: V
• On the origins of Riemann-Hilbert problems in mathematics
• Moments of the Riemann zeta function on short intervals of the critical line
• The Fyodorov-Hiary-Keating Conjecture. I
• Maximum of the characteristic polynomial of the Ginibre ensemble
• Planar Brownian motion and Gaussian multiplicative chaos
• A note on the maximum of the Riemann zeta function, and log-correlated random variables
• On the moments of the moments of the characteristic polynomials of random unitary matrices
• Moments of characteristic polynomials for compact symmetric spaces and Jack polynomials
• Random sorting networks: local statistics via random matrix laws
• Statistics of extremes in eigenvalue-counting staircases
• On the moments of the moments of $\zeta (1/2+it)$
• Extreme values of CUE characteristic polynomials: a numerical study
• The Riemann zeta function in short intervals [after Najnudel, and Arguin, Belius, Bourgade, Radziwiłł, and Soundararajan]
• On the moments of the moments of the characteristic polynomials of Haar distributed symplectic and orthogonal matrices
• Uniform point variance bounds in classical beta ensembles
• On the moments of characteristic polynomials
• Asymptotics for averages over classical orthogonal ensembles
• A new approach to the characteristic polynomial of a random unitary matrix
• On the critical-subcritical moments of moments of random characteristic polynomials: a GMC perspective
• Moments of Moments and Branching Random Walks
• Symmetric Function Theory and Unitary Invariant Ensembles

Cited by in corpus (2)

• A model problem for multiplicative chaos in number theory
• A conjectural asymptotic formula for multiplicative chaos in number theory