Random matrices and determinantal processes

Abstract

We survey recent results on determinantal processes, random growth, random tilings and their relation to random matrix theory.

40 pages. Lectures given at the summer school on Mathematical statistical mechanics in July 05 at Ecole de Physique, Les Houches

References in corpus (10)

• Random Matrix Theories in Quantum Physics: Common Concepts
• Level-Spacing Distributions and the Airy Kernel
• Universal Distributions for Growth Processes in 1+1 Dimensions and Random Matrices
• Discrete polynuclear growth and determinantal processes
• A variational principle for domino tilings
• Local statistics of lattice dimers
• Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges
• Distribution functions for largest eigenvalues and their applications
• Correlation functions for symmetrized increasing subsequences
• Dynamical Correlations among Vicious Random Walkers

Cited by in corpus (113)

• Determinantal point processes for machine learning
• Fluctuation properties of the TASEP with periodic initial configuration
• The one-dimensional KPZ equation and its universality class
• Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques
• Finite temperature free fermions and the Kardar-Parisi-Zhang equation at finite time
• A note on biorthogonal ensembles
• Airy processes and variational problems
• The ASEP and determinantal point processes
• From interacting particle systems to random matrices
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Nonintersecting random walks in the neighborhood of a symmetric tacnode
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
• Point Processes with Gaussian Boson Sampling
• Asymptotic domino statistics in the Aztec diamond
• Superdiffusivity of the 1D lattice Kardar-Parisi-Zhang equation
• Edge scaling limits for a family of non-Hermitian random matrix ensembles
• Fermi distribution of semicalssical non-eqilibrium Fermi states
• A multi-layer extension of the stochastic heat equation
• The periodic Schur process and free fermions at finite temperature
• The free boundary Schur process and applications I
• Non-colliding Brownian bridges and the asymmetric tacnode process
• Large Complex Correlated Wishart Matrices: The Pearcey Kernel and Expansion at the Hard Edge
• Determinantal point processes and fermions on complex manifolds: Bulk universality
• The two-parameter Poisson--Dirichlet point process
• Simple derivation of the $(-\lambda H{)}^{5/2}$ tail for the 1D KPZ equation
• Local universality in biorthogonal Laguerre ensembles
• Reflected Brownian motions in the KPZ universality class
• The universal Airy_1 and Airy_2 processes in the Totally Asymmetric Simple Exclusion Process
• CLT for biorthogonal ensembles and related combinatorial identities
• A multi-dimensional Markov chain and the Meixner ensemble
• Large gap asymptotics for Airy kernel determinants with discontinuities
• Determinantal Probability: Basic Properties and Conjectures
• Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
• Statistics of the maximal distance and momentum in a trapped Fermi gas at low temperature
• Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Stationary half-space last passage percolation
• Polynomial Ensembles and Recurrence Coefficients
• TASEP and generalizations: Method for exact solution
• Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble
• On two ways to use determinantal point processes for Monte Carlo integration
• A determinantal point process for column subset selection
• Growth of balls of holomorphic sections and energy at equilibrium
• Central Limit Theorems for Biorthogonal Ensembles and Asymptotics of Recurrence Coefficients
• A Survey on the Eigenvalues Local Behavior of Large Complex Correlated Wishart Matrices
• Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights
• Random partitions and instanton counting
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices
• Monotonous subsequences and the descent process of invariant random permutations
• Brownian Gibbs property for Airy line ensembles
• Brownian bridges for late time asymptotics of KPZ fluctuations in finite volume
• The hard-edge tacnode process for Brownian motion
• Noninteracting trapped Fermions in double-well potentials: inverted parabola kernel
• Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach
• Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices
• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• Series Jackson networks and non-crossing probabilities
• Why random matrices share universal processes with interacting particle systems?
• Determinantal probability measures on Grassmannians
• The Dyson Brownian minor process
• The Gaussian free field in an interlacing particle system with two jump rates
• Stationary cocycles for the corner growth model
• KPZ fluctuations in finite volume
• Matrix models and stochastic growth in Donaldson-Thomas theory
• Hole probability for noninteracting fermions in a $d$-dimensional trap
• A new approach to the characteristic polynomial of a random unitary matrix
• Universality properties of Gelfand-Tsetlin patterns
• On the deformed Pearcey determinant
• Decorated Young Tableaux and the Poissonized Robinson-Schensted Process
• Large time asymptotics of growth models on space-like paths I: PushASEP
• Large deviation bounds for the Airy point process
• Empirical Distributions of Eigenvalues of Product Ensembles
• The variance of the number of sums of two squares in ${\mathbb{F}}_q[T]$ in short intervals
• Ewens measures on compact groups and hypergeometric kernels
• Hard-edge asymptotics of the Jacobi growth process
• Correlation functions for determinantal processes defined by infinite block Toeplitz minors
• Limit shape and fluctuations for exactly solvable inhomogeneous corner growth models
• Edge Universality for Nonintersecting Brownian Bridges
• Multiplication operator and exceptional Jacobi polynomials
• Dimers and orthogonal polynomials: connections with random matrices
• Nonintersecting paths with a staircase initial condition
• On the mean projection theorem for determinantal point processes
• A quadratic identity in the shuffle algebra and an alternative proof for de Bruijn's formula
• Stationary cocycles and Busemann functions for the corner growth model
• Equilibrium measures in the presence of weak rational external fields
• Markov Properties of Discrete Determinantal Point Processes
• The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles
• Sharp asymptotics for Toeplitz determinants, fluctuations and the gaussian free field on a Riemann surface
• From Gumbel to Tracy-Widom
• Sums of squares from elliptic pfaffians
• The Laguerre Unitary Process
• Signal reconstruction using determinantal sampling
• Stochastic and Quantum Dynamics of Repulsive Particles: from Random Matrix Theory to Trapped Fermions
• Phase transitions and equilibrium measures in random matrix models
• Generalized Laguerre Unitary Ensembles and an interacting particles model with a wall
• Eigenvalues of GUE Minors
• Introduction to determinantal point processes from a quantum probability viewpoint
• Kernel quadrature with DPPs
• The umpteen operator and its Lifshitz tails
• On sampling determinantal and Pfaffian point processes on a quantum computer
• The half-space Airy stat process
• Asymptotics for Recurrence Coefficients of X1-Jacobi Polynomials and Christoffel Function
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"
• Painlevé kernels in Hermitian matrix models
• Limits of determinantal processes near a tacnode
• A graph-based equilibrium problem for the limiting distribution of non-intersecting Brownian motions at low temperature
• On some special directed last-passage percolation models
• Interlaced processes on the circle
• Coupling Functions for Domino tilings of Aztec diamonds
• Opposite sign correlations in fermion or boson gases
• Universal edge fluctuations of discrete interlaced particle systems