Airy processes and variational problems

Abstract

We review the Airy processes; their formulation and how they are conjectured to govern the large time, large distance spatial fluctuations of one dimensional random growth models. We also describe formulas which express the probabilities that they lie below a given curve as Fredholm determinants of certain boundary value operators, and the several applications of these formulas to variational problems involving Airy processes that arise in physical problems, as well as to their local behaviour.

Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI Proceedings: Topics in percolative and disordered systems"

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Cited by in corpus (34)

• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• The KPZ fixed point
• Nonlinear Fluctuating Hydrodynamics in One Dimension: the Case of Two Conserved Fields
• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• On Time Correlations for KPZ Growth in One Dimension
• A Pfaffian representation for flat ASEP
• Time-time covariance for last passage percolation with generic initial profile
• Reflected Brownian motions in the KPZ universality class
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• One-sided reflected Brownian motions and the KPZ fixed point
• How flat is flat in random interface growth?
• Exact formulas for random growth with half-flat initial data
• Crossover from droplet to flat initial conditions in the KPZ equation from the replica Bethe ansatz
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Multiplicative functionals on ensembles of non-intersecting paths
• Optimal Exponent for Coalescence of Finite Geodesics in Exponential Last Passage Percolation
• Maximum of an Airy process plus Brownian motion and memory in KPZ growth
• Kardar-Parisi-Zhang Interfaces with Curved Initial Shapes and Variational Formula
• Convergence of the Environment Seen from Geodesics in Exponential Last-Passage Percolation
• Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains
• Extreme statistics of non-intersecting Brownian paths
• Midpoint distribution of directed polymers in the stationary regime: exact result through linear response
• Upper tail decay of KPZ models with Brownian initial conditions
• Stochastic growth in time dependent environments
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• KPZ fluctuations in finite volume
• Cutoff profile of the Metropolis biased card shuffling
• Fluctuations of TASEP and LPP with general initial data
• Exact solution of interacting particle systems related to random matrices
• Initial perturbation matters: implications of geometry-dependent universal Kardar-Parisi-Zhang statistics for spatiotemporal chaos
• Tracy-Widom fluctuations in 2D random Schrodinger operators
• TASEP with a general initial condition and a deterministically moving wall
• Ergodicity of the Airy line ensemble
• On the average of the Airy process and its time reversal