The KPZ fixed point

Abstract

An explicit Fredholm determinant formula is derived for the multipoint distribution of the height function of the totally asymmetric simple exclusion process (TASEP) with arbitrary right-finite initial condition. The method is by solving the biorthogonal ensemble/non-intersecting path representation found by [Sas05; BFPS07]. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data. In the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula, in terms of analogous kernels based on Brownian motion, for the transition probabilities of the scaling invariant Markov process at the centre of the KPZ universality class. The formula readily reproduces known special self-similar solutions such as the Airy${}_1$ and Airy${}_2$ processes. The process takes values in real valued functions which look locally like Brownian motion, and is Hölder $1/3-$ in time. Both the KPZ fixed point and TASEP are shown to be stochastic integrable systems in the sense that the time evolution of their transition probabilities can be linearized through a new Brownian scattering transform and its discrete analogue.

Final version. Minor edits and corrections, some additional detail. 51 pages

References in corpus (9)

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• Fluctuations of the one-dimensional polynuclear growth model with external sources
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Cited by in corpus (51)

• An appetizer to modern developments on the Kardar-Parisi-Zhang universality class
• The directed landscape
• The KPZ Limit of ASEP with Boundary
• Exponents governing the rarity of disjoint polymers in Brownian last passage percolation
• Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems
• From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics
• Time-time covariance for last passage percolation with generic initial profile
• Generalizations of TASEP in discrete and continuous inhomogeneous space
• Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition
• Three-halves variation of geodesics in the directed landscape
• One-sided reflected Brownian motions and the KPZ fixed point
• Markov limits of steady states of the KPZ equation on an interval
• The heat and the landscape I
• Riemann surfaces for KPZ with periodic boundaries
• Multi-point distribution of periodic TASEP
• From the asymmetric simple exclusion processes to the stationary measures of the KPZ fixed point on an interval
• TASEP and generalizations: Method for exact solution
• The stationary horizon and semi-infinite geodesics in the directed landscape
• Lyapunov exponents of the SHE for general initial data
• Convergence of the Environment Seen from Geodesics in Exponential Last-Passage Percolation
• Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison
• Fluctuation Exponents of the KPZ equation on a large torus
• The stationary AKPZ equation: logarithmic superdiffusivity
• Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique
• Finite GUE distribution with cut-off at a shock
• Upper tail decay of KPZ models with Brownian initial conditions
• Differential equations for the KPZ and periodic KPZ fixed points
• Universality of deterministic KPZ
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• Disjoint optimizers and the directed landscape
• Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point
• Cutoff profile of the Metropolis biased card shuffling
• Periodic TASEP with general initial conditions
• Random polymers via orthogonal Whittaker and symplectic Schur functions
• Fluctuations of the Arctic curve in the tilings of the Aztec diamond on restricted domains
• Large-scale limit of interface fluctuation models
• Local Behavior of Airy Processes
• Fractal Geometry of the Valleys of the Parabolic Anderson Equation
• Exact solution of interacting particle systems related to random matrices
• KPZ on torus: Gaussian fluctuations
• Comments on David Aldous and Persi Diaconis' "Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem"
• Short- and long-time path tightness of the continuum directed random polymer
• Mapping TASEP back in time
• Nonstationary generalized TASEP in KPZ and jamming regimes
• Temporal increments of the KPZ equation with general initial data
• Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation
• Asymptotic fluctuations of geometric q-TASEP, geometric q-PushTASEP and q-PushASEP
• Point Fields of Last Passage Percolation and Coalescing Fractional Brownian Motions
• Waiting Times for Ties in Random Competitions
• When the geodesic becomes rigid in the directed landscape
• Multi-point distribution of discrete time periodic TASEP