The One-dimensional KPZ Equation and the Airy Process

Abstract

Our previous work on the one-dimensional KPZ equation with sharp wedge initial data is extended to the case of the joint height statistics at n spatial points for some common fixed time. Assuming a particular factorization, we compute an n-point generating function and write it in terms of a Fredholm determinant. For long times the generating function converges to a limit, which is established to be equivalent to the standard expression of the n-point distribution of the Airy process.

15 pages

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• The Physicist's Companion to Current Fluctuations: One-Dimensional Bulk-Driven Lattice Gases
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• Renormalization fixed point of the KPZ universality class
• Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
• Airy processes and variational problems
• Local behavior and hitting probabilities of the Airy1 process
• Continuum statistics of the Airy2 process
• Spectral theory for the q-Boson particle system
• Finite time corrections in KPZ growth models
• Endpoint distribution of directed polymers in 1+1 dimensions
• A multi-layer extension of the stochastic heat equation
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• On the equal time two-point distribution of the one-dimensional KPZ equation by replica
• How flat is flat in random interface growth?
• Exact formulas for random growth with half-flat initial data
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• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
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• Height fluctuations for the stationary KPZ equation
• Non-intersecting Brownian bridges and the Laguerre Orthogonal Ensemble
• Multiplicative functionals on ensembles of non-intersecting paths
• Tails of the endpoint distribution of directed polymers
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• Supremum of the Airy2 process minus a parabola on a half line
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• Field theory of disordered systems -- Avalanches of an elastic interface in a random medium
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• Crossing probability for directed polymers in random media: exact tail of the distribution
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• Analytical Methods and Field Theory for Disordered Systems
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