A Riemann-Hilbert approach to the lower tail of the KPZ equation

Abstract

Fredholm determinants associated to deformations of the Airy kernel are closely connected to the solution to the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, and they also appear as largest particle distribution in models of positive-temperature free fermions. We show that logarithmic derivatives of the Fredholm determinants can be expressed in terms of a 2x2 Riemann-Hilbert problem, and we use this to derive asymptotics for the Fredholm determinants. As an application of our result, we derive precise lower tail asymptotics for the solution of the KPZ equation with narrow wedge initial data, refining recent results by Corwin and Ghosal.

37 pages, 1 figure

References in corpus (4)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Asymptotics in ASEP with Step Initial Condition
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
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Cited by in corpus (11)

• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
• Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
• Short time large deviations of the KPZ equation
• On the origins of Riemann-Hilbert problems in mathematics
• Lyapunov exponents of the SHE for general initial data
• Uniform tail asymptotics for Airy kernel determinant solutions to KdV and for the narrow wedge solution to KPZ
• Lyapunov exponents of the half-line SHE
• The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem
• KPZ fluctuations in finite volume
• Fractal Geometry of the Valleys of the Parabolic Anderson Equation
• Upper-tail large deviation principle for the ASEP