Exact short-time height distribution in 1D KPZ equation and edge fermions at high temperature

Abstract

We consider the early time regime of the Kardar-Parisi-Zhang (KPZ) equation in $1+1$ dimensions in curved (or droplet) geometry. We show that for short time $t$, the probability distribution $P(H,t)$ of the height $H$ at a given point $x$ takes the scaling form $P(H,t)\sim \exp \left(-{\Phi }_{\mathrm{drop}}(H)/\sqrt{t}\right)$ where the rate function ${\Phi }_{\mathrm{drop}}(H)$ is computed exactly. While it is Gaussian in the center, i.e., for small $H$, the PDF has highly asymmetric non-Gaussian tails which we characterize in detail. This function ${\Phi }_{\mathrm{drop}}(H)$ is surprisingly reminiscent of the large deviation function describing the stationary fluctuations of finite size models belonging to the KPZ universality class. Thanks to a recently discovered connection between KPZ and free fermions, our results have interesting implications for the fluctuations of the rightmost fermion in a harmonic trap at high temperature and the full couting statistics at the edge.

5 pages + 7 pages of supplemental material, 3 figures, typos corrected

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Cited by in corpus (53)

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• The inverse scattering of the Zakharov-Shabat system solves the weak noise theory of the Kardar-Parisi-Zhang equation
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• Multicritical edge statistics for the momenta of fermions in non-harmonic traps
• Systematic time expansion for the Kardar-Parisi-Zhang equation, linear statistics of the GUE at the edge and trapped fermions
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• Inverse scattering solution of the weak noise theory of the Kardar-Parisi-Zhang equation with flat and Brownian initial conditions
• Landau theory of the short-time dynamical phase transitions of the Kardar-Parisi-Zhang interface
• The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion
• Height distribution tails in the Kardar-Parisi-Zhang equation with Brownian initial conditions
• Simple derivation of the $(-\lambda H{)}^{5/2}$ tail for the 1D KPZ equation
• Initial pseudo-steady state & asymptotic KPZ universality in semiconductor on polymer deposition
• Exact short-time height distribution for the flat Kardar-Parisi-Zhang interface
• Large fluctuations of the KPZ equation in a half-space
• Optimal paths of non-equilibrium stochastic fields: the Kardar-Parisi-Zhang interface as a test case
• Finite-size effects in the short-time height distribution of the Kardar-Parisi-Zhang equation
• Short time large deviations of the KPZ equation
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line
• The Kardar-Parisi-Zhang exponents for the $2+1$ dimensions
• The fractal geometry of growth: fluctuation-dissipation theorem and hidden symmetry
• Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
• Replica Bethe Ansatz solution to the Kardar-Parisi-Zhang equation on the half-line
• Probing the large deviations of the Kardar-Parisi-Zhang equation at short time with an importance sampling of directed polymers in random media
• A Riemann-Hilbert approach to the lower tail of the KPZ equation
• Large fluctuations of a Kardar-Parisi-Zhang interface on a half-line: the height statistics at a shifted point
• Riemann surfaces for KPZ with periodic boundaries
• Time-averaged height distribution of the Kardar-Parisi-Zhang interface
• 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
• KPZ equation with a small noise, deep upper tail and limit shape
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• The Fluctuation-Dissipation Relations: Growth, Diffusion, and Beyond
• Integrability in the weak noise theory
• Universal scaling relation for growth phenomena
• Density profile of noninteracting fermions in a rotating $2d$ trap at finite temperature
• Restoring the fluctuation-dissipation theorem in Kardar-Parisi-Zhang universality class through a new emergent fractal dimension
• KPZ fluctuations in finite volume
• Short-time large deviations of the spatially averaged height of a KPZ interface on a ring
• Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equations
• Width scaling of an interface constrained by a membrane
• Bessel kernel determinants and integrable equations
• Exact short-time height distribution and dynamical phase transition in the relaxation of a Kardar-Parisi-Zhang interface with random initial condition
• High moments of the SHE in the clustering regimes
• Upper-tail large deviation principle for the ASEP
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"