Exact height distributions for the KPZ equation with narrow wedge initial condition

By Tomohiro Sasamoto, Herbert Spohn. 2010

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We consider the KPZ equation in one space dimension with narrow wedge initial condition, h(x, t = 0) = −|x|/δ, δ � 1, evolving into a parabolic profile with superimposed fluctuations. Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution h(x, t) valid for any x and t> 0. The corresponding distribution function converges in the long time limit, t →∞ , to the Tracy–Widom distribution. The first order correction is a shift of order t −1/3 . We provide numerical computations based on the exact formula.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/43a3ab2fc960e3cadc7bf8373c9cabcb9dc316f6

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Citations in the corpus

Height Fluctuations for the Stationary KPZ Equation (2014, background): “The physically relevant solution to the KPZ equation [17] is defined indirectly via the well-posed stochastic heat equation (SHE) with multiplicative noise [2, 5, 16], ∂tZ(t, x) = 1 2∂ 2 xZ(t, x) + Z(t, x)ξ(t, x), with initial condition Z(0, x) = Z0(x) = e .”
Stochastic resetting and applications (2019, background): “Here, it is known that for times Tmicro t T ∗, and for a flat initial profile, the interface height H(x, t) has a deterministic linear growth with stochastic t fluctuations [129–137]: H(x, t) = v∞t+ (Γt) χ(x) .”
An appetizer to modern developments on the Kardar–Parisi–Zhang universality class (2017, methodology): “By following Prähofer and Spohn’s celebrated paper [50], it will be illustrated how the model is related to a number of different problems, how the exact results were derived, and what the outcome is from the physics viewpoint.”
Evidence for Geometry-Dependent Universal Fluctuations of the Kardar-Parisi-Zhang Interfaces in Liquid-Crystal Turbulence (2012, background): “4, although these do not contradict any analytical results for the solvable models [4, 26, 84, 85], only some of the properties are confirmed to be shared.”
Energy solutions of KPZ are unique (2015, background): “lusion process (there the drift is −t/24 because Bertini and Giacomin consider diﬀerent parameters for the equation). The same drift also appears in [ACQ11, Theorem 1.1], in the key formula (4.17) of [SS10], and in [FQ15, Theorem 1.1]. In [GJ14, Theorem 3] it is claimed that the Cole-Hopf solution i”
The Kardar-Parisi-Zhang equation and universality class (2011, methodology): “The formula was independently and in parallel discovered in mathematics by Amir-Corwin-Quastel [5] and mathematical physics by Sasamoto-Spohn [147, 148, 149] in the fall of 2009 and posted early in 2010.”
Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions (2010, background): “Spohn [25, 26, 27] independently obtained a formula eq uivalent to (1.”
Introduction to KPZ (2011, methodology): “The second key point is that [108] deform contours during the asymptotic analysis in order to end up with a nice formula (the limiting contours should be the nice ones for the Airy functions).”
A pedestrian's view on interacting particle systems, KPZ universality and random matrices (2008, methodology): “Both approaches have recently been used to prove the correct order of fluctuations in the KPZ equation [111, 13] as well as refined universality in the sense of the Tracy-Widom distribution [113, 114, 3].”
Emergent statistical mechanics of entanglement in random unitary circuits (2018, unknown): “[70] Tomohiro Sasamoto and Herbert Spohn.”
The One-Dimensional KPZ Equation and Its Universality Class (2015, background): “2, for the curved case there is an exact solution for the distribution of h(0, t) for all t > 0 [20, 21].”
Directed polymers and the quantum Toda lattice (2009, background): “It will be interesting to relate this, in a suitable scaling limit, to the “crossover distributions” recently introduced in the context of KPZ and the stochastic heat equation by Sasamoto and Spohn [49–52] and Amir, Corwin and Quastel [1], building on recent work of Tracy and Widom [58–61] on the as”
Operator Spreading in Random Unitary Circuits (2017, background): “[66] Tomohiro Sasamoto and Herbert Spohn, “Exact height distributions for the {KPZ} equation with narrow wedge initial condition,” Nuclear Physics B 834, 523 – 542 (2010).”