On global solutions of the random Hamilton-Jacobi equations and the KPZ problem

Yuri Bakhtin, Konstantin Khanin

Research output: Contribution to journalArticle

Abstract

In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton-Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension 1 + 1. In the case of general viscous Hamilton-Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.

Original languageEnglish (US)
Pages (from-to)R93-R121
JournalNonlinearity
Volume31
Issue number4
DOIs
StatePublished - Feb 19 2018

Fingerprint

Hamilton-Jacobi equation
Hamilton-Jacobi Equation
Global Solution
Directed Polymers
Hamiltonians
Universality
Polymers
Interlacing
polymers
Statistics
Renormalization
Shock
Fixed point
shock
statistics
Scaling
Partial
scaling
Term
Model

Keywords

  • global solutions
  • Hamilton-Jacobi equations
  • KPZ problem

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

On global solutions of the random Hamilton-Jacobi equations and the KPZ problem. / Bakhtin, Yuri; Khanin, Konstantin.

In: Nonlinearity, Vol. 31, No. 4, 19.02.2018, p. R93-R121.

Research output: Contribution to journalArticle

@article{646ffb4533074cf199598b916a6268c0,
title = "On global solutions of the random Hamilton-Jacobi equations and the KPZ problem",
abstract = "In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton-Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension 1 + 1. In the case of general viscous Hamilton-Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.",
keywords = "global solutions, Hamilton-Jacobi equations, KPZ problem",
author = "Yuri Bakhtin and Konstantin Khanin",
year = "2018",
month = "2",
day = "19",
doi = "10.1088/1361-6544/aa99a6",
language = "English (US)",
volume = "31",
pages = "R93--R121",
journal = "Nonlinearity",
issn = "0951-7715",
publisher = "IOP Publishing Ltd.",
number = "4",

}

TY - JOUR

T1 - On global solutions of the random Hamilton-Jacobi equations and the KPZ problem

AU - Bakhtin, Yuri

AU - Khanin, Konstantin

PY - 2018/2/19

Y1 - 2018/2/19

N2 - In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton-Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension 1 + 1. In the case of general viscous Hamilton-Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.

AB - In this paper, we discuss possible qualitative approaches to the problem of KPZ universality. Throughout the paper, our point of view is based on the geometrical and dynamical properties of minimisers and shocks forming interlacing tree-like structures. We believe that the KPZ universality can be explained in terms of statistics of these structures evolving in time. The paper is focussed on the setting of the random Hamilton-Jacobi equations. We formulate several conjectures concerning global solutions and discuss how their properties are connected to the KPZ scalings in dimension 1 + 1. In the case of general viscous Hamilton-Jacobi equations with non-quadratic Hamiltonians, we define generalised directed polymers. We expect that their behaviour is similar to the behaviour of classical directed polymers, and present arguments in favour of this conjecture. We also define a new renormalisation transformation defined in purely geometrical terms and discuss conjectural properties of the corresponding fixed points. Most of our conjectures are widely open, and supported by only partial rigorous results for particular models.

KW - global solutions

KW - Hamilton-Jacobi equations

KW - KPZ problem

UR - http://www.scopus.com/inward/record.url?scp=85045255622&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045255622&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/aa99a6

DO - 10.1088/1361-6544/aa99a6

M3 - Article

VL - 31

SP - R93-R121

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 4

ER -