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 language | English (US) |
|---|---|
| Pages (from-to) | R93-R121 |
| Journal | Nonlinearity |
| Volume | 31 |
| Issue number | 4 |
| DOIs | |
| State | Published - Feb 19 2018 |
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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 journal › Article
}
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 -