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

TL;DR

This paper explores the geometric and dynamical structures underlying the KPZ universality class through the lens of random Hamilton-Jacobi equations, proposing conjectures on global solutions and their relation to KPZ scaling.

Contribution

It introduces a geometric framework for understanding KPZ universality, formulates new conjectures on global solutions, and defines generalized directed polymers and a novel renormalization transformation.

Findings

Conjectures on global solutions linked to KPZ scalings

Definition of generalized directed polymers for non-quadratic Hamiltonians

Proposal of a new geometrical renormalization transformation

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.