The Kardar-Parisi-Zhang equation and universality class

Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the Kardar-Parisi-Zhang (KPZ) universality class and underlying it is, again, a continuum object -- a non-linear stochastic partial differential equation -- known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with {\it narrow wedge} initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact one-point distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.Comment: 57 pages, survey article, 7 figures, addition physics ref. added and typo's correcte

Soliton approach to the noisy Burgers equation: Steepest descent method

The noisy Burgers equation in one spatial dimension is analyzed by means of the Martin-Siggia-Rose technique in functional form. In a canonical formulation the morphology and scaling behavior are accessed by mean of a principle of least action in the asymptotic non-perturbative weak noise limit. The ensuing coupled saddle point field equations for the local slope and noise fields, replacing the noisy Burgers equation, are solved yielding nonlinear localized soliton solutions and extended linear diffusive mode solutions, describing the morphology of a growing interface. The canonical formalism and the principle of least action also associate momentum, energy, and action with a soliton-diffusive mode configuration and thus provides a selection criterion for the noise-induced fluctuations. In a ``quantum mechanical'' representation of the path integral the noise fluctuations, corresponding to different paths in the path integral, are interpreted as ``quantum fluctuations'' and the growth morphology represented by a Landau-type quasi-particle gas of ``quantum solitons'' with gapless dispersion and ``quantum diffusive modes'' with a gap in the spectrum. Finally, the scaling properties are dicussed from a heuristic point of view in terms of a``quantum spectral representation'' for the slope correlations. The dynamic eponent z=3/2 is given by the gapless soliton dispersion law, whereas the roughness exponent zeta =1/2 follows from a regularity property of the form factor in the spectral representation. A heuristic expression for the scaling function is given by spectral representation and has a form similar to the probability distribution for Levy flights with index $z$.Comment: 30 pages, Revtex file, 14 figures, to be submitted to Phys. Rev.

Heuristic derivation of continuum kinetic equations from microscopic dynamics

We present an approximate and heuristic scheme for the derivation of continuum kinetic equations from microscopic dynamics for stochastic, interacting systems. The method consists of a mean-field type, decoupled approximation of the master equation followed by the `naive' continuum limit. The Ising model and driven diffusive systems are used as illustrations. The equations derived are in agreement with other approaches, and consequences of the microscopic dependences of coarse-grained parameters compare favorably with exact or high-temperature expansions. The method is valuable when more systematic and rigorous approaches fail, and when microscopic inputs in the continuum theory are desirable.Comment: 7 pages, RevTeX, two-column, 4 PS figures include

On the universality of the incompressible Euler equation on compact manifolds

The incompressible Euler equations on a compact Riemannian manifold $(M,g)$ take the form \begin{align*} \partial_t u + \nabla_u u &= - \mathrm{grad}_g p \mathrm{div}_g u &= 0. \end{align*} We show that any quadratic ODE $\partial_t y = B(y,y)$, where $B : {\bf R}^n \times {\bf R}^n \to {\bf R}^n$ is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold $M$ if and only if $B$ obeys the cancellation condition $\langle B(y,y), y \rangle = 0$ for some positive definite inner product $\langle,\rangle$ on ${\bf R}^n$. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.Comment: 14 pages, no figures, to appear, Discrete and Continuous Dynamical Systems. This is the final versio

Fisher Hartwig determinants, conformal field theory and universality in generalised XX models

We discuss certain quadratic models of spinless fermions on a 1D lattice, and their corresponding spin chains. These were studied by Keating and Mezzadri in the context of their relation to the Haar measures of the classical compact groups. We show how these models correspond to translation invariant models on an infinite or semi-infinite chain, which in the simplest case reduce to the familiar XX model. We give physical context to mathematical results for the entanglement entropy, and calculate the spin-spin correlation functions using the Fisher-Hartwig conjecture. These calculations rigorously demonstrate universality in classes of these models. We show that these are in agreement with field theoretic and renormalization group arguments that we provide

Infrared Behaviour of Systems With Goldstone Bosons

We develop various complementary concepts and techniques for handling quantum fluctuations of Goldstone bosons.We emphasise that one of the consequences of the masslessness of Goldstone bosons is that the longitudinal fluctuations also have a diverging susceptibility characterised by an anomalous dimension $(d-2)$ in space-time dimensions $2<d<4$.In $d=4$ these fluctuations diverge logarithmically in the infrared region.We show the generality of this phenomenon by providing three arguments based on i). Renormalization group flows, ii). Ward identities, and iii). Schwinger-Dyson equations.We obtain an explicit form for the generating functional of one-particle irreducible vertices of the O(N) (non)--linear $\sigma$--models in the leading 1/N approximation.We show that this incorporates all infrared behaviour correctly both in linear and non-linear $\sigma$-- models. Our techniques provide an alternative to chiral perturbation theory.Some consequences are discussed briefly.Comment: 28 pages,2 Figs, a new section on some universal features of multipion processes has been adde

Geometric Path Integrals. A Language for Multiscale Biology and Systems Robustness

In this paper we suggest that, under suitable conditions, supervised learning can provide the basis to formulate at the microscopic level quantitative questions on the phenotype structure of multicellular organisms. The problem of explaining the robustness of the phenotype structure is rephrased as a real geometrical problem on a fixed domain. We further suggest a generalization of path integrals that reduces the problem of deciding whether a given molecular network can generate specific phenotypes to a numerical property of a robustness function with complex output, for which we give heuristic justification. Finally, we use our formalism to interpret a pointedly quantitative developmental biology problem on the allowed number of pairs of legs in centipedes