Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

Spohn, Herbert

doi:10.1016/j.physa.2006.04.006

Cited by 90 publications

(112 citation statements)

References 47 publications

(56 reference statements)

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“…[95][96][97][98][99][100][101][102][103])) and vehicular traffic [104][105][106]. Enjoying considerable attention, it is the focus of several comprehensive reviews [107][108][109][110][111]. Of course, one-dimensional chains cannot support many of the interesting features discovered in the KLS, e.g., anisotropic correlations, discontinuity singularities in S(k), and "icicles."…”

Section: Discussionmentioning

confidence: 99%

Twenty Five Years After KLS: A Celebration of Non-equilibrium Statistical Mechanics

Zia

2009

J Stat Phys

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When Lenz proposed a simple model for phase transitions in magnetism, he couldn't have imagined that the "Ising model" was to become a jewel in field of equilibrium statistical mechanics. Its role spans the spectrum, from a good pedagogical example to a universality class in critical phenomena. A quarter century ago, Katz, Lebowitz and Spohn found a similar treasure. By introducing a seemingly trivial modification to the Ising lattice gas, they took it into the vast realms of non-equilibrium statistical mechanics. An abundant variety of unexpected behavior emerged and caught many of us by surprise. We present a brief review of some of the new insights garnered and some of the outstanding puzzles, as well as speculate on the model's role in the future of non-equilibrium statistical physics.

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Section: Discussionmentioning

confidence: 99%

Twenty Five Years After KLS: A Celebration of Non-equilibrium Statistical Mechanics

Zia

2009

J Stat Phys

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“…A number of books and review articles have been written about these models in mathematics and physics, including [18,44,82,86,100,102,106,108,115,117,120,158,157]. Though not directly addressed here, the study of these systems is closely related to and influenced by problems in random matrix theory, non-intersecting path ensembles, random tilings and certain combinatorial problems involving asymptotic representation theory [28,71,95,158]. In particular, many (but not all) of the statistics which arise in these systems were first analytically discovered in the context of random matrix theory.…”

Section: 1mentioning

confidence: 99%

The Kardar–parisi–zhang Equation and Universality Class

Corwin

2012

Random Matrices: Theory Appl.

680

824

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Abstract. 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 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.

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“…This topic, launched by Calabrese & Cardy in 2005/2006 [29,30], received a lot of attention in recent years [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. The basic idea is the following: the usual quantum/classical correspondence tells us that, modulo Wick rotation, the real-time evolution of a onedimensional quantum model is equivalent to a two-dimensional classical statistical model.…”

Section: Introduction and Overviewmentioning

confidence: 99%

Inhomogeneous field theory inside the arctic circle

Allegra¹,

Dubail²,

Stéphan³

et al. 2016

J. Stat. Mech.

121

252

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Abstract. Motivated by quantum quenches in spin chains, a one-dimensional toy-model of fermionic particles evolving in imaginary-time from a domain-wall initial state is solved. The main interest of this toymodel is that it exhibits the arctic circle phenomenon, namely a spatial phase separation between a critically fluctuating region and a frozen region. Large-scale correlations inside the critical region are expressed in terms of correlators in a (euclidean) two-dimensional massless Dirac field theory. It is observed that this theory is inhomogenous: the metric is position-dependent, so it is in fact a Dirac theory in curved space. The technique used to solve the toy-model is then extended to deal with the transfer matrices of other models: dimers on the honeycomb and square lattice, and the six-vertex model at the free fermion point (∆ = 0). In all cases, explicit expressions are given for the long-range correlations in the critical region, as well as for the underlying Dirac action. Although the setup developed here is heavily based on fermionic observables, the results can be translated into the language of height configurations and of the gaussian free field, via bosonization. Correlations close to the phase boundary and the generic appearance of Airy processes in all these models are also briefly revisited in the appendix.

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Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes of crystals

Cited by 90 publications

References 47 publications

Twenty Five Years After KLS: A Celebration of Non-equilibrium Statistical Mechanics

Twenty Five Years After KLS: A Celebration of Non-equilibrium Statistical Mechanics

The Kardar–parisi–zhang Equation and Universality Class

Inhomogeneous field theory inside the arctic circle

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