## THE KARDAR-PARISI-ZHANG EQUATION AND UNIVERSALITY CLASS (2011)

Citations: | 8 - 1 self |

### BibTeX

@MISC{Corwin11thekardar-parisi-zhang,

author = {Ivan Corwin},

title = {THE KARDAR-PARISI-ZHANG EQUATION AND UNIVERSALITY CLASS},

year = {2011}

}

### OpenURL

### 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.

### Citations

746 |
Interacting particle systems
- Liggett
- 1985
(Show Context)
Citation Context ...chastic heat equation (SHE) with multiplicative noise and stochastic Burgers equation. 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 randommatrix theory, non-intersecting path ensembles, randomtilings andcertain comb... |

637 | Partial Differential Equations
- Evans
- 1998
(Show Context)
Citation Context ...k solution to the viscous Burgers equation. This is equivalent to imposing the so called entropy condition and the resulting solution can be solved via the method of characteristics (see Chapter 3 of =-=[66]-=-). If γ we taken to go to zero like ǫ, and time were sped up like ǫ−2T (not ǫ−1T) then the above argument shows that the resulting PDE for the limit shape is, in fact, the viscous Burgers equation. Li... |

345 | On the distribution of the length of the longest increasing subsequence in a random permutation
- Baik, Deift, et al.
- 1999
(Show Context)
Citation Context ...up of mathematicians determined theexact formulafor the one-point statistics of the KPZ class (in the wedge growth geometry corresponding to Figure 0a). That seminal work of Baik, Deift and Johansson =-=[9, 93]-=- dealt with two closely related discrete models (polynuclear, and corner growth) predicted to have the KPZ scaling. 2 By using exact formulas (arising from combinatorics and representation theory) and... |

320 |
Quantum inverse scattering method and correlation functions
- Korepin, Bogoliugov, et al.
- 1993
(Show Context)
Citation Context ...gas on R with n particles. The eigenfunctions for this system were found by McGuire [123] in 1964, and the norms of these eigenfunctions were determined using the algebraic Bethe Ansatz (for instance =-=[104, 25, 155]-=-) in [40] in 2007. Going from the moments of Z to its distribution (or the distribution of its logarithm) is mathematically unsound since one readily checks that the moments grow like ecn3 which means... |

296 |
Large Scale Dynamics of Interacting Particles
- Spohn
- 1991
(Show Context)
Citation Context ...chastic heat equation (SHE) with multiplicative noise and stochastic Burgers equation. 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 randommatrix theory, non-intersecting path ensembles, randomtilings andcertain comb... |

244 |
An introduction to stochastic partial differential equations, École d’été de probabilités de
- Walsh
- 1986
(Show Context)
Citation Context ... all the same. 2.2.2. The Stochastic Heat equation. Since the KPZ equation is now defined in terms of the SHE, it is important to understand what is means to solve the SHE. The lecture notes of Walsh =-=[170]-=- are a good reference for the study of linear stochastic PDEs and even though he does not deal with multiplicative noise, most of the theorems he states can be immediately translated into the setting ... |

237 | Shape fluctuations and random matrices
- Johansson
(Show Context)
Citation Context ...ial scale of t 2/3 (in sharp contrast to random deposition). This prediction has not been proved. However, in the case of the corner growth model (see Figure 0a) the same predictions have been proved =-=[93, 137]-=-. We too will focus on the corner growth model – not ballistic deposition. 1.1.2. Kardar, Parisi and Zhang’s prediction. The KPZ universality class was introduced in the context of studying the motion... |

229 |
Scaling Limits of Interacting Particle Systems
- Kipnis, Landim
- 1999
(Show Context)
Citation Context ...chastic heat equation (SHE) with multiplicative noise and stochastic Burgers equation. 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 randommatrix theory, non-intersecting path ensembles, randomtilings andcertain comb... |

156 |
Inverse scattering on the line
- Deift, Trubowitz
- 1979
(Show Context)
Citation Context ...VERSALITY CLASS 21 estimates of thesortdescribedabove, norhas theconnection withintegrable systems been sufficiently understood (for instance whether it arises in relation to inverse scattering as in =-=[53]-=-). In [49] some upper tail estimates are derived for various KPZ initial conditions. (5) Universality: The universality conjecture roughly says that (within reason) changing local rules of a model wil... |

152 |
Symmetric functions and P-recursiveness
- Gessel
- 1990
(Show Context)
Citation Context ...pohn [135] who recognized the connection of poissonized Ulams problem to the polynuclear growth model. 3 [9] studied the combinatorial problem in terms of Toeplitz matrices (a reduction due to Gessel =-=[75]-=-) and the associated Riemann-Hilbert problem method. Their asymptotics therefore were those of Riemann-Hilbert steepest descent and their limiting formula’s were in terms of Riemann-Hilbert problems a... |

139 |
Dynamic scaling of growing interfaces
- Kardar, Parisi, et al.
- 1986
(Show Context)
Citation Context ...EQUATION AND UNIVERSALITY CLASS 2 1. Introduction The overall goal of this review is to give a clear overview of over twenty-five years of work – startingevenbeforetheseminalpaperofKardar-Parisi-Zhang=-=[98]-=-in1986–whichhasculminated in early 2010 with the discovery of the probability distribution for the solution to the KPZ stochastic PDE (not to be confused with the KPZ of quantum gravity) and the under... |

136 | Longest increasing subsequences: from patience sorting to the Baik-DeiftJohansson theorem
- ALDOUS, P
- 1999
(Show Context)
Citation Context ...s of the longest increasing subsequence of a random permutation (Ulam’s problem). The Poissonize version of Ulam’s problem is related to an interacting particle system known as the Hammersley process =-=[83, 3]-=- (see [131, 7] for relevant numerical experiments). It was Prähofer and Spohn [135] who recognized the connection of poissonized Ulams problem to the polynuclear growth model. 3 [9] studied the combin... |

131 |
M.: “The Non-Linear Diffusion Equation
- Burgers
- 1974
(Show Context)
Citation Context ...ty distributions or local rules. These exponents had first been identified in 1977 by [73] in the study of the stochastic Burgers equation (stochastic space-time noise, not initial data as studied in =-=[39]-=-). In 1985 the model of a directed polymer in random media was first formulated in [87] and in the same year the scaling exponents for the driven lattice gas (the asymmetric simple exclusion process) ... |

93 | A Brownian-Motion Model for the Eigenvalues of a Random - Dyson - 1962 |

92 | Scale invariance of the PNG droplet and the Airy process
- Prähofer, Spohn
- 2002
(Show Context)
Citation Context ...ial scale of t 2/3 (in sharp contrast to random deposition). This prediction has not been proved. However, in the case of the corner growth model (see Figure 0a) the same predictions have been proved =-=[93, 137]-=-. We too will focus on the corner growth model – not ballistic deposition. 1.1.2. Kardar, Parisi and Zhang’s prediction. The KPZ universality class was introduced in the context of studying the motion... |

88 |
Fractal concepts in surface growth
- Barabâasi, Stanley
- 1995
(Show Context)
Citation Context |

85 | Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Birkhäuser
- Holden, Øksendal, et al.
- 1996
(Show Context)
Citation Context ...Hopf-Cole interpretation had been used previously (though without the justification provided by [23]) in the physics literature for some time. Other attempts at interpreting the KPZ equation (such as =-=[85]-=-) have resulted in solutions which are considered to be physically irrelevant as their solutions have very different long-time scaling behavior [43]. Since the KPZ equation is a fundamental mathematic... |

84 |
Fractals, scaling and growth far from equilibrium, volume 5 of Cambridge Nonlinear Science Series
- Meakin
- 1998
(Show Context)
Citation Context |

81 |
A few seedlings of research
- Hammersley
- 1972
(Show Context)
Citation Context ...s of the longest increasing subsequence of a random permutation (Ulam’s problem). The Poissonize version of Ulam’s problem is related to an interacting particle system known as the Hammersley process =-=[83, 3]-=- (see [131, 7] for relevant numerical experiments). It was Prähofer and Spohn [135] who recognized the connection of poissonized Ulams problem to the polynuclear growth model. 3 [9] studied the combin... |

80 | The Parisi formula - Talagrand - 2006 |

76 | Discrete polynuclear growth and determinantal processes
- Johansson
- 2003
(Show Context)
Citation Context ... KPZ Universality Geometry Limit shape Fluctuations KPZ/SHE • Wedge ❅ ❅❅❅❅� � ��� ◦ ◦ ◦ ◦ ◦ 0 • • • • • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 |X| ≤ 1 X X > 1 • One pt: FGUE [9, 93] • Multi pt: Airy 2 =-=[137, 94, 29]-=- Z(0,X) = δ {X=0} • Converges:[5] • One pt:[5, 147] (bounds and stats) • Brownian �❅ ❅�❅❅� � �❅ • ◦ ◦ • ◦ 0 ◦ • • • ◦ ¯h(T,TX) = T/2 • One pt: F0 [11, 69] • Multi pt: Airy stat [10] Z(0,X) = e B(X) • ... |

73 |
Interaction of Markov processes
- Spitzer
- 1970
(Show Context)
Citation Context ... function and find the long-time asymptotics to be Tracy-Widom GUE, only [41] extracted finite time statistics.THE KARDAR-PARISI-ZHANG EQUATION AND UNIVERSALITY CLASS 12 The simple exclusion process =-=[99, 156]-=- (and its integrated version – the corner growth model) is the poster child for all such particle systems and serve as a paradigm for non-equilibrium statistical mechanics (with thousands of articles ... |

56 | Limiting distributions for a polynuclear growth model with external sources
- Baik, Rains
(Show Context)
Citation Context ... One pt: FGUE [9, 93] • Multi pt: Airy 2 [137, 94, 29] Z(0,X) = δ {X=0} • Converges:[5] • One pt:[5, 147] (bounds and stats) • Brownian �❅ ❅�❅❅� � �❅ • ◦ ◦ • ◦ 0 ◦ • • • ◦ ¯h(T,TX) = T/2 • One pt: F0 =-=[11, 69]-=- • Multi pt: Airy stat [10] Z(0,X) = e B(X) • Converges:[23] • One pt:[15, 49] (bounds, NO stats) • Flat ❅�❅�❅�❅�❅� ◦ • ◦ • ◦ 0 • ◦ • ◦ • ¯h(T,TX) = T/2 • One pt: FGOE [12, 13, 70, 146] • Multi pt: Ai... |

49 | Current fluctuations for the totally asymmetric simple exclusion process
- Prähofer, Spohn
- 2002
(Show Context)
Citation Context ... 32] Z(0,X) = 1 • Converges:[23] • One pt: OPEN (NO bounds / stats) • Wedge→Brownian ❅ ❅❅❅ ❅�❅ ❅� � ◦ ◦ ◦ ◦ ◦ 0 • ◦ ◦ • • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 X ∈ [−1,0] T/2 X > 0 • One pt: (FGOE) 2 =-=[11, 8, 136, 19]-=- • Multi pt: Airy 2→BM [90, 46] Z(0,X) = e B(X) 1X≥0 • Converges:[49] • One pt:[49] (bounds and stats) • Wedge→Flat ❅ ❅❅❅❅�❅�❅� ◦ ◦ ◦ ◦ ◦ 0 • ◦ • ◦ • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 X ∈ [−1,0] T... |

49 |
Hydrodynamic limit for attractive particle systems on
- Rezakhanlou
- 1991
(Show Context)
Citation Context ...and consider whether ¯h(T,X) = lim ǫ→0 ǫhγ(t/γ,x) exists. If we assume that such a limit exists (in a suitable sense) for T = 0 (and calling the limit ¯h 0 (X)) then it is a theorem (see for instance =-=[143, 144, 173]-=-) that the limit exists for all T > 0 and that ¯ h(T,X) is the unique weak solution to the PDE known as the inviscid Burgers equation ∂T ¯ h = 1−(∂X ¯ h) 2 , 2 with initial data ¯ h0 (X) and subject t... |

48 | The asymptotics of monotone subsequences of involutions
- Baik, Rains
(Show Context)
Citation Context ...,TX) = T/2 • One pt: F0 [11, 69] • Multi pt: Airy stat [10] Z(0,X) = e B(X) • Converges:[23] • One pt:[15, 49] (bounds, NO stats) • Flat ❅�❅�❅�❅�❅� ◦ • ◦ • ◦ 0 • ◦ • ◦ • ¯h(T,TX) = T/2 • One pt: FGOE =-=[12, 13, 70, 146]-=- • Multi pt: Airy 1 [31, 32] Z(0,X) = 1 • Converges:[23] • One pt: OPEN (NO bounds / stats) • Wedge→Brownian ❅ ❅❅❅ ❅�❅ ❅� � ◦ ◦ ◦ ◦ ◦ 0 • ◦ ◦ • • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 X ∈ [−1,0] T/2 X... |

47 |
Maximal Displacement of Branching Brownian motion
- Bramson
- 1978
(Show Context)
Citation Context ...problems, including important problems in bio-statistics [120, 89, 121, 145] and operations research [6]; they arise in the study of branching Brownian motions and random walks in random environments =-=[37]-=-; they serve as paradigms for the study of other disordered systems [57]; and phenomena like pinning and wetting [4, 86]. On top of that, their relationship to random growth models and interacting par... |

46 |
Aspects of first-passage percolation
- Kesten
- 1986
(Show Context)
Citation Context |

40 |
Polymers on disordered trees, spin glasses, and traveling waves
- Derrida, Spohn
- 1988
(Show Context)
Citation Context ...145] and operations research [6]; they arise in the study of branching Brownian motions and random walks in random environments [37]; they serve as paradigms for the study of other disordered systems =-=[57]-=-; and phenomena like pinning and wetting [4, 86]. On top of that, their relationship to random growth models and interacting particle systems means that any exact solvability developed in the framewor... |

38 |
Kinetic roughening of growing surfaces
- Krug, Spohn
- 1991
(Show Context)
Citation Context |

37 | Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process
- Ferrari, Spohn
(Show Context)
Citation Context ... One pt: FGUE [9, 93] • Multi pt: Airy 2 [137, 94, 29] Z(0,X) = δ {X=0} • Converges:[5] • One pt:[5, 147] (bounds and stats) • Brownian �❅ ❅�❅❅� � �❅ • ◦ ◦ • ◦ 0 ◦ • • • ◦ ¯h(T,TX) = T/2 • One pt: F0 =-=[11, 69]-=- • Multi pt: Airy stat [10] Z(0,X) = e B(X) • Converges:[23] • One pt:[15, 49] (bounds, NO stats) • Flat ❅�❅�❅�❅�❅� ◦ • ◦ • ◦ 0 • ◦ • ◦ • ¯h(T,TX) = T/2 • One pt: FGOE [12, 13, 70, 146] • Multi pt: Ai... |

37 | Random matrices and determinantal processes
- Johansson
(Show Context)
Citation Context ...ms is closely related to and influenced by problems in randommatrix theory, non-intersecting path ensembles, randomtilings andcertain 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. Inall of thesetypes of models therearecert... |

37 |
Universal distributions for growth processes in 1+1 dimensions and random matrices
- Prähofer, Spohn
(Show Context)
Citation Context ...n discovered in the early 90s by Tracy and Widom in the context of random matrix theory [162]. The previous numerical work of the 1990s agreed with the values readily computed from the exact formulas =-=[135]-=-. Experimental work has shown that the scalings and the statistics for the KPZ class are excellent fits for certain 2 While [93] dealt directly with the corner growth model, [9] considered the fluctua... |

31 |
A note on the diffusion of directed polymers in a random environment
- Bolthausen
- 1989
(Show Context)
Citation Context ...cted polymers was by Imbrie and Spencer [92] in 1988 where (by use of an elaborate expansion) they proved that in dimensions d ≥ 3 and with small enough β, the walk is diffusive (ξ = 1/2). Bolthausen =-=[26]-=- strengthened the result (under same same d ≥ 3, β small assumptions) to a central limit theorem for the endpoint of the walk. His work relied on the now fundamental observation that renormalized part... |

31 |
Probabilistic analysis of directed polymers in a random environment: a review. In Stochastic analysis on large scale interacting systems
- Comets, Shiga, et al.
(Show Context)
Citation Context |

30 | E.M.Rains, On longest increasing subsequences in random permutations, Analysis, geometry, number theory: the mathematics of Leon Ehrenpreis
- Odlyzko
- 1998
(Show Context)
Citation Context ...est increasing subsequence of a random permutation (Ulam’s problem). The Poissonize version of Ulam’s problem is related to an interacting particle system known as the Hammersley process [83, 3] (see =-=[131, 7]-=- for relevant numerical experiments). It was Prähofer and Spohn [135] who recognized the connection of poissonized Ulams problem to the polynuclear growth model. 3 [9] studied the combinatorial proble... |

30 | Exact solution of the master equation for the asymmetric exclusion process
- Schütz
- 1997
(Show Context)
Citation Context ...l 1, though rescaling time like β 4 T and space like β 2 X effectively reintroduces this factor 0THE KARDAR-PARISI-ZHANG EQUATION AND UNIVERSALITY CLASS 20 The first approach greatly extends work of =-=[151]-=-, while the third approach likewise extends the solvability of the zero temperature polymer models (last passage percolation) via the RobinsonSchensted-Knuth correspondence (studied at length since th... |

29 | Fluctuations of the one-dimensional polynuclear growth model with external sources
- Imamura, Sasamoto
(Show Context)
Citation Context ...pt: OPEN (NO bounds / stats) • Wedge→Brownian ❅ ❅❅❅ ❅�❅ ❅� � ◦ ◦ ◦ ◦ ◦ 0 • ◦ ◦ • • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 X ∈ [−1,0] T/2 X > 0 • One pt: (FGOE) 2 [11, 8, 136, 19] • Multi pt: Airy 2→BM =-=[90, 46]-=- Z(0,X) = e B(X) 1X≥0 • Converges:[49] • One pt:[49] (bounds and stats) • Wedge→Flat ❅ ❅❅❅❅�❅�❅� ◦ ◦ ◦ ◦ ◦ 0 • ◦ • ◦ • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 X ∈ [−1,0] T/2 X > 0 • Multi pt: Airy 2→1 [... |

28 | First passage percolation has sublinear distance variance - Benjamini, Kalai, et al. |

28 |
Spatial correlations of the 1D KPZ surface on a flat substrate
- Sasamoto
- 2005
(Show Context)
Citation Context ...,TX) = T/2 • One pt: F0 [11, 69] • Multi pt: Airy stat [10] Z(0,X) = e B(X) • Converges:[23] • One pt:[15, 49] (bounds, NO stats) • Flat ❅�❅�❅�❅�❅� ◦ • ◦ • ◦ 0 • ◦ • ◦ • ¯h(T,TX) = T/2 • One pt: FGOE =-=[12, 13, 70, 146]-=- • Multi pt: Airy 1 [31, 32] Z(0,X) = 1 • Converges:[23] • One pt: OPEN (NO bounds / stats) • Wedge→Brownian ❅ ❅❅❅ ❅�❅ ❅� � ◦ ◦ ◦ ◦ ◦ 0 • ◦ ◦ • • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 X ∈ [−1,0] T/2 X... |

25 | Symmetrized random permutations
- Baik, Rains
- 2001
(Show Context)
Citation Context ...,TX) = T/2 • One pt: F0 [11, 69] • Multi pt: Airy stat [10] Z(0,X) = e B(X) • Converges:[23] • One pt:[15, 49] (bounds, NO stats) • Flat ❅�❅�❅�❅�❅� ◦ • ◦ • ◦ 0 • ◦ • ◦ • ¯h(T,TX) = T/2 • One pt: FGOE =-=[12, 13, 70, 146]-=- • Multi pt: Airy 1 [31, 32] Z(0,X) = 1 • Converges:[23] • One pt: OPEN (NO bounds / stats) • Wedge→Brownian ❅ ❅❅❅ ❅�❅ ❅� � ◦ ◦ ◦ ◦ ◦ 0 • ◦ ◦ • • ¯h(T,TX) = ⎧ ⎪⎨ −X X < −1 T ⎪⎩ 1+X2 2 X ∈ [−1,0] T/2 X... |

25 | 2007 Non equilibrium steady states: fluctuations and large deviations of the density and of the current - Derrida |

25 |
Stochastic partial differential equations for some measure-valued diffusions, Probab. Theory Related Fields 79
- Konno, Shiga
- 1988
(Show Context)
Citation Context ... and Burkholder-Davis-Gundy yield the necessary estimates. TheuniquenessofthelimitofthelawsQǫ andthecoincidencewiththelawofthesolutiontothe SHE is shown by way of the method of the martingale problem =-=[105]-=-. In the following fT(X) denotes the canonical coordinate in C([0,∞),C(R + )) (i.e., the space-time function fT(X) is a function of ω ∈ Ω corresponding to the probability measure Q below) and (g,h) = ... |

24 |
Excess noise for driven diffusive systems
- Beijeren, Kutner, et al.
(Show Context)
Citation Context ...l of a directed polymer in random media was first formulated in [87] and in the same year the scaling exponents for the driven lattice gas (the asymmetric simple exclusion process) were discovered by =-=[20]-=-. 1 The connections between polymers and lattice gases were understood quickly [97, 88]. The big step of Kardar, Parisi and Zhang was therefore to relate these models and calculations to interface mot... |

24 |
Exact Analysis of an Interacting Bose Gas
- Lieb, Liniger
- 1963
(Show Context)
Citation Context ...ze the Hamiltonian and respect the boundary condition. The eigenfunctions we written down in 1963 for the repulsive delta interaction (the +1 in the boundary condition becomes −1) by Lieb and Liniger =-=[114]-=- by Bethe ansatz and their completeness was proved by Dorlas [61] (on [0,1]) and Tracy and Widom [169] (on R as we are considering presently). For the attractive case, McGuire [123] wrote the eigenfun... |

23 | Exact large deviation function in the asymmetric exclusion process - Derrida, Lebowitz - 1998 |

23 |
Large-distance and long-time properties of a randomly stirred fluid. Phys. Rev. A
- Forster, DR, et al.
- 1977
(Show Context)
Citation Context ...ath rate p and total asymmetry q −p = γ > 0 Figure 0b: Ballistic deposition model. Growth occurs when blocks stick to first point of contact (denoted in gray). Employing Forster, Nelson and Stephen’s =-=[73]-=- 1977 dynamical renormalization group techniques (highly non-rigorous from a mathematical perspective), [98] predicted that scaling exponent of 1/3 and 2/3 should describe the fluctuations and correla... |

23 |
Study of exactly soluble one-dimensional N-body problems
- McGuire
- 1964
(Show Context)
Citation Context ...tisfies a closed evolution equation which coincide with the quantum many body system known as the attractive δ-Bose gas on R with n particles. The eigenfunctions for this system were found by McGuire =-=[123]-=- in 1964, and the norms of these eigenfunctions were determined using the algebraic Bethe Ansatz (for instance [104, 25, 155]) in [40] in 2007. Going from the moments of Z to its distribution (or the ... |

22 | Directed polymers in random environment are diffusive at weak disorder
- Comets, Yoshida
(Show Context)
Citation Context ... that βc = 0 for d ∈ {1,2} [44] and 0 < βc ≤ ∞ for d ≥ 3. In d ≥ 3 and weak disorder the walk converges to a Brownian motion, and the limiting diffusion matrix is the same as for standard random walk =-=[45]-=-. On the other hand, in strong disorder it is known (see [44]) that there exist (random) points at which the path π has a positive probability (under dP β Q ) of ending. This is certainly different be... |

22 | Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation Phys - Gwa, Spohn - 1992 |