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176
Discrete Polynuclear Growth and Determinantal processes
 Comm. Math. Phys
, 2003
"... Abstract. We consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F1 GOE TracyWidom distribution in terms of the Airy process. ..."
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Cited by 162 (11 self)
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Abstract. We consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F1 GOE TracyWidom distribution in terms of the Airy process. We also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem. 1. Introduction and
THE KARDARPARISIZHANG EQUATION AND UNIVERSALITY CLASS
, 2011
"... 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 univ ..."
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Cited by 97 (15 self)
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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 KardarParisiZhang (KPZ) universality class and underlying it is, again, a continuum object – a nonlinear 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 onepoint 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.
The arctic circle boundary and the Airy process
 Ann. Prob
, 2005
"... Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the struc ..."
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Cited by 88 (6 self)
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Abstract. We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the structure of the tiling at the center of the Aztec diamond. 1. Introduction and
Random matrices and determinantal processes
 Mathematical Statistical Physics, Session LXXXIII: Lecture Notes of the Les Houches Summer School 2005
"... Eigenvalues of random matrices have a rich mathematical structure and are a source of interesting distributions and processes. These distributions are natural statistical models in many problems in quantum physics, [15]. They occur for example, at least conjecturally, in the statistics of spectra of ..."
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Cited by 85 (5 self)
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Eigenvalues of random matrices have a rich mathematical structure and are a source of interesting distributions and processes. These distributions are natural statistical models in many problems in quantum physics, [15]. They occur for example, at least conjecturally, in the statistics of spectra of quantized models
Scaling limit for the spacetime covariance of the stationary totally asymmetric simple exclusion process
 Comm. Math. Phys
"... The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. Fo ..."
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Cited by 79 (27 self)
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The totally asymmetric simple exclusion process (TASEP) on the onedimensional lattice with the Bernoulli ρ measure as initial conditions, 0 < ρ < 1, is stationary in space and time. Let Nt(j) be the number of particles which have crossed the bond from j to j + 1 during the time span [0,t]. For j = (1 − 2ρ)t + 2w(ρ(1 − ρ)) 1/3 t 2/3 we prove that the fluctuations of Nt(j) for large t are of order t 1/3 and we determine the limiting distribution function Fw(s), which is a generalization of the GUE TracyWidom distribution. The family Fw(s) of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a RiemannHilbert problem. In our work we arrive at Fw(s) through the asymptotics of a Fredholm determinant. Fw(s) is simply related to the scaling function for the spacetime covariance of the stationary TASEP, equivalently to the asymptotic transition
Large time asymptotics of growth models on spacelike paths I: PushASEP
, 2008
"... We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom’s model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for ..."
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Cited by 71 (32 self)
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We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom’s model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any spacelike path are described by the Airy1 and Airy2 processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle’s trajectory as special cases.
Beta ensembles, stochastic Airy spectrum, and a diffusion
, 2008
"... We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the lowlying eigenvalues of the random Schrödinger operator − d2 dx 2 + x + 2 √ β b ′ x restricted to the positive halfline, where b ′ x is white noise. In doing so we extend the definit ..."
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Cited by 67 (9 self)
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We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the lowlying eigenvalues of the random Schrödinger operator − d2 dx 2 + x + 2 √ β b ′ x restricted to the positive halfline, where b ′ x is white noise. In doing so we extend the definition of the TracyWidom(β) distributions to all β> 0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a onedimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how TracyWidom laws arise from a functional central limit theorem.
Fluctuation properties of the TASEP with periodic initial configuration
, 2006
"... We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and det ..."
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Cited by 67 (34 self)
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We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us [27] and here we provide a selfcontained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.
Orthogonal polynomial ensembles in probability theory
 Prob. Surv
, 2005
"... Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary ..."
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Cited by 62 (1 self)
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Abstract: We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other wellknown ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, noncolliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther
Exact scaling functions for onedimensional stationary KPZ growth
, 2004
"... With deep appreciation dedicated to Giovanni JonaLasinio at the occasion of his 70th birthday. We determine the stationary twopoint correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a ..."
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Cited by 58 (11 self)
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With deep appreciation dedicated to Giovanni JonaLasinio at the occasion of his 70th birthday. We determine the stationary twopoint correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a RiemannHilbert problem related to the Painlevé II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode coupling approximation. 1