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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 91 (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
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 78 (35 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.
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 63 (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
Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
, 2004
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PDEs for the joint distributions of the Dyson, Airy and Sine processes
 Ann. Probab
, 2005
"... In a celebrated paper, Dyson shows that the spectrum of a n × n random Hermitian matrix, diffusing according to an OrnsteinUhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the ..."
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Cited by 37 (11 self)
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In a celebrated paper, Dyson shows that the spectrum of a n × n random Hermitian matrix, diffusing according to an OrnsteinUhlenbeck process, evolves as n noncolliding Brownian motions held together by a drift term. The universal edge and bulk scalings for Hermitian random matrices, applied to the Dyson process, lead to the Airy and Sine processes. In particular, the Airy process is a continuous stationary process, describing the motion of the outermost particle of the Dyson Brownian motion, when the number of particles gets large, with space and time appropriately rescaled. In this paper, we answer a question posed by Kurt Johansson, to find a PDE for the joint distribution of the Airy Process at two different times. Similarly we find a PDE satisfied by the joint distribution of the Sine process. This hinges on finding a PDE for the joint distribution of the Dyson process, which itself is based on the joint probability
Transition between Airy1 and Airy2 processes and TASEP fluctuations
 Comm. Pure Appl. Math
"... We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuat ..."
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Cited by 31 (15 self)
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We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. For large time t, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy1 and Airy2 processes, whose onepoint distributions are the GOE and GUE TracyWidom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its onepoint distribution is a new interpolation between GOE and GUE edge distributions. 1
Dyson's Nonintersecting Brownian motions with a few outliers
, 2008
"... Consider n nonintersecting Brownian particles on R (Dyson Brownian motions), all starting from the origin at time t = 0, and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ± √ 2nt(1 −t). The Air ..."
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Cited by 28 (5 self)
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Consider n nonintersecting Brownian particles on R (Dyson Brownian motions), all starting from the origin at time t = 0, and forced to return to x = 0 at time t = 1. For large n, the average mean density of particles has its support, for each 0 < t < 1, on the interval ± √ 2nt(1 −t). The Airy process A (τ) is defined as the motion of these nonintersecting Brownian motions for large n, but viewed from the curve C: y = √ 2nt(1 −t) with an appropriate spacetime rescaling. Assume now a finite number r of these particles are forced to a different target point, say a = ρ0 n/2> 0. Does it affect the Brownian fluctuations along the curve C for large n? In this paper, we show that no new process appears as long as one considers points (y,t) ∈ C, such that 0 < t < (1 + ρ2 0)−1, which is the tcoordinate of the point of tangency of the tangent to the curve passing through (ρ0 n/2,1). At this point of tangency the fluctuations obey a
All orders asymptotic expansion of large partitions
, 2008
"... The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and ..."
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Cited by 28 (6 self)
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The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and also in algebraic geometry. In particular we compute the GromovWitten invariants of the Xp = O(p − 2) ⊕ O(−p) → P1 CalabiYau 3fold, and we prove a conjecture of M. Mariño, that the generating functions Fg of Gromov–Witten invariants of Xp, come from a matrix model, and are the symplectic invariants of the mirror spectral curve.