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Seibergwitten theory and random partitions
"... We study N = 2 supersymmetric four dimensional gauge theories, in a certain N = 2 supergravity background, called Ωbackground. The partition function of the theory in the Ωbackground can be calculated explicitly. We investigate various representations for this partition function: a statistical sum ..."
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Cited by 97 (6 self)
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We study N = 2 supersymmetric four dimensional gauge theories, in a certain N = 2 supergravity background, called Ωbackground. The partition function of the theory in the Ωbackground can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, a free fermion correlator. These representations allow to derive rigorously the SeibergWitten geometry, the curves, the differentials, and the prepotential. We study pure N = 2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five dimensional theory compactified
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
, 2008
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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 76 (6 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
Correlation function of Schur process with application to local geometry of a random 3dimensional Young Diagram
, 2001
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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]. For j = ..."
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Cited by 37 (9 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
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 37 (3 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
equations for Hurwitz numbers
"... We consider ramified coverings of P 1 with arbitrary ramification type over 0, ∞ ∈ P 1 and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a τfunction for the Toda lattice hierarchy of Ueno and Takasaki. ..."
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Cited by 32 (3 self)
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We consider ramified coverings of P 1 with arbitrary ramification type over 0, ∞ ∈ P 1 and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a τfunction for the Toda lattice hierarchy of Ueno and Takasaki.
Toeplitz determinants, random growth and determinantal processes
 In Proc. Internat. Cong. Mathematicians, vol. III (Beijing 2002), Higher Ed
, 2002
"... We summarize some of the recent developments which link certain problems in combinatorial theory related to random growth to random matrix theory. ..."
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Cited by 21 (2 self)
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We summarize some of the recent developments which link certain problems in combinatorial theory related to random growth to random matrix theory.
Generalized Riffle Shuffles and Quasisymmetric Functions
, 2001
"... Let x i be a probability distribution on a totally ordered set I, i.e., the probability of i 2 I is x i . (Hence x i 0 and x i = 1.) Fix n 2 P = f1; 2; : : :g, and define a random permutation w 2 S n as follows. For each 1 j n, choose independently an integer i j (from the distribution x i ). ..."
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Cited by 20 (0 self)
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Let x i be a probability distribution on a totally ordered set I, i.e., the probability of i 2 I is x i . (Hence x i 0 and x i = 1.) Fix n 2 P = f1; 2; : : :g, and define a random permutation w 2 S n as follows. For each 1 j n, choose independently an integer i j (from the distribution x i ). Then standardize the sequence i = i 1 \Delta \Delta \Delta i n in the sense of [34, p. 322], i.e., let ff 1 ! \Delta \Delta \Delta ! ff k be the elements of I actually appearing in i, and let a i be the number of ff i 's in i. Replace the ff 1 's in i by 1; 2; : : : ; a 1 from lefttoright, then the ff 2 's in i by a 1 + 1; a 1 + 2; : : : ; a 1 + a 2 from lefttoright, etc. For instance, if I = P and i = 311431, then w = 412653. This defines a probability distribution on the symmetric group S n , which we call the QSdistribution (because of the close connection with quasisymmetric functions explained below). If we need to be explicit about the parameters x = (x i ) i2I , t
Generating functions for intersection numbers on moduli spaces of curves
, 2000
"... Using the connection between intersection theory on the DeligneMumford spaces Mg,n and the edge scaling of the GUE matrix model (see [12, 14]), we express the npoint functions for the intersection numbers as ndimensional errorfunctiontype integrals and also give a derivation of Witten’s KdV equ ..."
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Cited by 18 (4 self)
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Using the connection between intersection theory on the DeligneMumford spaces Mg,n and the edge scaling of the GUE matrix model (see [12, 14]), we express the npoint functions for the intersection numbers as ndimensional errorfunctiontype integrals and also give a derivation of Witten’s KdV equations using the higher Fay identities of Adler, Shiota, and van Moerbeke.