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128
Statistical mechanics of complex networks
 Rev. Mod. Phys
"... Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as ra ..."
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Cited by 1188 (8 self)
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Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real
Levelspacing distributions and the Airy kernel
, 1993
"... Scaling levelspacing distribution functions in the "bulk of the spectrum " in random matrix models of N x N hermitian matrices and then going to the limit N, o¢, leads to the Fredholm determinant of the sine kernel sin n(xy)/n(xy). Similarly a double scaling limit at the "edge of ..."
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Cited by 244 (25 self)
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Scaling levelspacing distribution functions in the "bulk of the spectrum " in random matrix models of N x N hermitian matrices and then going to the limit N, o¢, leads to the Fredholm determinant of the sine kernel sin n(xy)/n(xy). Similarly a double scaling limit at the "edge of the spectrum " leads to the Airy kernel [Ai (x)Ai ' (y) Ai ' (x)Ai (y) ] / (x y). We announce analogies for this Airy kernel of the following properties of the sine kernel: the completely integrable system of PDE's found by Jimbo, Miwa, M6ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev6 transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general n, of the probability that an interval contains precisely n eigenvalues.
Discrete orthogonal polynomial ensembles and the Plancherel measure
, 2001
"... We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble i ..."
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Cited by 140 (8 self)
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We consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory. These ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions. The Meixner ensemble is related to a twodimensional directed growth model, and the Charlier ensemble is related to the lengths of weakly increasing subsequences in random words. The Krawtchouk ensemble occurs in connection with zigzag paths in random domino tilings of the Aztec diamond, and also in a certain simplified directed firstpassage percolation model. We use the Charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of Tracy and Widom. As a limit of the Meixner ensemble or the Charlier ensemble we obtain the Plancherel measure on partitions, and using this we prove a conjecture of Baik, Deift and Johansson that under the Plancherel measure, the distribution of the lengths of the first k rows in the partition, appropriately scaled, converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the Gaussian Unitary Ensemble. In this problem a certain discrete kernel, which we call the discrete Bessel kernel, plays an important role.
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 80 (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
Nonintersecting paths, random tilings and random matrices
 Probab. Theory Related Fields
, 2002
"... Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the s ..."
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Cited by 78 (8 self)
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Abstract. We investigate certain measures induced by families of nonintersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abchexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact can be obtained from nonintersecting Brownian motions. The derivations of the measures are based on the KarlinMcGregor or LindströmGesselViennot method. We use the measures to show some asymptotic results for the models. 1.
Limit theorems for height fluctuations in a class of discrete space and time growth models
 J. Statist. Phys
, 2001
"... We introduce a class of onedimensional discrete spacediscrete time stochastic growth models described by a height function h t(x) with corner initialization. We prove, with one exception, that the limiting distribution function of h t(x) (suitably centered and normalized) equals a Fredholm determi ..."
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Cited by 69 (9 self)
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We introduce a class of onedimensional discrete spacediscrete time stochastic growth models described by a height function h t(x) with corner initialization. We prove, with one exception, that the limiting distribution function of h t(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppalainen Johansson model. Hence some of our results are not new, but the proofs are.
A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices
 J. Statist. Phys
, 2002
"... Recently Johansson (21) and Johnstone (16) proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X g X(X t X) converges to the Tracy–Widom law as n, p (the dimensions of X) tend to. in some ratio n/p Q c>0.We extend these results in ..."
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Cited by 60 (3 self)
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Recently Johansson (21) and Johnstone (16) proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X g X(X t X) converges to the Tracy–Widom law as n, p (the dimensions of X) tend to. in some ratio n/p Q c>0.We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with nonGaussian entries, provided n − p=O(p 1/3). We also prove that l max [ (n 1/2 +p 1/2) 2 +O(p 1/2 log(p)) (a.e.) for general c>0. KEY WORDS: Sample covariance matrices; principal component; Tracy– Widom distribution.
Flux compactification
"... Contents We review recent work in which compactifications of string and M theory are constructed in which all scalar fields (moduli) are massive, and supersymmetry is broken with a small positive cosmological constant, features needed to reproduce real world physics. We explain how this work implies ..."
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Cited by 53 (2 self)
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Contents We review recent work in which compactifications of string and M theory are constructed in which all scalar fields (moduli) are massive, and supersymmetry is broken with a small positive cosmological constant, features needed to reproduce real world physics. We explain how this work implies that there is a “landscape ” of string/M theory vacua, perhaps containing many candidates
On the distributions of the lengths of the longest monotone subsequences in random words
"... We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant rep ..."
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Cited by 50 (9 self)
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We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k×k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N → ∞ limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k × k hermitian matrices of trace zero. I.