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75
On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
, 2010
"... In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) β-ensembles and their var ..."
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Cited by 36 (4 self)
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In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) β-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
Discrete gap probabilities and discrete Painlevé equations
- DUKE MATH J
, 2003
"... We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1-kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm ..."
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Cited by 29 (6 self)
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We prove that Fredholm determinants of the form det(1 − Ks), where Ks is the restriction of either the discrete Bessel kernel or the discrete 2F1-kernel to {s, s + 1,...}, can be expressed, respectively, through solutions of discrete Painlevé II (dPII) and Painlevé V (dPV) equations. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a z-measure, or as normalized Toeplitz determinants with symbols eη(ζ +ζ −1) and (1 + ξζ)
Painléve III and a singular linear statistics in Hermitian random matrix ensembles I.
, 2008
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Fredholm determinants, Jimbo-Miwa-Ueno tau-functions, and representation theory
, 2001
"... The authors show that a wide class of Fredholm determinants arising in the representation theory of “big ” groups such as the infinite–dimensional unitary group, solve Painlevé equations. Their methods are based on the theory of integrable operators and the theory of Riemann–Hilbert problems. ..."
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Cited by 23 (5 self)
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The authors show that a wide class of Fredholm determinants arising in the representation theory of “big ” groups such as the infinite–dimensional unitary group, solve Painlevé equations. Their methods are based on the theory of integrable operators and the theory of Riemann–Hilbert problems.
Increasing subsequences and the hard-to-soft-edge transition in matrix ensembles
- J. Phys. A
"... Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the ..."
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Cited by 23 (4 self)
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Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) ≤ l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains. 1
Recurrences for elliptic hypergeometric integrals
, 2006
"... In [15], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generali ..."
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Cited by 22 (7 self)
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In [15], the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.
Hermite and Laguerre β-ensembles: asymptotic corrections to the eigenvalue density
"... We consider Hermite and Laguerre β-ensembles of large N × N random matrices. For all β even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density whi ..."
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Cited by 21 (11 self)
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We consider Hermite and Laguerre β-ensembles of large N × N random matrices. For all β even, corrections to the limiting global density are obtained, and the limiting density at the soft edge is evaluated. We use the saddle point method on multidimensional integral representations of the density which are based on special realizations of the generalized (multivariate) classical orthogonal polynomials. The corrections to the bulk density are oscillatory terms that depends on β. At the edges, the density can be expressed as a multiple integral of the Konstevich type which constitutes a β-deformation of the Airy function. This allows us to obtain the main contribution to the soft edge density when the spectral parameter
Moduli spaces of d-connections and difference Painlevé equations
- DUKE MATH. J
, 2004
"... We show that difference Painlevé equations can be interpreted as isomorphisms of moduli spaces of d-connections on P1 with given singularity structure. We also derive a new difference equation that lifts to an isomorphism between A (1)∗ 2 –surfaces in Sakai’s classification [25]. It is the most gene ..."
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Cited by 19 (1 self)
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We show that difference Painlevé equations can be interpreted as isomorphisms of moduli spaces of d-connections on P1 with given singularity structure. We also derive a new difference equation that lifts to an isomorphism between A (1)∗ 2 –surfaces in Sakai’s classification [25]. It is the most general difference Painlevé equation known so far, and it degenerates to both difference Painlevé V and classical (differential) Painlevé VI equations.
Random matrix theory and entanglement in quantum spin chains
- Comm. Math. Phys
, 2004
"... We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entrop ..."
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Cited by 16 (1 self)
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We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first N spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as N → ∞. This is shown to grow logarithmically with N. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.
