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1.
Scaling models of randomN×N hermitian matrices and passing to the limitN→∞ leads to integral operators whose Fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order. For the Gaussian Unitary Ensemble, and for many others'as well, the kernel one obtains by scaling in the “bulk” of the spectrum is the “sine kernel” $\frac{{\sin \pi (x - y)}}{{\pi (x - y)}}$ . Rescaling the GUE at the “edge” of the spectrum leads to the kernel $\frac{{Ai(x)Ai'(y) - Ai'(x)Ai(y)}}{{x - y}}$ , where Ai is the Airy function. In previous work we found several analogies between properties of this “Airy kernel” and known properties of the sine kernel: a system of partial differential equations associated with the logarithmic differential of the Fredholm determinant when the underlying domain is a union of intervals; a representation of the Fredholm determinant in terms of a Painlevé transcendent in the case of a single interval; and, also in this case, asymptotic expansions for these determinants and related quantities, achieved with the help of a differential operator which commutes with the integral operator. In this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the Laguerre or Jacobi ensembles at the edge of the spectrum, namely $$\frac{{J_\alpha (\sqrt x )\sqrt y J'_\alpha (\sqrt y ) - \sqrt x J'_\alpha (\sqrt x )J_\alpha (\sqrt y )}}{{2(x - y)}},$$ , whereJ α(z) is the Bessel function of order α. In the cases α=?1/2 these become, after a variable change, the kernels which arise when taking scaling limits in the bulk of the spectrum for the Gaussian orthogonal and symplectic ensembles. In particular, an asymptotic expansion we derive will generalize ones found by Dyson for the Fredholm determinants of these kernels. 相似文献
2.
The calculation of correlation functions for β=1 random matrix ensembles, which can be carried out using Pfaffians, has the peculiar feature of requiring a separate calculation
depending on the parity of the matrix size N. This same complication is present in the calculation of the correlations for the Ginibre Orthogonal Ensemble of real Gaussian
matrices. In fact the methods used to compute the β=1, N odd, correlations break down in the case of N odd real Ginibre matrices, necessitating a new approach to both problems. The new approach taken in this work is to deduce
the β=1, N odd correlations as limiting cases of their N even counterparts, when one of the particles is removed towards infinity. This method is shown to yield the correlations
for N odd real Gaussian matrices.
The work of P.J.F. was supported by the Australian Research Council, and A.M. was supported by an Australian Postgraduate
Award. 相似文献
3.
By scientific standards, the accuracy of short-term economic forecasts has been poor, and shows no sign of improving over time. We form a delay matrix of time-series data on the overall rate of growth of the economy, with lags spanning the period over which any regularity of behaviour is postulated by economists to exist. We use methods of random matrix theory to analyse the correlation matrix of the delay matrix. This is done for annual data from 1871 to 1994 for 17 economies, and for post-war quarterly data for the US and the UK. The properties of the eigenvalues and eigenvectors of these correlation matrices are similar, though not identical, to those implied by random matrix theory. This suggests that the genuine information content in economic growth data is low, and so forecasting failure arises from inherent properties of the data. 相似文献
4.
Marcel Novaes 《Annals of Physics》2011,326(4):828-838
We obtain explicit expressions for positive integer moments of the probability density of eigenvalues of the Jacobi and Laguerre random matrix ensembles, in the asymptotic regime of large dimension. These densities are closely related to the Selberg and Selberg-like multidimensional integrals. Our method of solution is combinatorial: it consists in the enumeration of certain classes of lattice paths associated to the solution of recurrence relations. 相似文献
5.
Sheldon Goldstein Joel L. Lebowitz Roderich Tumulka Nino Zanghì 《Journal of statistical physics》2006,125(5-6):1193-1221
For a quantum system, a density matrix ρ that is not pure can arise, via averaging, from a distribution μ of its wave function, a normalized vector belonging to its Hilbert space ?. While ρ itself does not determine a unique μ, additional facts, such as that the system has come to thermal equilibrium, might. It is thus not unreasonable to ask, which μ, if any, corresponds to a given thermodynamic ensemble? To answer this question we construct, for any given density matrix ρ, a natural measure on the unit sphere in ?, denoted GAP(ρ). We do this using a suitable projection of the Gaussian measure on ? with covariance ρ. We establish some nice properties of GAP(ρ) and show that this measure arises naturally when considering macroscopic systems. In particular, we argue that it is the most appropriate choice for systems in thermal equilibrium, described by the canonical ensemble density matrix ρβ = (1/Z) exp (?β H). GAP(ρ) may also be relevant to quantum chaos and to the stochastic evolution of open quantum systems, where distributions on ? are often used. 相似文献
6.
We show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved Bernoulli measures , for the golden mean. In this first part we establish some large-deviation results for random products of matrices, using perturbation theory of quasicompact operators. 相似文献
7.
P. J. Forrester 《Journal of statistical physics》1988,51(3-4):457-479
The Coulomb system consisting of an equal number of positive and negative charged rods confined to a one-dimensional lattice is studied. The grand partition function can be calculated exactly at two values of the coupling constant=q
2/k
B
T (q denoting the magnitude of the charges). The exact results lead to the conjecture that in the complex scaled fugacity plane, all the zeros of the grand partition function lie on the negative real axis for<2, on the point=–1 for=2, and on the unit circle for>2. In addition, for>4, we conjecture in general and prove at=4 that the zeros pinch the real axis in the thermodynamic limit, with an essential singularity in the pressure at the reduced density 1/2. 相似文献
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9.
In this paper the transport of quantum particles in time-dependent random media is studied. In the white noise limit, a quantum
model for collisions is obtained. At the level of Wigner equation, this limit is described by a linear Wigner-Boltzmann equation.
AMS subject classifications: 35Q40, 35S10, 81Q99, 81V99
á Fredo. Frédéric Poupaud deceased October 13th 2004.
This research was partially supported by the EU financed network IHP-HPRN-CT-2002-00282 and by MCYT (Spain), Proyecto BFM2002–00831. 相似文献
10.
11.
Anna Porzio 《Journal of statistical physics》1998,91(1-2):17-29
In previous work we developed a thermodynamic formalism for the Bernoulli convolution associated with the golden mean, and we obtained by perturbative analysis the existence, regularity, and strict convexity of the pressure F() in a neighborhood of =0. This gives the existence of a multifractal spectrum f() in a neighborhood of the almost sure value =f()=0, 9957.... In the present paper, by a direct study of the Ruelle–Perron–Frobenius operator associated with the random unbounded matrix product arising in our problem, we can prove the regularity of the pressure F() for (at least) (–1/2,+). This yields the interval of the singularity spectrum between the minimal value of the dimension of v, min=0.94042..., and the almost sure value, a.s.=0.9957.... 相似文献
12.
We show how the formalism developed in a previous paper allows us to exhibit the multifractal nature of the infinitely convolved Bernoulli measures for the golden mean. In this second part we show how the Hausdorff dimension of the set where the measure has a power law singularity of strength is related to the large-deviation function given in Part I. 相似文献
13.
We investigate the parameter dynamics of eigenvalues of Hamiltonians (‘level dynamics’) defined on symmetric spaces relevant to condensed matter and particle physics. In particular we: (1) identify the appropriate reduced manifold on which the motion takes place, (2) identify the correct Poisson structure ensuring the Hamiltonian character of the reduced dynamics, (3) determine the canonical measure on the reduced space, (4) calculate the resulting eigenvalue density. 相似文献
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15.
Construction of Laguerre polynomial's photon-added squeezing vacuum state and its quantum properties 下载免费PDF全文
Laguerre polynomial's photon-added squeezing vacuum state is constructed by operation of Laguerre polynomial's photon-added operator on squeezing vacuum state. By making use of the technique of integration within an ordered product of operators, we derive the normalization coefficient and the calculation expression of (a^1a^+). Its statistical properties, such as squeezing, the anti-bunching effect, the sub-Poissonian distribution property, the negativity of Wigner function, etc., are investigated. The influences of the squeezing parameter on quantum properties are discussed. Numerical results show that,firstly, the squeezing effect of the 1-order Laguerre polynomial's photon-added operator exciting squeezing vacuum state is strengthened, but its anti-bunching effect and sub-Poissonian statistical property are weakened with increasing squeezing parameter;secondly, its squeezing effect is similar to that of squeezing vacuum state, but its anti-bunching effect and subPoissonian distribution property are stronger than that of squeezing vacuum state. These results show that the operation of Laguerre polynomial's photon-added operator on squeezing vacuum state can enhance its non-classical properties. 相似文献
16.
For the unitary ensembles of N×N Hermitian matrices associated with a weight function w there is a kernel, expressible in terms of the polynomials orthogonal with respect to the weight function, which plays an important role. For the orthogonal and symplectic ensembles of Hermitian matrices there are 2×2 matrix kernels, usually constructed using skew-orthogonal polynomials, which play an analogous role. These matrix kernels are determined by their upper left-hand entries. We derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles. We also show that whenever w/w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w/w. General formulas are obtained for these extra terms. We do not use skew-orthogonal polynomials in the derivations 相似文献
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18.
提出量子力学算符Hermite多项式方法,即将若干常用的特殊函数的宗量由普通数变为算符,并用它来发现涉及Hermite多项式(单变数和双变数)的二项式定理和涉及Laguerre多项式的负二项式定理,它们在计算若干量子光场的物理性质时有实质性的应用. 该方法不但具有简捷的优点,而且能导出很多新的算符恒等式,成为发展数学物理理论的一个重要分支.
关键词:
量子力学
Hermite多项式
二项式定理
Laguerre多项式 相似文献
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20.
We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises
in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to
an ensemble of random matrices. 相似文献