Random-matrix theories in quantum physics: common concepts

Guhr, Thomas; Müller–Groeling, Axel; Weidenmüller, Hans A.

doi:10.1016/s0370-1573(97)00088-4

Cited by 2,209 publications

(2,927 citation statements)

References 707 publications

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“…This large number of elements (1 million) does not frighten a physicist with a computer. Eugene Wigner applied random matrix theory 50 years ago to interpret the complex spectrum of energy levels in nuclear physics (36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48). We do exactly the same thing and apply random matrix theory to the matrix C. We find that certain eigenvalues of that 1,000 ϫ 1,000 matrix deviate from the predictions of random matrix theory, which has not eigenvalues greater than an upper bound of Ϸ2.0.…”

Section: Cross-correlations In Price Fluctuations Of Different Stocksmentioning

confidence: 86%

Self-organized complexity in economics and finance

Stanley

Amaral

Buldyrev

et al. 2002

Proc. Natl. Acad. Sci. U.S.A.

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This article discusses some of the similarities between work being done by economists and by physicists seeking to contribute to economics. We also mention some of the differences in the approaches taken and seek to justify these different approaches by developing the argument that by approaching the same problem from different points of view, new results might emerge. In particular, we review two newly discovered scaling results that appear to be universal, in the sense that they hold for widely different economies as well as for different time periods: (i) the fluctuation of price changes of any stock market is characterized by a probability density function, which is a simple power law with exponent ؊4 extending over 10 2 SDs (a factor of 10 8 on the y axis); this result is analogous to the Gutenberg-Richter power law describing the histogram of earthquakes of a given strength; and (ii) for a wide range of economic organizations, the histogram shows how size of organization is inversely correlated to fluctuations in size with an exponent Ϸ0.2. Neither of these two new empirical laws has a firm theoretical foundation. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behavior of the response function at the critical point (zero magnetic field) leads to large fluctuations.

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Section: Cross-correlations In Price Fluctuations Of Different Stocksmentioning

confidence: 86%

Self-organized complexity in economics and finance

Stanley

Amaral

Buldyrev

et al. 2002

Proc. Natl. Acad. Sci. U.S.A.

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“…Today's applications cover all areas in Physics, and we refer to [1] for a review. However, many years before Wigner and Dyson, John Wishart had already introduced a random matrix ensemble in his studies on samples from a multivariate population.…”

Section: Introductionmentioning

confidence: 99%

Power law deformation of Wishart–Laguerre ensembles of random matrices

Akemann¹,

Vivo²

2008

J. Stat. Mech.

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We introduce a one-parameter deformation of the Wishart-Laguerre or chiral ensembles of positive definite random matrices with Dyson index β = 1, 2 and 4. Our generalised model has a fat-tailed distribution while preserving the invariance under orthogonal, unitary or symplectic transformations. The spectral properties are derived analytically for finite matrix size N × M for all three β, in terms of the orthogonal polynomials of the standard Wishart-Laguerre ensembles. For large-N in a certain double scaling limit we obtain a generalised Marčenko-Pastur distribution on the macroscopic scale, and a generalised Bessel-law at the hard edge which is shown to be universal. Both macroscopic and microscopic correlations exhibit power-law tails, where the microscopic limit depends on β and the difference M − N . In the limit where our parameter governing the power-law goes to infinity we recover the correlations of the Wishart-Laguerre ensembles. To illustrate these findings the generalised Marčenko-Pastur distribution is shown to be in very good agreement with empirical data from financial covariance matrices.

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“…For large N dot + N the distribution of wave-function components is asymptotically Gaussian [48,54]:…”

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confidence: 99%

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

2017

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The Sachdev-Ye-Kitaev (SYK) model describes a collection of randomly interacting Majorana fermions that exhibits profound connections to quantum chaos and black holes. We propose a solid-state implementation based on a quantum dot coupled to an array of topological superconducting wires hosting Majorana zero modes. Interactions and disorder intrinsic to the dot mediate the desired random Majorana couplings, while an approximate symmetry suppresses additional unwanted terms. We use random-matrix theory and numerics to show that our setup emulates the SYK model (up to corrections that we quantify) and discuss experimental signatures. DOI: 10.1103/PhysRevB.96.121119 Introduction. Majorana fermions provide building blocks for many novel phenomena. As one notable example, Majorana-fermion zero modes [1,2] capture the essence of non-Abelian statistics and topological quantum computation [3,4], and correspondingly now form the centerpiece of a vibrant experimental effort [5][6][7][8][9][10][11][12][13][14][15][16]. More recently, randomly interacting Majorana fermions governed by the "Sachdev-YeKitaev (SYK) model" [17][18][19] were shown to exhibit sharp connections to chaos, quantum-information scrambling, and black holes-naturally igniting broad interdisciplinary activity (see, e.g., [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]). The goal of this Rapid Communication is to exploit hardware components of a Majorana-based topological quantum computer for a tabletop implementation of the SYK model, thus uniting these very different topics.The SYK Hamiltonian reads

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Random-matrix theories in quantum physics: common concepts

Cited by 2,209 publications

References 707 publications

Self-organized complexity in economics and finance

Self-organized complexity in economics and finance

Power law deformation of Wishart–Laguerre ensembles of random matrices

Approximating the Sachdev-Ye-Kitaev model with Majorana wires

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