Results 1 - 10
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36
A pedestrian’s view on interacting particle systems, KPZ universality, and random matrices
, 2010
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Numerical solution of Riemann–Hilbert problems
- Painlevé II. Found. Comput. Math
, 2011
"... In recent developments, a general approach for solving Riemann–Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with an approach to compute Fredholm dete ..."
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Cited by 15 (6 self)
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In recent developments, a general approach for solving Riemann–Hilbert problems numerically has been developed. We review this numerical framework, and apply it to the calculation of orthogonal polynomials on the real line. Combining this numerical algorithm with an approach to compute Fredholm determinants, we are able to calcu-late level densities and gap statistics for general finite-dimensional unitary ensembles. We also include a description of how to compute the Hastings–McLeod solution of the homogeneous Painleve ́ II equation. 1
On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart Matrix, in preparation
, 2010
"... The ratio of the largest eigenvalue divided by the trace of a p × p random Wishart matrix with n degrees of freedom and identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate expli ..."
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Cited by 13 (2 self)
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The ratio of the largest eigenvalue divided by the trace of a p × p random Wishart matrix with n degrees of freedom and identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p, n → ∞ with p/n → c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p, n. 1
Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals
- Foundations of Computational Mathematics
, 2011
"... Abstract. High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of conver-gence is equal, numerical stability strongly depends ..."
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Cited by 13 (3 self)
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Abstract. High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of conver-gence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that mini-mizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman–Valiron and Levin– Pfluger theory of entire functions, and by the saddle-point method of asymptotic
Accuracy of the Tracy–Widom limits for the extreme eigenvalues in white Wishart matrices
- BERNOULLI
, 2012
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Endpoint distribution of directed polymers in 1+1 dimensions
, 2012
"... We give an explicit formula for the joint density of the max and argmax of the Airy2 process minus a parabola. The argmax has a universal distribution which governs the rescaled endpoint for large time or temperature of directed polymers in 1 + 1 dimensions. ..."
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Cited by 9 (2 self)
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We give an explicit formula for the joint density of the max and argmax of the Airy2 process minus a parabola. The argmax has a universal distribution which governs the rescaled endpoint for large time or temperature of directed polymers in 1 + 1 dimensions.
Asymptotics of eigenbased collaborative sensing
- Proc. IEEE Information Theory Workshop (ITW 2009
, 2009
"... Abstract—In this contribution, we propose a new technique for collaborative sensing based on the analysis of the normalized (by the trace) largest eigenvalues of the sample covariance matrix. Assuming that several base stations are cooperating and without the knowledge of the noise variance, the tes ..."
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Cited by 5 (0 self)
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Abstract—In this contribution, we propose a new technique for collaborative sensing based on the analysis of the normalized (by the trace) largest eigenvalues of the sample covariance matrix. Assuming that several base stations are cooperating and without the knowledge of the noise variance, the test is able to determine the presence of mobile users in a network when only few samples are available. Unlike previous heuristic techniques, we show that the test has roots within the Generalized Likelihood Ratio Test and provide an asymptotic random matrix analysis enabling to determine adequate threshold detection values (probability of false alarm). Simulations sustain our theoretical claims. I.
