The Experts below are selected from a list of 7266 Experts worldwide ranked by ideXlab platform
Jack H Koolen - One of the best experts on this subject based on the ideXlab platform.
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augmenting the delsarte bound a forbidden interval for the order of maximal cliques in strongly regular graphs
European Journal of Combinatorics, 2021Co-Authors: Gary R W Greaves, Jack H Koolen, Jongyook ParkAbstract:Abstract In this paper, we study the order of a maximal clique in an amply regular graph with a fixed Smallest Eigenvalue by considering a vertex that is adjacent to some (but not all) vertices of the maximal clique. As a consequence, we show that if a strongly regular graph contains a Delsarte clique, then the parameter μ is either small or large. Furthermore, we obtain a cubic polynomial that assures that a maximal clique in an amply regular graph is either small or large (under certain assumptions). Combining this cubic polynomial with the claw-bound, we rule out an infinite family of feasible parameters ( v , k , λ , μ ) for strongly regular graphs. Lastly, we provide tables of parameters ( v , k , λ , μ ) for nonexistent strongly regular graphs with Smallest Eigenvalue − 4 , − 5 , − 6 or − 7 .
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sesqui regular graphs with Smallest Eigenvalue at least 3
arXiv: Combinatorics, 2021Co-Authors: Qianqian Yang, Brhane Gebremichel, Masood Ur Rehman, Jae Young Yang, Jack H KoolenAbstract:Koolen et al. showed that if a graph with Smallest Eigenvalue at least $-3$ has large minimal valency, then it is $2$-integrable. In this paper, we will focus on the sesqui-regular graphs with Smallest Eigenvalue at least $-3$ and study their integrability.
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recent progress on graphs with fixed Smallest Eigenvalue
arXiv: Combinatorics, 2020Co-Authors: Jack H Koolen, Mengyue Cao, Qianqian YangAbstract:We give a survey on graphs with fixed Smallest Eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed Smallest Eigenvalue and large minimal valency; (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed Smallest Eigenvalue and the characterizations of certain families of distance-regular graphs. At the end of the survey, we also discuss signed graphs with fixed Smallest Eigenvalue and present some new findings.
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problems on graphs with fixed Smallest Eigenvalue
Algebra Colloquium, 2020Co-Authors: Jack H Koolen, Qianqian YangAbstract:In this note we give several problems and conjectures on graphs with fixed Smallest Eigenvalue.
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non bipartite distance regular graphs with a small Smallest Eigenvalue
Electronic Journal of Combinatorics, 2019Co-Authors: Zhi Qiao, Yifan Jing, Jack H KoolenAbstract:In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $\theta_{\min}\leq -\alpha k$, where $0<\alpha<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $\theta_{\min}$ compared with $k$. In particular, we will show that if $\theta_{\min}$ is relatively close to $-k$, then the odd girth $g$ must be large. Also we will classify the non-bipartite distance-regular graphs with $\theta_{\min} \leq \frac{D-1}{D}$ for $D =4,5$.
Tim Wirtz - One of the best experts on this subject based on the ideXlab platform.
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the Smallest Eigenvalue distribution in the real wishart laguerre ensemble with even topology
arXiv: Mathematical Physics, 2015Co-Authors: Tim Wirtz, Thomas Guhr, Gernot Akemann, Mario Kieburg, R WegnerAbstract:We consider rectangular random matrices of size $p\times n$ belonging to the real Wishart-Laguerre ensemble also known as the chiral Gaussian orthogonal ensemble. This ensemble appears in many applications like QCD, mesoscopic physics, and time series analysis. We are particularly interested in the distribution of the Smallest non-zero Eigenvalue and the gap probability to find no Eigenvalue in an interval $[0,t]$. While for odd topology $\nu=n-p$ explicit closed results are known for finite and infinite matrix size, for even $\nu>2$ only recursive expressions in $p$ are available.The Smallest Eigenvalue distribution as well as the gap probability for general even $\nu$ is equivalent to expectation values of characteristic polynomials raised to a half-integer. The computation of such averages is done via a combination of skew-orthogonal polynomials and bosonisation methods. The results are given in terms of Pfaffian determinants both at finite $p$ and in the hard edge scaling limit ($p\to\infty$ and $\nu$ fixed) for an arbitrary even topology $\nu$. Numerical simulations for the correlated Wishart ensemble illustrate the universality of our results in this particular limit. These simulations point to a validity of the hard edge scaling limit beyond the invariant case.
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completing the picture for the Smallest Eigenvalue of real wishart matrices
Physical Review Letters, 2014Co-Authors: Gernot Akemann, Thomas Guhr, Mario Kieburg, R Wegner, Tim WirtzAbstract:Rectangular real $N\ifmmode\times\else\texttimes\fi{}(N+\ensuremath{\nu})$ matrices $W$ with a Gaussian distribution appear very frequently in data analysis, condensed matter physics, and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of $W{W}^{T}$. The extreme Eigenvalues of $W{W}^{T}$ are of particular interest. We explicitly compute the distribution and the gap probability of the Smallest nonzero Eigenvalue in this ensemble, both for arbitrary fixed $N$ and $\ensuremath{\nu}$, and in the universal large $N$ limit with $\ensuremath{\nu}$ fixed. We uncover an integrable Pfaffian structure valid for all even values of $\ensuremath{\nu}\ensuremath{\ge}0$. This extends previous results for odd $\ensuremath{\nu}$ at infinite $N$ and recursive results for finite $N$ and for all $\ensuremath{\nu}$. Our mathematical results include the computation of expectation values of half-integer powers of characteristic polynomials.
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distribution of the Smallest Eigenvalue in complex and real correlated wishart ensembles
Journal of Physics A, 2014Co-Authors: Tim Wirtz, Thomas GuhrAbstract:For the correlated Gaussian–Wishart ensemble we compute the distribution of the Smallest Eigenvalue and a related gap probability. We obtain exact results for the complex (β = 2) and for the real case (β = 1). For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity. We calculate the asymptotic and universal results for the gap probability and the distribution of the Smallest Eigenvalue. We use the supersymmetry method, in particular the generalized Hubbard–Stratonovich transformation and superbosonization.
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distribution of the Smallest Eigenvalue in complex and real correlated wishart ensembles
arXiv: Mathematical Physics, 2013Co-Authors: Tim Wirtz, Thomas GuhrAbstract:For the correlated Gaussian Wishart ensemble we compute the distribution of the Smallest Eigenvalue and a related gap probability.We obtain exact results for the complex (\beta=2) and for the real case (\beta=1). For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity. We calculate the asymptotic and universal results for the gap probability and the distribution of the Smallest Eigenvalue. We use the Supersymmetry method, in particular the generalized Hubbard-Stratonovich transformation and superbosonization.
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distribution of the Smallest Eigenvalue in the correlated wishart model
Physical Review Letters, 2013Co-Authors: Tim Wirtz, Thomas GuhrAbstract:Wishart random matrix theory is of major importance for the analysis of correlated time series. The distribution of the Smallest Eigenvalue for Wishart correlation matrices is particularly interesting in many applications. In the complex and in the real case, we calculate it exactly for arbitrary empirical Eigenvalues, i.e., for fully correlated Gaussian Wishart ensembles. To this end, we derive certain dualities of matrix models in ordinary space. We thereby completely avoid the otherwise unsurmountable problem of computing a highly nontrivial group integral. Our results are compact and much easier to handle than previous ones. Furthermore, we obtain a new universality for the distribution of the Smallest Eigenvalue on the proper local scale.
Thomas Guhr - One of the best experts on this subject based on the ideXlab platform.
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the Smallest Eigenvalue distribution in the real wishart laguerre ensemble with even topology
arXiv: Mathematical Physics, 2015Co-Authors: Tim Wirtz, Thomas Guhr, Gernot Akemann, Mario Kieburg, R WegnerAbstract:We consider rectangular random matrices of size $p\times n$ belonging to the real Wishart-Laguerre ensemble also known as the chiral Gaussian orthogonal ensemble. This ensemble appears in many applications like QCD, mesoscopic physics, and time series analysis. We are particularly interested in the distribution of the Smallest non-zero Eigenvalue and the gap probability to find no Eigenvalue in an interval $[0,t]$. While for odd topology $\nu=n-p$ explicit closed results are known for finite and infinite matrix size, for even $\nu>2$ only recursive expressions in $p$ are available.The Smallest Eigenvalue distribution as well as the gap probability for general even $\nu$ is equivalent to expectation values of characteristic polynomials raised to a half-integer. The computation of such averages is done via a combination of skew-orthogonal polynomials and bosonisation methods. The results are given in terms of Pfaffian determinants both at finite $p$ and in the hard edge scaling limit ($p\to\infty$ and $\nu$ fixed) for an arbitrary even topology $\nu$. Numerical simulations for the correlated Wishart ensemble illustrate the universality of our results in this particular limit. These simulations point to a validity of the hard edge scaling limit beyond the invariant case.
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completing the picture for the Smallest Eigenvalue of real wishart matrices
Physical Review Letters, 2014Co-Authors: Gernot Akemann, Thomas Guhr, Mario Kieburg, R Wegner, Tim WirtzAbstract:Rectangular real $N\ifmmode\times\else\texttimes\fi{}(N+\ensuremath{\nu})$ matrices $W$ with a Gaussian distribution appear very frequently in data analysis, condensed matter physics, and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of $W{W}^{T}$. The extreme Eigenvalues of $W{W}^{T}$ are of particular interest. We explicitly compute the distribution and the gap probability of the Smallest nonzero Eigenvalue in this ensemble, both for arbitrary fixed $N$ and $\ensuremath{\nu}$, and in the universal large $N$ limit with $\ensuremath{\nu}$ fixed. We uncover an integrable Pfaffian structure valid for all even values of $\ensuremath{\nu}\ensuremath{\ge}0$. This extends previous results for odd $\ensuremath{\nu}$ at infinite $N$ and recursive results for finite $N$ and for all $\ensuremath{\nu}$. Our mathematical results include the computation of expectation values of half-integer powers of characteristic polynomials.
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distribution of the Smallest Eigenvalue in complex and real correlated wishart ensembles
Journal of Physics A, 2014Co-Authors: Tim Wirtz, Thomas GuhrAbstract:For the correlated Gaussian–Wishart ensemble we compute the distribution of the Smallest Eigenvalue and a related gap probability. We obtain exact results for the complex (β = 2) and for the real case (β = 1). For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity. We calculate the asymptotic and universal results for the gap probability and the distribution of the Smallest Eigenvalue. We use the supersymmetry method, in particular the generalized Hubbard–Stratonovich transformation and superbosonization.
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distribution of the Smallest Eigenvalue in complex and real correlated wishart ensembles
arXiv: Mathematical Physics, 2013Co-Authors: Tim Wirtz, Thomas GuhrAbstract:For the correlated Gaussian Wishart ensemble we compute the distribution of the Smallest Eigenvalue and a related gap probability.We obtain exact results for the complex (\beta=2) and for the real case (\beta=1). For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity. We calculate the asymptotic and universal results for the gap probability and the distribution of the Smallest Eigenvalue. We use the Supersymmetry method, in particular the generalized Hubbard-Stratonovich transformation and superbosonization.
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distribution of the Smallest Eigenvalue in the correlated wishart model
Physical Review Letters, 2013Co-Authors: Tim Wirtz, Thomas GuhrAbstract:Wishart random matrix theory is of major importance for the analysis of correlated time series. The distribution of the Smallest Eigenvalue for Wishart correlation matrices is particularly interesting in many applications. In the complex and in the real case, we calculate it exactly for arbitrary empirical Eigenvalues, i.e., for fully correlated Gaussian Wishart ensembles. To this end, we derive certain dualities of matrix models in ordinary space. We thereby completely avoid the otherwise unsurmountable problem of computing a highly nontrivial group integral. Our results are compact and much easier to handle than previous ones. Furthermore, we obtain a new universality for the distribution of the Smallest Eigenvalue on the proper local scale.
Monique Barel - One of the best experts on this subject based on the ideXlab platform.
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a schur based algorithm for computing bounds to the Smallest Eigenvalue of a symmetric positive definite toeplitz matrix
Linear Algebra and its Applications, 2008Co-Authors: Nicola Mastronardi, Monique Barel, Raf VandebrilAbstract:Abstract Recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the Smallest Eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. They compute the Smallest Eigenvalue in an iterative fashion, many of them relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the Smallest singular value of upper triangular matrices, respectively, an algorithm for computing bounds to the Smallest Eigenvalue of a symmetric positive definite Toeplitz matrix is derived. The algorithm relies on the computation of the R factor of the QR -factorization of the Toeplitz matrix and the inverse of R . The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm.
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Computing a Lower Bound of the Smallest Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix
IEEE Transactions on Information Theory, 2008Co-Authors: Teresa Laudadio, Nicola Mastronardi, Monique BarelAbstract:In this correspondence, several algorithms to compute a lower bound of the Smallest Eigenvalue of a symmetric positive-definite Toeplitz matrix are described and compared in terms of accuracy and computational efficiency. Exploiting the Toeplitz structure of the considered matrix, new theoretical insights are derived and an efficient implementation of some of the aforementioned algorithms is provided.
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A Schur-based algorithm for computing the Smallest Eigenvalue of a symmetric positive definite Toeplitz matrix
2006Co-Authors: Nicola Mastronardi, Monique Barel, Raf VandebrilAbstract:Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the Smallest Eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. Many of them compute the Smallest Eigenvalue in an iterative fashion, relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the Smallest singular value of upper triangular matrices, respectively, an algorithm for computing the Smallest Eigenvalue of a symmetric positive definite Toeplitz matrix is derived. The algorithm relies on the computation of the R factor of the QR–factorization of the Toeplitz matrix and the inverse of R. The latter computation is efficiently accomplished by the generalized Schur algorithm.
Gernot Akemann - One of the best experts on this subject based on the ideXlab platform.
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the Smallest Eigenvalue distribution in the real wishart laguerre ensemble with even topology
arXiv: Mathematical Physics, 2015Co-Authors: Tim Wirtz, Thomas Guhr, Gernot Akemann, Mario Kieburg, R WegnerAbstract:We consider rectangular random matrices of size $p\times n$ belonging to the real Wishart-Laguerre ensemble also known as the chiral Gaussian orthogonal ensemble. This ensemble appears in many applications like QCD, mesoscopic physics, and time series analysis. We are particularly interested in the distribution of the Smallest non-zero Eigenvalue and the gap probability to find no Eigenvalue in an interval $[0,t]$. While for odd topology $\nu=n-p$ explicit closed results are known for finite and infinite matrix size, for even $\nu>2$ only recursive expressions in $p$ are available.The Smallest Eigenvalue distribution as well as the gap probability for general even $\nu$ is equivalent to expectation values of characteristic polynomials raised to a half-integer. The computation of such averages is done via a combination of skew-orthogonal polynomials and bosonisation methods. The results are given in terms of Pfaffian determinants both at finite $p$ and in the hard edge scaling limit ($p\to\infty$ and $\nu$ fixed) for an arbitrary even topology $\nu$. Numerical simulations for the correlated Wishart ensemble illustrate the universality of our results in this particular limit. These simulations point to a validity of the hard edge scaling limit beyond the invariant case.
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completing the picture for the Smallest Eigenvalue of real wishart matrices
Physical Review Letters, 2014Co-Authors: Gernot Akemann, Thomas Guhr, Mario Kieburg, R Wegner, Tim WirtzAbstract:Rectangular real $N\ifmmode\times\else\texttimes\fi{}(N+\ensuremath{\nu})$ matrices $W$ with a Gaussian distribution appear very frequently in data analysis, condensed matter physics, and quantum field theory. A central question concerns the correlations encoded in the spectral statistics of $W{W}^{T}$. The extreme Eigenvalues of $W{W}^{T}$ are of particular interest. We explicitly compute the distribution and the gap probability of the Smallest nonzero Eigenvalue in this ensemble, both for arbitrary fixed $N$ and $\ensuremath{\nu}$, and in the universal large $N$ limit with $\ensuremath{\nu}$ fixed. We uncover an integrable Pfaffian structure valid for all even values of $\ensuremath{\nu}\ensuremath{\ge}0$. This extends previous results for odd $\ensuremath{\nu}$ at infinite $N$ and recursive results for finite $N$ and for all $\ensuremath{\nu}$. Our mathematical results include the computation of expectation values of half-integer powers of characteristic polynomials.
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compact Smallest Eigenvalue expressions in wishart laguerre ensembles with or without a fixed trace
Journal of Statistical Mechanics: Theory and Experiment, 2011Co-Authors: Gernot Akemann, Pierpaolo VivoAbstract:The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt Eigenvalues. For a bipartition of size M ≥ N, these are distributed according to a Wishart–Laguerre ensemble (WL) of random matrices of size N × M, with a fixed-trace constraint. We first compute the distribution and moments of the Smallest Eigenvalue in the fixed-trace orthogonal WL ensemble for arbitrary M ≥ N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble given by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large N limit with M − N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.
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compact Smallest Eigenvalue expressions in wishart laguerre ensembles with or without fixed trace
arXiv: Mathematical Physics, 2011Co-Authors: Gernot Akemann, Pierpaolo VivoAbstract:The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt Eigenvalues. For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint. We first compute the distribution and moments of the Smallest Eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.
