Project A4: Asymptotics of spectral distributions
Summary:
The main focus of this project is the investigation of asymptotic distributions of eigenvalues and eigenvectors of matrices of high dimensional random matrix ensembles as well as spectral distributions of infinite dimensional operators in free probability theory. Another focus is the connection of matrix-valued stochastic processes and their induced spectral processes to representation theory and the limits of related combinatorial structures like Young diagrams and partitions. Furthermore, representation theoretic methods will be used to compute asymptotic approximations to higher correlations of characteristic polynomials. The limiting local and global distributions of eigenvalues appearing in this context are often universal and appear as limiting objects in various contexts of mathematics as well as mathematical physics. A rather incomplete list contains representation theory, asymptotic combinatorics, nuclear growth models in probability, free probability and operator algebras, determinantal point processes, integrable systems as well as the correlations of zeros of L-functions. These similarities ask for an explanation in a more general framework. In this project, we intend to concentrate on some of these connections, connecting probability, algebraic combinatorics and complex analysis. In cooperation with a number of other projects of the CRC we hope to advance the understanding of these surprising connections between different fields.
Links:
Probability Theory and Mathematical Statistics
Recent Preprints:
16049
Friedrich Götze, Alexey Naumov, Alexander Tikhomirov, Dimitry Timushev PDF
On the local semicircular law for Wigner ensembles
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A4
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On the local semicircular law for Wigner ensembles
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16048
Friedrich Götze, Holger Sambale PDF
Second order concentration via logarithmic Sobolev inequalities
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A4
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Second order concentration via logarithmic Sobolev inequalities
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16047
Sergey Bobkov, Gennadiy P. Chistyakov, Friedrich Götze PDF
Rényi divergence and the central limit theorem
Project:
A4
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Rényi divergence and the central limit theorem
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16045
Mario Kieburg, Holger Kösters PDF
Products of random matrices from polynomial ensembles
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A4
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Products of random matrices from polynomial ensembles
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16044
Mario Kieburg, Holger Kösters PDF
Exact relation between singular value and eigenvalue statistics
Project:
A4
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Exact relation between singular value and eigenvalue statistics
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16006
Zhi Shui Hu, Vladimir Ulyanov, Qun Qiang Feng PDF
Limit theorems for number of edges in the generalized random graphs with random vertex weights
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A4
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Limit theorems for number of edges in the generalized random graphs with random vertex weights
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15106
Thomas Kriecherbauer, Martin Venker PDF
Edge Statistics for a Class of Repulsive Particle Systems
Project:
A4
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Edge Statistics for a Class of Repulsive Particle Systems
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15105
Kristina Schubert, Martin Venker PDF
Empirical Spacings of Unfolded Eigenvalues
Project:
A4
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Empirical Spacings of Unfolded Eigenvalues
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15104
Holger Kösters, Alexander Tikhomirov PDF
Limiting Spectral Distributions of Sums of Products of non-Hermitian Random Matrices
Project:
A4
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Limiting Spectral Distributions of Sums of Products of non-Hermitian Random Matrices
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15087
Friedrich Götze, Anna Reshetenko PDF
Asymptotic Expansions in Free Limit Theorems
Project:
A4
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Asymptotic Expansions in Free Limit Theorems
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