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 National Science
Foundation Award #0505680 |
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Asymptotic Problems in the Theory of Random Spectra |
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| Investigator(s): |
Brian Rider (PI)
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| Sponsor: |
University of Colorado at Boulder, CO 80309 3034926221
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| Start Date/Expiration Date |
2005-08-01 to 2006-07-31 (amended 2005-07-21) |
| Awarded Amount to Date: |
$28,068 |
| Abstract: The PI will investigate several problems concerning the spectra of random
matrices and random Schroedinger operators. Foremost, the PI will
continue his study of random matrices with no symmetry, establishing
fluctuation results for linear spectral statistics, limit theorems for the
spectral edge, and further investigating the connections between these
ensembles and roots of random polynomials. In the realm of Hermitian
random matrices, the PI will employ the Riemann-Hilbert Problem method to
investigate the behavior of the Janossy densities for large dimensional
'Unitary Ensembles'. The goal is to then use these results to establish
the speed of convergence of the largest eigenvalue distribution to its
limiting Tracy-Widom distribution. Finally, the PI will generalize his
recent results on the distribution of the ground state eigenvalue of a
one-dimensional periodic Schroedinger operator with White Noise potential
to a broader class of random potentials.
The random matrix models under study in this proposal have important
applications to such disparate areas as multivariate statistics (principal
component analysis), theoretical physics (energy levels and resonances of
quantum systems) and electrical engineering (signal processing and
wireless communication). Schroedinger operators with random potential on
the other hand have long been studied as fundmental models in disordered
solids. The PI brings to bear a variety of techniques from Probability
and Analysis to study the detailed behavior of basic classes of random
matrices and random Schroedinger operators in physically relevant limiting
regimes. An overall goal of this proposal is to uncover new
commonalities, or universal properties, of these models through asymptotic
analysis. |
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| NSF Org: |
DMS - Division of Mathematical Sciences |
| Award Number: |
0505680 |
| Award Instrument: |
Continuing grant |
| Program Manager: |
Wen C. Masters
DMS Division of Mathematical Sciences
MPS Directorate for Mathematical & Physical Sciences
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| NSF Program(s): |
PROBABILITY |
| Field Application(s): |
Other nsf.applications NEC |
| Program Reference Code(s): |
UNASSIGNED, 0000 |
| Program Element Code(s): |
1263 |
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