Closest Spacing of Eigenvalues

Mathematics – Spectral Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

This is my Ph.D. thesis from 2001. It is posted here for archival value, and has not been updated to include more recent work

Scientific paper

We study the distribution of the minimum spacing between eigenvalues of a random n by n unitary matrix. The minimum spacing scales as $n^{-4/3}$, not $n^{-2}$ as would be the case for n independent points on the unit circle, illustrating the well known phenomenon that the eigenvalues of random matrices 'repel each other'. We derive the distribution for the rescaled minimum spacing in the limit as $n\to\infty$. To find the minimum spacing, we count the number of eigenvalue pairs closer than $xn^{-4/3}$. We use heuristics to guess that this integer-valued random variable is Poisson, calculate the actual moments of the limiting distribution, and find that the actual moments match those of the guess. The matching moments prove that the heuristic guess is correct, and lead directly to the main result. We prove analogous results for the Gaussian unitary ensemble (GUE) and, with restrictions, a universal class of unitary ensembles (UUE) studied by Deift, Kreicherbauer, McLaughlin, Venakides, and Zhou.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Closest Spacing of Eigenvalues does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Closest Spacing of Eigenvalues, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Closest Spacing of Eigenvalues will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-139706

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.