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Eigenvalues of random normal matrices Random normal matrix model: - - PowerPoint PPT Presentation
Eigenvalues of random normal matrices Random normal matrix model: - - PowerPoint PPT Presentation
Eigenvalues of random normal matrices Random normal matrix model: Droplets Example: concentric ellipses Local droplets 1 0.5 -1.5 -1 -0.5 0.5 -0.5 -1 hypotrochoids deltoid Definition: K is a local droplet if Hele-Shaw flow 1 0.5
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Example:
Droplets
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concentric ellipses
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- 1.5
- 1
- 0.5
0.5
- 1
- 0.5
0.5 1
Local droplets
hypotrochoids deltoid Definition: K is a local droplet if
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Hele-Shaw flow
- 1.5
- 1
- 0.5
0.5
- 1
- 0.5
0.5 1
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S-function and local droplets
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Two pieces of the dynamics
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?-function as conformal welding (mating)
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centaurs
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Examples of QDs
bounded
unbounded
d=0 d=1 d=1 limacons Neumann ovals wings d=2 inversion
bounded
. . . . .
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Example: 7 cardioids in ellipse
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Connectivity bounds
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Examples
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Comments:
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Bers slice for hexagonal torus
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Inside the droplet
- Convergence of fluctuations to a Gaussian field. Use of
RNM model to approximate various objects related to GFF
- Universality laws at regular points in the bulk and
boundary, some types of singular points in the bulk
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References
- Based on joint work with Seung Yeop Lee. I thank
Seung Yeop for pictures and crucial contributions
- Use of conformal dynamics was inspired by a paper by
Khavinson and Swiatek
- Interesting analogy with some work in dimer model