Volume distribution of nodal domains of random bandlimited
Volume distribution of nodal domains of random bandlimited functions Dmitry Beliaev, Oxford Igor Wigman, KCL Indian Institute of Science April 27, 2017
1. Motivation & Background
Chladni plates video
General Setup
Nodal components & domains
Questions – local vs. nonlocal I. Local (“easier” via Kac-Rice): a. Total volume, nodal intersections, curvature… b. Critical points (values) c. Euler characteristic (Gauss-Bonnet). II. Nonlocal (semilocal): a. Nodal count (Nazarov-Sodin `09, `12, `15) b. Topology: genus, area…; Nesting (Sarnak. W) c. Geometry: Lengths/areas/volumes d*. Total Betti Number (Gayet-Welschinger)
“Semilocality” (Nazarov-Sodin)
Nodal count (deterministic) Nodal picture for the square, arbitrarily high energy. A. Stern’s thesis, Gottingen, 1925. Courtesy of P. Sarnak.
Berry’s Random Wave Model
2. Random Band Limited Functions
Random Band-Limited Functions
Limiting ensembles
Limiting ensembles (cont. )
1. Random Spherical harmonics
Spherical harmonics vs. Planar waves Random spherical harmonics Alex Barnett Berry’s RWM
2. “Real Fubini Study”
Some pictures (Alex Barnett) Berry’s Random
Nodal count (Nazarov-Sodin)
3. Universal law for nodal volume distribution (Dmitry Beliaev-IW `16)
Scale-invariant (Euclidean) case
Scale-invariant (Euclidean) case
Random spherical harmonics
Discussion
Coupling area + boundary length
Dido Building Cartphage / J. M. W. Turner The National Gallery
The Embarkation of the Queen of Sheba / Claude Lorrain The National Gallery
General Case
General case (cont. )
- Slides: 28