1. 4.a The random waves model
4. Eigenvectors
4.a The random waves model
The random waves conjecture (M. Berry (1977))
The statistics of eigenfunctions of chaotic billiards
in the semi-classical limit are reproduced by the
Gaussian statistics of random waves.

2. Monochromatic random waves on d-regular graphs (Yehonatan Elon)
Theorem: Guassian random vectors on the vertices of the infinite d- regular tree with covariance are almost surely eigenvectors of the infinite tree graph! (Elon 2010).
The random wave conjecture: same is true for d- regular graphs for k<log V
G(4000,3) d = 3-regular graph

3. Nodal domains on graphs
Interior vertex
Nodal domains on graphs

4. Nodal domains of an eigenvector f :
Counting nodal domains
Nodal domains of an eigenvector f :
Connected sub-graphs where the components of f have the same sign.
Courant : the number of nodal domains of the n’th eigenvector cannot exceed n.
Courant bound
Assuming the validity of the random wave conjecture Elon (2009) was able to reconstruct the figure based on the following arguments:.

5. Assuming the validity of the random wave conjecture

6. 3. Nodal domains vs Percolation on random v-regular graphs
We consider random regular G(V,d) graphs in the limit

7. Phase transition at pc = 1/(d-1):
Percolation on Graphs
(Erdos and Renyi, Alon et. al., Nachmias and Peres)
Start with a random d-regular graph on V vertices.
Perform an independent p-bond percolation :
Retain a bond with a probability p
Delete a bond with a probability 1- p
Phase transition at pc = 1/(d-1):
Denote by L the number of vertices in the largest connected component
p < pc pc
p > pc
1/V1/3
E(L) ~o(V2/3)
E(L) ~ V2/3
E(L)~o(V2/3) V1/3
Bond percolation ~ Vertex percolation

8. 5 3 1 4 2

10. The shape of the critical surface

11. Critical percolation on the infinite d-regular tree