Magic Carpet

Setup:

The magic carpet is a modification of the Sierpinski carpet. We start with a square with torus boundary condition. For each inductive step, whenever we leave out the middle cell, we identify its boundaries with torus boundary condition. In other words, whenever we make a cut, we sew it up.

For each level n magic carpet M_n we associate a graph G_n = (V_n,E_n) with it the same way as we did to the Sierpinski carpet. Note now the difference is that the obtained graph is 4-regular. Due to the identification of boundaries, each cell has exactly four neighbors. Below is an example of G_2 obtained from M_2.

Define a random potential of level m on the magic carpet the same way as we did to Sierpinski carpet. As usual we look for solutions of Hu = 1 and Hv = \lambda v where H = \triangle + P, \triangle is the graph laplacian and P acts multiplicatively on each vertex. We showcase some eigenfunctions for each value of P_max, as well as the loglog of eigenvalue counting function.

Results: Click here for results