Sierpinski Carpet

Setup:

Start with a square S defined to be the level 0 Sierpinski carpet. We partition the square into 3*3 = 9 equal sized smaller squares, and define the level 1 carpet to be the union of 8 of the 9 squares, with the middle square omitted. We call each of these 8 squares a cell, let them be disjoint (by assigning to the boundaries to one of the neighboring square). With level n carpet defined, we obtain level n+1 carpet by splitting each cell in level n into 3*3 = 9 equal sized square cells and dsicard the middle.

Given a Sierpinski carpet C_n of level n, define a random potential of level m on by considering C_n as a subset of the full square S to inherit a potential defined on S. Observe that the random potential so inherited is constant on each cells of C_n. We create an undirected graph G_n = (V_n,E_n) to represent C_n. Let V_n be the set of cells in C_n. Let there be an edge between u,v\in V_n if the two cells u,v are neighbors, i.e their closure intersect non-trivially. The following shows how to obtain V_2 from C_2.

As before define operator H: V_n^*\to V_n^* by H = \triangle+P where P acts multiplicatively on each vertex, and \triangle is the level n graph laplacian. Although the existence of a self similar laplacian on Sierpinski Carpet is unkown, the uniqueness is established by Barlow et. al. We must take note that the graph laplacian of level n defers with the point-wise laplacian on the carpet given by Kigami by a multiplicative factor (8c)^n.

We obtain the solutions to the eigen problem Hv = \lambda v and Hu = 1 via matlab eigensolver and linear system solver of the graph laplacian matrix. We demonstrate selected eigenfunctions with varying P_{max} in the result section, included also is a plot of the eigenvalue counting function and its loglog plot.

Results:
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