Consider the adjacency matrix $\mathbf{A}$ of a one dimensional lattice of size $N$. That is, $A$ is a $N\times N$ matrix with $A_{ij}=1$ if vertex $i$ adjacent to vertex $j$ (there exists an edge between $i$ and $j$). Whether there are boundary conditions or not (1D line or a closed ring), the spetrum of $A$ is very well known: there exists an analytic formula for both its eigenvalues and eigenvectors for any finite $N$.
Furthermore, let $\mathbf{w}$ be a random matrix of size $N\times N$ such that $\left[\mathbf{w}\right]_{ij}=w_{ij}$ is normally distributed with mean $0$ and variance $1/N$. As $N\to\infty$, the spectrum of $\mathbf{w}$ is also well known: if $\mathbf{w}$ has symmetric entries then its eigenvalues will follow the wigner semi-circle distribution. I am not sure about the distribution of the eigenvectors.
Now, let $\mathbf{B}$ be a $N\times N$ matrix such that $B_{ij}=w_{ij}A_{ij}$, i.e $\mathbf{B}$ represents the 1D lattice with random weights between links.
Here are a few questions I had:
- Is the distribution of the eigenvalues known?
- What will be the joint distribution of eigenvalues?
- What will be the joint distribution of eigenvectors?
- In the wigner case, eigenvectors and eigenvalues are independent, is it still the case for $\mathbf{B}$?
Best Answer
I presume this will depend on the connectivity of the 1D lattice. Let me consider the simple case of nearest neighbor connections, when the adjacency matrix $A$ is tridiagonal with the same values on each diagonal. The statistics of the matrix $B$ when the nonzero elements are i.i.d. random variables was studied in General Tridiagonal Random Matrix Models, Limiting Distributions and Fluctuations (2006).