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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:

  1. Is the distribution of the eigenvalues known?
  2. What will be the joint distribution of eigenvalues?
  3. What will be the joint distribution of eigenvectors?
  4. In the wigner case, eigenvectors and eigenvalues are independent, is it still the case for $\mathbf{B}$?
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1 Answer 1

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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).

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