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I was numerically playing with tridiagonal symmetric matrix (zero on diagonal) of the form

\begin{pmatrix} 0 & b_1 & 0 & 0 & 0 & \ldots & 0 \\ b_1 & 0 & b_2 & 0 & 0 & \ldots & 0 \\ 0 & b_2 & 0 & b_3 & 0 & \ldots & 0 \\ \vdots & \vdots \\ 0 & \ldots & & & b_{n-1}& 0 & b_n \\ 0 & \ldots & & & 0 & b_n &0 \\ \end{pmatrix}

$b_i$ numbers I generated from Normal distribution with mean $\mu = 0$ and variance $ \sigma^{2} = 25$. Then I calculated spectrum of 200 matrices of order 100 (so n=99). I show plot of the spectrum spectrum of 200 matrices.

Then I wanted to examine distribution of maximal eigenvalue. I tried to fit shifted Poisson distribution to histogram generated from 10000 matrices. Fitted line is given by $ p(k) = ld^{k-shift}\dfrac{\exp(-ld)}{(k-shift)!} $

where good looking fit was with $ld=4$, $shift=12$ histogram of maximal eigenvalue of 10000 matrices. I don't know any theory about this topic - if this could be Poisson distribution or any other, so it is just my guess. I don't even know if my question is difficult or easy to answer. Can you say something about distribution of highest eigenvalue? Or if it is easier to answer - can you say estimate of highest eigenvalue for given form of matrix?

I have read question: Eigenvalues of Symmetric Tridiagonal Matrices

and I just understood spectrum given by cosine function, where off-diagonal numbers are the same, but I could not find spectrum for arbitrary $b_i$. I computed spectrum using online Mathematica for $n=3$ and it was pretty complicated so I doubt there is a way to obtain analytical form for high $n$.

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  • $\begingroup$ Are you familiar with quantum spin chains with random couplings in Physics? I think your question is answered by the so-called real space renormalization-group. If you are still interested I can look up some references. $\endgroup$
    – aprendiz
    Apr 26, 2022 at 3:18
  • $\begingroup$ Well, you can look up, but it probably will not help me. Firstly, not familiar at all with quantum spin chains nor real space renormalization-group. Secondly, I am not quite sure what I meant with my question after time. I did not mention that symmetric matrix comes from length of vectors. So, $b_i= |v_{i+1}-v_{i}|$. It is length of following states ($v_i$) in phase space. Simply said deterministic system is described by time evolution of $v_i$. So I probably wanted distinguish experimental data - deterministic and more and more random data by spectrum of its $v_i$. $\endgroup$
    – weatherman
    May 1, 2022 at 20:17

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