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 $n$.