Exercise 1: Eigenproblem
Consider a random Hermitian matrix A of size N.
- Diagonalize A and store the N eigenvalues λi in crescent order.
- Compute the normalized spacings between eigenvaluessi = ∆λi/∆¯λ where
∆λi = λi+1− λi,
and ∆¯λ is the average ∆λi.
- Optional: Compute the average spacing ∆¯λ locally, i.e., over a di erent number of levels around λi (i.e. N/100,N/50,N/10…N) and compare the results of next exercise for the di erent choices.
Exercise 2: Random Matrix Theory
Study P(s), the distribution of the si de ned in the previous exercise, accumulating values of si from di erent random matrices of size at least N = 1000.
- Compute P(s) for a random HERMITIAN matrix.
- Compute P(s) for a DIAGONAL matrix with random real entries.
- Fit the corresponding distributions with the function:
P(s) = asα exp(−bsβ)
and report α,β,a,b.
- Optional: Compute and report the average hri of the following quantity
for the cases considered above. Compare the average hri that you obtain in the di erent cases.
Hint: if necessary neglect the rst matrix eigenvalue.