## Description

Exercise 1: Eigenproblem

Consider a random Hermitian matrix *A *of size *N*.

- Diagonalize
*A*and store the*N*eigenvalues*λ*in crescent order._{i } - Compute the normalized spacings between eigenvalues
*s*= ∆_{i }*λ*∆_{i}/^{¯}*λ*where

∆*λ**i *= *λ**i*+1− *λ**i**,*

and ∆^{¯}*λ *is the average ∆*λ _{i}*.

- Optional: Compute the average spacing ∆
^{¯}*λ*locally, i.e., over a di erent number of levels around*λ*(i.e._{i }*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 *s _{i }*de ned in the previous exercise, accumulating values of

*s*from di erent random matrices of size at least

_{i }*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*) = *as ^{α }*exp(−

*bs*)

^{β}and report *α,β,a,b*.

- Optional: Compute and report the average h
*r*i of the following quantity

for the cases considered above. Compare the average h*r*i that you obtain in the di erent cases.

Hint: if necessary neglect the rst matrix eigenvalue.