Simulate Wigner and Wishart spectra

CodingIntermediate⏱ 1 hourRandom Matricessimulationwignersemicircleuniversality

Theory: Spectral Norm of Random Matrices →

Problem

Generate symmetric $d\times d$ matrices with i.i.d. $\Ncal(0,1)$ entries above the diagonal and plot the histogram of the eigenvalues of $W/\sqrt d$ against the semicircle density.
Repeat with Rademacher entries, and with heavy-tailed but unit-variance entries. Comment on universality of the bulk.
Track $\lambda_{\max}$ across $d$ and compare with the prediction $2$ .

Implement it, then check the implementation against something you can verify independently.

Write it

Solve it here7 tests · Python runs in your browser

import numpy as np

def wigner_eigenvalues(d, seed=0, dist='gaussian'):
    """Eigenvalues of a Wigner matrix, scaled so the bulk has a limit.

Build a symmetric d x d matrix W whose entries on and above the diagonal
    are iid with mean 0 and variance 1, drawn from a generator seeded with
    `seed`, then return the eigenvalues of W / sqrt(d).

`dist` selects the entry law and must support at least:
      'gaussian'    standard normal entries
      'rademacher'  entries uniform on {-1, +1}
      'heavy'       any heavier-tailed law you like, rescaled to unit variance

The point of the last two is that the limiting bulk does not care which
    one you picked.
    """
    raise NotImplementedError

def semicircle_cdf(x):
    """CDF of the semicircle law on [-2, 2], in closed form.

Its density is sqrt(4 - x**2) / (2*pi). Accept a scalar or an array and
    return the same shape; clamp outside [-2, 2] to 0 and 1.
    """
    raise NotImplementedError

def wishart_eigenvalues(n, p, seed=0):
    """Eigenvalues of the sample covariance (1/n) X^T X for X ~ n x p standard
    normal. For p/n -> gamma these fill the Marchenko-Pastur bulk with edges
    (1 -+ sqrt(gamma))**2.
    """
    raise NotImplementedError

⌘/Ctrl + ↵ to run · limit 60s · draft saved in this browser

7 tests will run against your code. The first run downloads the Python runtime — several megabytes, once per browser.

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Problem	Field	Type	Difficulty	Time
Simulate distance contrast run	Classical ML	Coding	Beginner	30 min
Balls into bins, with one choice and with two run	AI Systems	Coding	Intermediate	1 hour
Cache hit rates under Zipf popularity run	AI Systems	Coding	Intermediate	1 hour
Frobenius versus operator norm for random matrices	Random Matrices	Numerical	Intermediate	30 min
Measure anisotropy in an embedding space run	RAG	Coding	Intermediate	1 hour

Problem

Write it

Solution

Related problems