Spectral radius of random matrices with independent entries

Understand

We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile.

• We show that the spectral radius of $X$ converges with very high probability to the square root of the spectral radius of the variance matrix of $X$ when $n$ tends to infinity.
• We also establish the optimal rate of convergence, that is a new result even for general i.i.d.
• matrices beyond the explicitly solvable Gaussian cases.