Condition number of random matrix gets worse as dimension grows

Question

I observed experimentally that, when generating (pseudo) random matrices, their condition number increases with the dimension. Why?

When using the norm induced by the Euclidean norm for vector spaces, the condition number is the absolute value of the ratio between the greatest and the smallest eigenvalue. It seems to me that, since increasing the dimension implies getting more eigenvalues, it is easier that this ratio increases.

Answer 1

This is correct, and is a (very famous) result of Alan Edelman's from his PhD thesis:

Edelman, Alan, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9, No. 4, 543-560 (1988). ZBL0678.15019.