Statistics of invertible matrices over finite fields

Let G_n be GL_n(F_q), the general linear group over a finite field. Choose a matrix g from G_n uniformly at random. We'll discuss recent progress towards understanding the distribution of the entries of g, the distribution of traces of g^k (powers of g), and other statistical questions, especially in the large-n limit (q is fixed while n tends to infinity). We'll motivate these problems and explain the main tools. Time permitting, we'll discuss similar questions over other finite classical groups.

 

Based on joint works with Rodgers and with Kovaleva, and works in progress with Zelingher.