Traces of powers of random p-adic matrices

What is the distribution of the traces of powers of a random matrix? A classical result of Diaconis and Shahshahani states that the traces tr(M^i) for a random unitary matrix M are independent and identically distributed (i.i.d.) complex Gaussians. Recently, Gorodetsky and Rodgers proved that the traces of powers of random unitary matrices over F_q are i.i.d. uniform random variables in F_q. In this talk, we will discuss the distribution of traces of powers of random p-adic matrices and prove that it converges to the distribution of i.i.d. uniformly random p-adic integers, as the size of the matrix goes to infinity.