ACTA issues

Large deviations for functions of two random projection matrices

Fumio Hiai, Dénes Petz

Acta Sci. Math. (Szeged) 72:3-4(2006), 581-609

Abstract. In this paper two independent and unitarily invariant projection matrices $P(N)$ and $Q(N)$ are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size $N$ converges to infinity. The result is formulated on the tracial state space $TS({\cal A})$ of the universal $C^*$-algebra ${\cal A}$ generated by two selfadjoint projections. The random pair $(P(N),Q(N))$ determines a random tracial state $\tau_N \in TS({\cal A})$ and $\tau_N$ satisfies the large deviation. The rate function is in close connection with Voiculescu's free entropy defined for pairs of projections.

AMS Subject Classification (1991): 15A52, 60F10, 46L54

Keyword(s): Eigenvalue density, large deviation, random matrices, free entropy, C^*, universal-algebra, tracial state space

Received October 18, 2005. (Registered under 5939/2009.)