Matrix Liberation Process I: Large Deviation Upper Bound and Almost Sure Convergence

Abstract

We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution and several properties on its rate function. As a simple consequence we obtain the almost sure convergence of the empirical distribution of the matrix liberation process to that of the corresponding liberation process as continuous processes in the large N limit.

A remark on orbital free entropy

“…In view of our work [6] and Voiculescu's liberation theory [9], a candidate constraint of selecting sequences of multi-matrices may be the way of approximating to X 1 ⊔· · ·⊔X n globally, though it probably does not satisfy the exact formula relating χ orb to χ in general.…”

Matrix liberation process II: Relation to orbital free entropy

“…As a consequence, we will clarify how the matrix liberation process might resolve several technical drawbacks around the definition of orbital free entropy. As another consequence, we will get a large deviation upper bound result by applying the established contraction principle at T = ∞ to the one for the matrix liberation process in our previous paper [29]. We will then make the resulting rate function up into a new microstate-free candiadate for free mutual information.…”

“…This paper is a sequel to our previous one [29] on the matrix liberation process, and devoted to explaining how the matrix liberation process is connected to the orbital free entropy χ orb . Here, the negative of orbital free entropy may be regarded as a possible microstate approach to mutual information in free probability.…”

“…(Some considerations looking for better variants of orbital free entropy were made by Biane and Dabrowski [5], and a direct generalization dropping the hyperfiniteness for given random multi-variables was then given by us [27].) The liberation process is exactly the large N limit of the matrix liberation process introduced in [29] and its 'invariant measure' (or its limit distribution as time goes to ∞) exactly arises as the 'distribution' of the adjoint actions of Haardistributed unitary random matrices. Thus it is natural to consider the matrix liberation process for the conjectural unification between Voiculescu's and our approaches to mutual information in free probability.…”

“…As a very first step, we proved in [29], following the idea of [4], the large deviation upper bound with a good rate function that completely characterizes the corresponding liberation process as a unique minimizer. The next ideal steps on this line of research should be: (1) proving the large deviation lower bound with the same rate function, (2) applying the contraction principle to the resulting large deviation upper/lower bounds at time T = ∞, and (3) identifying the resulting rate function with Voiculescu's free mutual information.…”