Liberation of Projections

Abstract

We study the liberation process for projections: $(p,q)\mapsto (p_t,q)=(u_{t}p{u}_t^{\ast },q)$ where $u_t$ is a free unitary Brownian motion freely independent from $\{p,q\}$. Its action on the operator-valued angle $q{p}_{t}q$ between the projections induces a flow on the corresponding spectral measures ${\mu }_t$; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure ${\mu }_t$ possesses a piecewise analytic density for any $t>0$ and any initial projections of trace $\frac{1}{2}$. We us this to prove the Unification Conjecture for free entropy and information in this trace $\frac{1}{2}$ setting.

53 pages

References in corpus (4)

• Product of random projections, Jacobi ensembles and universality problems arising from free probability
• Wigner chaos and the fourth moment
• Large deviations for functions of two random projection matrices
• On the free convolution with a free multiplicative analogue of the normal distribution

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• Strong Convergence of Unitary Brownian Motion
• Orbital free entropy, revisited
• Remarks on free mutual information and orbital free entropy
• Heat Kernel Empirical Laws on ${\mathbb{U}}_N$ and ${\mathbb{GL}}_N$
• On the free convolution with a free multiplicative analogue of the normal distribution
• Liberation, free mutual information and orbital free entropy
• Time Reversal of free diffusions I : Reversed Brownian motion, Reversed SDE and first order regularity of conjugate variables
• Asymptotically liberating sequences of random unitary matrices
• The Two-Parameter Free Unitary Segal-Bargmann Transform and its Biane-Gross-Malliavin Identification