Branching graph of infinite beta random matrices

Infinite-dimensional orthogonal/unitary/symplectic group acts on infinite Hermitian (real/complex/quaternion, corresponding to the random matrix parameter $\beta=1/2/4$) matrices by conjugations. Ergodic invariant measures of this action were classified by Pickrell and Olshanski–Vershik 25 years ago. The answer turns out to be given by sums of independent Gaussian Wigner and Wishart matrices, which are central objects of the modern random matrix theory.
I will introduce and discuss a novel part of the theory corresponding to $\beta=\infty$.