We give a free probabilistic proposal to compute the fine-grained radiation
entropy for an arbitrary bulk radiation state, in the context of the
Penington-Shenker-Stanford-Yang (PSSY) model where the gravitational path
integral can be implemented with full control. We observe that the replica
trick gravitational path integral is combinatorially matching the free
multiplicative convolution between the spectra of the gravitational sector and
the matter sector respectively. The convolution formula computes the radiation
entropy accurately even in cases when the island formula fails to apply. It
also helps to justify this gravitational replica trick as a soluble Hausdorff
moment problem. We then work out how the free convolution formula can be
evaluated using free harmonic analysis, which also gives a new free
probabilistic treatment of resolving the separable sample covariance matrix
spectrum. The free convolution formula suggests that the quantum information
encoded in competing quantum extremal surfaces can be modelled as free random
variables in a finite von Neumann algebra. Using the close tie between free
probability and random matrix theory, we show that the PSSY model can be
described as a random matrix model that is essentially a generalization of
Page’s model. It is then manifest that the island formula is only applicable
when the convolution factorizes in regimes characterized by the one-shot
entropies. We further show that the convolution formula can be reorganized to a
generalized entropy formula in terms of the relative entropy.
