Investigating the role of mutual information in the Page curve for a functional renormalization group improved Schwarzschild black hole

The present work delves into probing the importance of mutual information of relevant subsystems in obtaining the correct time-evolution of fine-grained entropy of Hawking radiation, as suggested by Page. This was done by considering a functional renormalization group improved or simply, quantum corrected Schwarzschild black solution which captures the flavour of an effective theory of quantum gravity. The mentioned black hole solution emerges from a asymptotically safe average effective action which describes a trajectory in momentum-space and satisfies a renormalizarion group equation. Furthermore, in the before Page time scenario, the behaviour of the mutual information between appropriate subsystems over time leads to the Hartman-Maldacena time. The observations made for the quantum corrected Schwarzschild black hole have been compared to the same made for the standard Schwarzschild black hole in order to draw some novel conclusions. Based upon our observations, we also propose a formula for computing fine-grained entropy of Hawking radiation for an eternal black hole, in presence of the island.