Emergent factorization of Hilbert space at large $N$ and black hole

Abstract: We investigate the emergent factorization of Hilbert space in the low-energy description of matrix models, addressing key aspects of the black hole information paradox. We examine the collective description for the low-energy sector of $SU(N)$ matrix model, characterized by a factorized Hilbert space composed of a finite number of boxes and anti-boxes. This factorization leads us to examine the emergence of thermofield dynamics (TFD) state in the low energy sector from a fine-tuned state. In addition, we study the collective Hamiltonian of the $U(N)$ matrix model for the semi-classical description of "particle-hole" fluctuations around a background Young tableau. Our investigation of these matrix models elucidates a concrete mechanism for constructing the truncated algebra of accessible observables, thereby facilitating an understanding of black hole complementarity. In the context of the black hole information paradox, we discuss the origin of the island appearing inside the black hole and provide a reinterpretation of the recent proposal -- the holography of information.