The Disordered Universe 2025: Statistical Systems Out of Equilibrium

These lecture notes provide an introduction to quantum spin glasses, with a particular emphasis on their application to open problems in cosmology and holography. Focusing on the spherical $p$-spin model, we develop a complete and self-contained analysis of its thermodynamic and dynamical properties. We introduce the replica method, discuss both replica-symmetric and one-step replica symmetry-broken solutions, and characterise the resulting glassy phase. The framework is then extended to the quantum model, where we study the emergence of slow dynamics and the role of quantum correlations through Euclidean and real-time correlation functions. Finally, we explore how concepts such as replica symmetry breaking and disorder naturally arise in cosmological and holographic systems related to black hole fragmentation, highlighting spin glass models as useful theoretical laboratories for these phenomena. These notes are based on a series of lectures held during the La Ricotta Summer School: The Disordered Universe 2025 from 21 July to 1 August 2025.