Soutenance de thèse : Alessandro Pacco

High-dimensional random landscapes underlie phenomena as diverse as glassy physics and machine learning, yet even their simplest toy models already display extraordinarily rich behavior. This thesis aims to deepen our understanding of that behavior, by combining landscape based approaches, via the Kac–Rice formalism, with dynamical approaches, paying special attention to both systems with reciprocal and with non-reciprocal interactions. After surveying core techniques and results through the pure spherical p-spin model, this thesis delivers three main advances: (i) exact dynamic–static comparison in a class of solvable non-reciprocal models, pinpointing differences and similarities of the two approaches; (ii) a stability-based calculations of the mean number of fixed points in the Sompolinsky–Crisanti–Sommers random neural network, for any level of non-reciprocity of the interactions; (iii) two approaches to probe the barriers and the distribution of deep local minima in the landscape of the p-spin model; (iv) some results on overlaps among eigenvectors of spiked, correlated random matrices, which are useful when exploring the geometry of energy landscapes. Together, these results sharpen our understanding of such systems, while opening new doors for future research directions.