Two Approaches towards Tensorial Complexity : From Functional Renormalization to Kac-Rice

Overview & Synopsis

This thesis investigates "tensorial complexity" through two lenses. The first part develops a renormalization-group framework for mean-field quantum spin glasses, focusing on the quantum (2+p)- and p-spin dynamics in the large-N limit. Coarse-graining is performed along a spectral variable associated with the disordered quadratic operator and the frequency respectively. The emergence of finite time-scale singularities in strong disorder regimes is interpreted as a precursor of glassy physics tied to metastability and symmetry breaking, and motivates improved truncations. In the 2+p model, a low-temperature phase space is constructed using the 2PI/Luttinger-Ward methods.The second part studies the injective norm of random skew-symmetric tensors or equivalently the geometric entanglement of random fermionic pure states. Using a Kac-Rice approach on Grassmann manifolds, it derives asymptotics and high-probability bounds in the regimes d → ∞ at fixed fermion number p, and in a double-scaling limit with fixed filling fraction and d and p → ∞. We also reveal particle-hole dualities linked to Hodge-star symmetry and support our results by numerics.