2025/05/15 by Soumendu Sundar Mukherjee, Mukherjee, Soumendu Sundar, Himasish Talukdar +1
Physics and Astronomy · #FOS: Mathematics #Probability (math.PR) #Quantum, superfluid, helium dynamics #Scientific Research and Discoveries
paper · pdf · doi:10.48550/arxiv.2505.10555
openalex publication_date 2025/05/15 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28
In this article, we study the spectra of matrix-valued contractions of the Gaussian Orthogonal Tensor Ensemble (GOTE). Let G denote a random tensor of order r and dimension n drawn from the density f(G) ∝ exp(-(1)/(2r)‖G‖2F). For w ∈ \mathbbSn - 1, the unit-sphere in ℝn, we consider the matrix-valued contraction G ⋅ w⊗ (r - 2) when both r and n go to infinity such that r / n → c ∈ [0, ∞]. We obtain semi-circle bulk-limits in all regimes, generalising the works of Goulart et al. (2022); Au and Garza-Vargas (2023); Bonnin (2024) in the fixed-r setting. We also study the edge-spectrum. We obtain a Baik-Ben Arous-Péché phase-transition for the largest and the smallest eigenvalues at r = 4, generalising a result of Mukherjee et al. (2024) in the context of adjacency matrices of random hypergraphs. We also show that the extreme eigenvectors of G ⋅ w⊗ (r - 2) contain non-trivial information about the contraction direction w. Finally, we report some results, in the case r = 4, on mixed contractions G ⋅ u ⊗ v, u, v ∈ \mathbbSn - 1. While the total variation distance between the joint distribution of the entries of G ⋅ u ⊗ v and that of G ⋅ u ⊗ u goes to 0 when ‖u - v‖ = o(n-1), the bulk and the largest eigenvalues of these two matrices have the same limit profile as long as ‖u - v‖ = o(1). Furthermore, it turns out that there are no outlier eigenvalues in the spectrum of G ⋅ u ⊗ v when ⟨ u, v ⟩ = o(1).