Sharp complexity asymptotics and topological trivialization for the (p, k) spiked tensor model

Using precise random matrix theory tools and the Kac-Rice formula, we provide sharp O(1) asymptotics for the average number of deep minima of the (p, k) spiked tensor model. These sharp estimates allow us to prove that, when the signal-to-noise ratio is large enough, the expected number of deep minima is asymptotically finite as N tends to infinity and to establish the occurrence of topological trivialization by showing that this number vanishes when the strength of the signal-to-noise ratio diverges. We also derive an explicit formula for the value of the absolute minimum (the limiting ground state energy) on the N-dimensional sphere, similar to the recent work of Jagannath, Lopatto, and Miolane [Ann. Appl. Probab. 4, 1910-1933 (2020)].