Extend tree-diagram accuracy for debiased power iteration up to sqrt(n) iterations (and no further)

Establish whether the tree diagram representation of the debiased power iteration x_{t+1} = A x_t - x_{t-1}, with initialization x_0 = 1 and A a Wigner random matrix with i.i.d. mean-0 variance-1 entries as defined in Assumption 1, remains asymptotically accurate for the trajectory up to n^{1/2} iterations, and prove that this accuracy cannot extend beyond n^{1/2} iterations.

Background

The paper develops a diagrammatic framework for analyzing first-order iterative algorithms on dense random symmetric matrices, proving a tree approximation that holds for a constant number of iterations and pushing it to n^{Ω(1)} iterations for debiased power iteration. The authors obtain rigorous control up to polynomially many iterations and identify √n as a natural barrier where collisions in walks become prevalent and cyclic diagrams proliferate.

Within this framework, they explicitly conjecture that the tree diagram representation can be extended up to n^{1/2} iterations but no further, marking a sharp threshold for the validity of the approximation in this specific algorithm.