Distinguish tree vs cyclic diagrams at the √n scale

Ascertain whether tree-shaped diagram contributions of size at least √n remain asymptotically distinguishable from cyclic diagram contributions in the dense Wigner matrix setting considered, specifically clarifying if tree diagrams of size ≥ √n differ significantly from cyclic diagrams in magnitude or effect on the algorithmic state.

Background

In analyzing the extension of the tree approximation to larger iteration counts, the authors argue that at T ≈ √n the contributions from tree diagrams become exponentially small while the number of non-tree (cyclic) diagrams grows rapidly. Conceptually, random walks of length ≳ √n in an n-vertex graph are likely to collide, potentially blurring the distinction between trees and cycles.

This raises a fundamental uncertainty about the diagrammatic separation at the √n scale, which directly impacts whether tree-based asymptotics can be sustained beyond this regime.