Finite-size effects on the hardness transition in the Wishart Planted Ensemble

Ascertain whether the high success probabilities observed in a subset of N = 16 Wishart Planted Ensemble (WPE) instances are due to finite-size effects, and determine how the system size N influences the location and sharpness of the hardness phase transition in the WPE.

Background

The authors benchmark dimensional annealing methods on WPE instances of size N = 16 and observe that about one sixth of instances are unexpectedly easy for all tested methods. The WPE exhibits an easy–hard–easy transition whose hardest region is known theoretically, but the authors’ empirical findings may be influenced by small system size.

They explicitly conjecture that these observations stem from finite-size effects that shift and smooth the hardness transition for small N, raising the need to quantify how the transition’s position and sharpness depend on system size.

References

We conjecture that this could be a finite-size effect, and that the location and sharpness of the hardness phase transition in the WPE is affected by the quite small size (N=16) of our WPE instances.

— Soft vector spins with dimensional annealing for combinatorial optimization  (2604.01003 - Syed et al., 1 Apr 2026) in Subsection “Comparison of annealing schemes for different problem classes and dimensions” in Section 3 (Simulations and optimization performance)