Massless-phase conjecture for the two-dimensional O(3) model

Determine whether the two-dimensional O(3) model is massive at every temperature, thereby resolving the widely believed conjecture that no massless phase exists in the two-dimensional O(3) model.

Background

The paper discusses work on the two-dimensional O(3) model by conditioning on the third spin coordinate, which produces an XY model with random conductances. The cited arguments concern whether the resulting random XY system can exhibit a massless phase and therefore whether the conventional expectation of massiveness for the two-dimensional O(3) model is correct. The paper presents this as a widely believed conjecture in the surrounding literature rather than as a problem resolved by its own random-geometric-graph results.

References

Patrascioiu--Seiler argue for the existence of a massless phase in the two-dimensional $O(3)$-model by examining the percolation properties of the resulting random $XY$-system, thereby arguing against a widely believed conjecture, which states that the tow-dimensional $O(3)$-model should be massive at all temperatures.

— Asymptotic long-range order for the XY-model on random geometric graphs  (2609.02618 - Disertori et al., 2 Sep 2026) in Introduction, paragraph beginning “In the recent years there has been increasing interest in the $XY$-model on random graphs”