relaxation exhibits slow dynamic features such as stretched exponential relaxation and breakdown of proportionality between diffusive time scale and structural relaxation time scale.

It was a common belief that the 2D UFXY model and the corresponding LCG belong to the same universality class with essentially the same phase transition properties, ground state symmetry, for example. However, recent work on LCG by Gupta, Teitel, and Gingras (GTG)[@gtg] and also another work on UFXY model by Denniston and Tang (DT)[@dt], showed that there exist some difference between the two model systems especially in the case of dense frustration. Both model systems exhibit first order transition but the low temperature vortex configurations in UFXY models are different from the charge configurations of the corresponding LCG for $f$ near $1-g \simeq 0.382$. The underlying cause for this breakdown of Villain approximation in the limit of dense frustration is not known, but probably it is related to application of spin-wave integration to systems having many metastable states with similar energies, that may cause neglect of multi-vortex correlations.

Special interest has been given to the case of $f$ approaching $1-g$ [@halsey; @mychoi]. Consider a system where $f$ equals $p_0 /q_0$ ($p_0$ and $q_0$ are relative primes) 