Macroscopic loops exist in the random loop model on sparse random graphs above an explicit edge density threshold depending on loop weight θ and cross parameter u.
Decay of correlations in 2D quantum systems with continuous symmetry.Annales Henri Poincaré, 18(9):2831–2847
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Macroscopic loops in the random loop model on sparse random graphs
Macroscopic loops exist in the random loop model on sparse random graphs above an explicit edge density threshold depending on loop weight θ and cross parameter u.