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Random loop representations for quantum spin systems
, 2013
"... We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1/2 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)invariance. Quantum spin correlations are given by loop correlati ..."
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We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1/2 Heisenberg models with possibly anisotropic spin interactions and certain spin 1 models with SU(2)invariance. Quantum spin correlations are given by loop correlations. Decay of correlations is proved in 2Dlike graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic longrange order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase.
RANDOM PERMUTATIONS OF A REGULAR LATTICE
"... Abstract. Spatial random permutations were originally studied due to their connections to BoseEinstein condensation, but they possess many interesting properties of their own. For random permutations of a regular lattice with periodic boundary conditions, we prove existence of the infinite volume ..."
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Abstract. Spatial random permutations were originally studied due to their connections to BoseEinstein condensation, but they possess many interesting properties of their own. For random permutations of a regular lattice with periodic boundary conditions, we prove existence of the infinite volume limit under fairly weak assumptions. When the dimension of the lattice is two, we give numerical evidence of a KosterlitzThouless transition, and of long cycles having an almost sure fractal dimension in the scaling limit. Finally we comment on possible connections to SchrammLöwner curves.