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### Abstract

We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 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 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase.

It is a pleasure to thank Michael Aizenman, Jürg Fröhlich, Martin Hairer, Bruno Nachtergaele, Charles-Édouard Pfister, Robert Seiringer, and Tom Spencer, for valuable discussions. I am grateful to Marek Biskup and Roman Kotecký for their invitation to give lectures on related material in Prague in September 2011, and to Christian Hainzl and Stefan Teufel for their invitation to Tübingen in July 2012. I am also indebted to Maria Esteban and Mathieu Lewin for organizing the valuable program *Variational and spectral methods in quantum mechanics* at the Institut Henri Poincaré in 2013. Hong-Hao Tu kindly pointed out relevant references for the spin 1 model. The referee made several useful observations. This work is partially supported by EPSRC Grant No. EP/G056390/1.

I. INTRODUCTION II. PROBABILISTIC MODELS A. Model of random loops B. Space-time spin configurations III. QUANTUM SPIN SYSTEMS A. Families of quantum spin systems B. Random loop representations IV. DECAY OF CORRELATIONS IN 2D-LIKE GRAPHS V. OCCURRENCE OF MACROSCOPIC LOOPS IN DIMENSION

*d*⩾ 3 A. Setting and results B. Reflection positivity of the model of random loops C. Infrared bound for the correlation function VI. REFLECTION POSITIVITY IN SPACE AND TIME VII. SPECIFIC MODELS OF INTEREST A. Spin models B. Spin 1 SU(2)-invariant model VIII. CONCLUSION AND OUTLOOK A. Joint distribution of the lengths of macroscopic loops B. Nature of pure Gibbs states

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### Abstract

We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 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 2D-like graphs, and occurrence of macroscopic loops is proved in the cubic lattice in dimensions 3 and higher. As a consequence, a magnetic long-range order is rigorously established for the spin 1 model, thus confirming the presence of a nematic phase.

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