Quantum Heisenberg models and their probabilistic representations

Details Quantum Heisenberg models and their probabilistic representations Quantum Heisenberg models and their probabilistic representations

2011-04-06

arxiv.org/abs/1104.0983v3

Entropy and the Quantum II, Contemporary Mathematics 552, 177-224 (2011)

Physics

Mathematical Physics

48 pages, 9 figures

Scientific paper

These notes give a mathematical introduction to two seemingly unrelated topics: (i) quantum spin systems and their cycle and loop representations, due to T\'oth and Aizenman-Nachtergaele; (ii) coagulation-fragmentation stochastic processes. These topics are nonetheless related, as we argue that the lengths of cycles and loops satisfy an effective coagulation-fragmentation process. This suggests that their joint distribution is Poisson-Dirichlet. These ideas are far from being proved, but they are backed by several rigorous results, notably of Dyson-Lieb-Simon and Schramm.

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