Quantum Heisenberg models and their probabilistic representations

TL;DR

This paper explores the connection between quantum spin systems and coagulation-fragmentation processes, proposing that cycle and loop lengths follow a Poisson-Dirichlet distribution, supported by rigorous mathematical results.

Contribution

It introduces a novel probabilistic perspective linking quantum Heisenberg models with coagulation-fragmentation processes, suggesting a Poisson-Dirichlet distribution for cycle and loop lengths.

Findings

Cycle and loop lengths in quantum spin systems behave like coagulation-fragmentation processes.

Rigorous results support the Poisson-Dirichlet distribution hypothesis.

The work connects quantum models with stochastic processes through mathematical analysis.

Abstract

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.