Abstract
We show that in any dimension dge 1, the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson–Dirichlet law mathsf {PD(1)}, as the size of the system grows to infinity. In the case of transient dimensions, dge 3, the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.
Highlights
Representations of the Bose gas in terms of random permutations date back to the classic [8], where the Feynman–Kac approach was first used in the context of quantum statistical physics
Due to Holstein–Primakoff transformations, quantum spin systems are reformulated as the lattice Bose gas with interactions, the Feynman–Kac approach can be transferred to the quantum
The random stirring (a.k.a. random interchange) process on a finite connected graph is a process of random permutations of its vertex-labels where elementary swaps are appended according to independent Poisson flows of rate one on unoriented edges
Summary
Representations of the Bose gas in terms of random permutations date back to the classic [8], where the Feynman–Kac approach was first used in the context of quantum statistical physics. Heisenberg ferromagnet in terms of random permutations appears in the unjustly forgotten paper [16] It looks like the stochastic permutation (or, random loop) approach to the Bose gas and quantum spin systems, based on Feynman–Kac, became main stream objects in mathematically rigorous quantum statistical physics and probability in the early nineties, with independent and essentially parallel works where the Bose gas in continuum space [18], the Communicated by Ivan Corwin.
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