We dedicate this paper to Joel Lebowitz on the occasion of his 90th birthday with deep respect for his scientific and moral accomplishment. AbstractWe show that in any dimension d ≥ 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 PD(1), as the size of the system grows to infinity. In the case of transient dimensions, d ≥ 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.independent and essentially parallel works where the Bose gas in continuum space [17], the quantum Heisenberg ferromagnet on Z d [5,18,19], and the quantum Heisenberg antiferromagnet in Z 1 [1], had been considered, via random loop representations. The latter paper contains a derivation of a general, Poisson processes based, functional integral representations of quantum spin states on finite graphs. We refer to [13] for a more recent exposition of this general approach.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. The process was first introduced by T. E. Harris, in [12] and since then, due to its manifold relevance and intrinsic beauty, has been the object of abundantly many research papers. In particular, it turned out that the asymptotics of the cycle structure dynamics of random stirring on the d-dimensional discrete tori T N , as N → ∞, is of paramount importance for understanding the emergence of so-called off-diagonal long range order in the spin-1 2 isotropic quantum Heisenberg ferromagnet (for dimensions d ≥ 3) -a Holy Grail of mathematically rigorous quantum statistical physics. For details, see [19] or the surveys [10,20].The main and best known conjecture in this context (see [19]) states that, for dimensions d ≥ 3, there exists a positive and finite critical time β c = β c (d) beyond which cycles of macroscopic size of the random stirring emerge. For precise formulation see Conjecture 1 in section 1.6 below.Note that in the Feynman-Kac (a.k.a. imaginary time) setting the time parameter corresponds to inverse temperature. Accordingly, the critical value of time, β c , corresponds, in physical terms, to critical inverse temperature. This is reflected by our choice of notation.Inspired by the exhaustive analysis of the Curie-Weiss mean field version of the problem by Schramm, cf [16], and supported by numerical evidence, a refinement of this conjecture (see [10]) claims that beyond the critical time β c , the macroscopically scaled cycle lengths converge in distribution to the Poisson-Dirichlet law PD(1). For precise formulation see Conjecture 2 in section 1.6 below.The work presented in this note is primarily motivated by the following f...