Universal Behaviour of 3D Loop Soup Models

Abstract

These notes describe several loop soup models and their universal behaviour in dimensions greater or equal to 3. These loop models represent certain classical or quantum statistical mechanical systems. These systems undergo phase transitions that are characterised by changes in the structures of the loops. Namely, long-range order is equivalent to the occurrence of macroscopic loops. There are many such loops, and the joint distribution of their lengths is always given by a Poisson-Dirichlet distribution.This distribution concerns random partitions and it is not widely known in statistical physics. We introduce it explicitly, and we explain that it is the invariant measure of a mean-field split-merge process. It is relevant to spatial models because the macroscopic loops are so intertwined that they behave effectively in mean-field fashion. This heuristics can be made exact and it allows to calculate the parameter of the Poisson-Dirichlet distribution. We discuss consequences about symmetry breaking in certain quantum spin systems.Contents arXiv:1703.09503v4 [cond-mat.stat-mech]

Quantum Spins and Random Loops on the Complete Graph

“…David Aldous formulated a conjecture concerning the convergence of the rescaled loop sizes to PD(1), and he explained the heuristics; Schramm then provided a proof [24] of Aldous' conjecture. Models on the complete graph are easier to analyse than the corresponding models on a lattice Z d , d ≥ 3; but the heuristics for the latter models is remarkably similar to the one for the former models; see [11,32]. The ideas sketched here are confirmed by the results of numerical simulations of various loop soups, including lattice permutations [12], loop O(N)-models [21], and the random interchange model [4].…”

“…For future reference we note here the following formula, which will turn out to be relevant for the spin-systems considered in this paper. In [32,Eq. (4.18)] it is shown that…”

Loop Correlations in Random Wire Models

“…The conclusion of this heuristic is that, as the volume becomes large, the effective splitmerge process on partitions behaves like the standard, mean-field process where two partition elements η, η merge at rate 2c √ α ηη and an element η splits at rate c √ α 2 η 2 ; moreover, the element is split uniformly. It is known that the Poisson-Dirichlet distribution with parameter θ = α 2 is the invariant measure for this process [35,7,37] (partial results about uniqueness can be found in [16]).…”

“…It is written in [29] where it is derived using "supersymmetry" calculations in a loop O(N ) model. A calculation within Poisson-Dirichlet can be found in [37].…”

Phase transition for loop representations of quantum spin systems on trees

“…In the case of random interchange it was shown by Schramm [10] that infinite loops occur on the complete graph. The reader is encouraged to consult the recent review of Ueltschi [13] and references therein for a more complete overview of current results in this direction.…”