The interchange process with reversals on the complete graph

Björnberg, Jakob E.; Kotowski, Michał; Lees, Benjamin; Miłoś, Piotr

doi:10.1214/19-ejp366

Cited by 12 publications

(20 citation statements)

References 26 publications

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“…There are related loop models that include 'double bars' as well as the transposition 'crosses', that represent further quantum spin systems [1,17]. Without weights, it was proved in [4] that the joint distribution of the lengths of long loops is PD( 1 /2). With weights 2 #loops , the result of [3] is that…”

Section: Introductionmentioning

confidence: 99%

Characterising random partitions by random colouring

Björnberg¹,

Mailler²,

Mörters³

et al. 2020

Electron. Commun. Probab.

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Let (X 1 , X 2 , ...) be a random partition of the unit interval [0, 1], i.e. X i ≥ 0 and i≥1 X i = 1, and let (ε 1 , ε 2 , ...) be i.i.d. Bernoulli random variables of parameter p ∈ (0, 1). The Bernoulli convolution of the partition is the random variable Z = i≥1 ε i X i . The question addressed in this article is: Knowing the distribution of Z for some fixed p ∈ (0, 1), what can we infer about the random partition? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter p is not equal to 1 /2.

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Section: Introductionmentioning

confidence: 99%

Characterising random partitions by random colouring

Björnberg¹,

Mailler²,

Mörters³

et al. 2020

Electron. Commun. Probab.

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“…In the Curie-Weiss mean field case, phase transition and Poisson-Dirichlet structure of P 0,u , for θ = 1 and u ∈ [0, 1], was worked out recently in [4], extending the study of the pure random stirring case, θ = 1 and u = 1, in [16]. However, even in the mean-field case (Curie-Weiss), there are no direct matching results for P θ,β 0,u when θ = 1.…”

Section: Random Loops In the Quantum Heisenberg Modelmentioning

confidence: 99%

Split-and-Merge in Stationary Random Stirring on Lattice Torus

Ioffe

Tóth

2020

J Stat Phys

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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...

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“…For graphs of sufficiently high vertex degree one expects a phase transition in the sense that there is a critical (loop) parameter β c > 0 such that (a) for β < β c there are only finite loops and (b) for β > β c there are infinite loops almost surely. Resolving (b) is subject of ongoing research: Results have been obtained for the complete graph (Schramm, 2005;Berestycki, 2011 for u = 1, Björnberg et al, 2019 for u ∈ [0, 1]), the hypercube (Kotecký et al, 2016 for u = 1), trees (Angel, 2003;Hammond, 2013Hammond, , 2015 for u = 1, Björnberg and Ueltschi, 2018a; Hammond and Hegde, 2019 for u ∈ [0, 1]) and the Hamming graph (Miłoś and Şengül, 2019 for u = 1) and it remains an open problem for G = Z d with d ≥ 2. We can, however, easily show (a) as follows: Loop models possess a natural percolation structure when viewing any edge e with X e not empty as opened; this occurs independently for all e ∈ E with probability 1 − e −β .…”

Section: Introductionmentioning

confidence: 99%

Critical parameters for loop and Bernoulli percolation

Mühlbacher¹

2021

ALEA

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We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter β ≥ 0. Intuitively, larger β means more randomness. In particular, at β = 0 we start with loops of length 1 and as β crosses a critical value β c , infinite loops start to occur almost surely. Our random loop models admit a natural comparison to bond percolation with p = 1−e −β on the same graph to obtain a lower bound on β c. For those graphs of diverging vertex degree where β c and the critical parameter for percolation have been calculated explicitly, that inequality has been found to be an equality. In contrast, we show in this paper that for graphs of bounded degree the inequality is strict, i.e. we show existence of an interval of values of β where there are no infinite loops, but infinite percolation clusters almost surely.

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The interchange process with reversals on the complete graph

Cited by 12 publications

References 26 publications

Characterising random partitions by random colouring

Characterising random partitions by random colouring

Split-and-Merge in Stationary Random Stirring on Lattice Torus

Critical parameters for loop and Bernoulli percolation

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