Random Cluster Model Research Articles

Connection probabilities of multiple FK-Ising interfaces

We find the scaling limits of a general class of boundary-to-boundary connection probabilities and multiple interfaces in the critical planar FK-Ising model, thus verifying predictions from the physics literature. We also discuss conjectural formulas using Coulomb gas integrals for the corresponding quantities in general critical planar random-cluster models with cluster-weight q∈[1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${q \\in [1,4)}$$\\end{document}. Thus far, proofs for convergence, including ours, rely on discrete complex analysis techniques and are beyond reach for other values of q than the FK-Ising model (q=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=2$$\\end{document}). Given the convergence of interfaces, the conjectural formulas for other values of q could be verified similarly with relatively minor technical work. The limit interfaces are variants of SLEκ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SLE}_\\kappa $$\\end{document} curves (with κ=16/3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa = 16/3$$\\end{document} for q=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=2$$\\end{document}). Their partition functions, that give the connection probabilities, also satisfy properties predicted for correlation functions in conformal field theory (CFT), expected to describe scaling limits of critical random-cluster models. We verify these properties for all q∈[1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q \\in [1,4)$$\\end{document}, thus providing further evidence of the expected CFT description of these models.

Pore structure-transport relationships in high-temperature shift catalyst pellets studied by integrated multiscale porosimetry and X-ray tomography

Diffusion-limited, heterogeneously-catalysed processes mean choices influencing pore structure-transport relationships, made during pellet fabrication, affect product performance. This work shows how the ‘sifting strategy’ can identify the critical aspects of a highly complex catalyst pellet pore structure that control mass transport to construct an idiosyncratic, minimalist model. This is implemented using fully-integrated gas overcondensation, mercury porosimetry and X-ray tomography experiments. It showed high temperature shift (HTS) catalyst pellets had a trimodal pore structure. The second mode, consisting of macropores within the roll-compacted feed particles, controlled mass transport. Knudsen-regime mass transport was shown to be critically-controlled by an incipiently-percolating cluster of these intermediate-sized macropores, as its rate could be drastically reduced via introduction of very few blockages via mercury entrapment. This incipiently percolating network could be represented by a lattice-based, random cluster model. X-ray tomographic images, analysed with an AI segmentation algorithm, validated the proposed model of interpretation for the indirect characterization data.

Mixing time of random walk on dynamical random cluster

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} between open and closed, following a Glauber dynamics for the random cluster model with parameters p, q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n2/μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^2/\\mu $$\\end{document}. In our proof we construct a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.

Approximating the chromatic polynomial is as hard as computing it exactly

We show that for any non-real algebraic number q, such that|q-1|>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|q-1|>1$$\\end{document} or ℜ(q)>32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Re(q)>\\frac{3}{2}$$\\end{document} it is #P-hard to computea multiplicative (resp. additive) approximation to the absolutevalue (resp. argument) of the chromatic polynomial evaluated at q on planar graphs. This implies #P-hardness for allnon-real algebraic q on the family of all graphs. We, moreover,prove several hardness results for q, such that |q-1|≤1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|q-1|\\leq 1$$\\end{document}.Our hardness results are obtained by showing that a polynomial timealgorithm for approximately computing the chromaticpolynomial of a planar graph at non-real algebraic q (satisfyingsome properties) leads to a polynomial time algorithm forexactly computing it, which is known to be hard by a resultof Vertigan. Many of our results extend in fact to the more generalpartition function of the random cluster model, a well-knownreparametrization of the Tutte polynomial.

Sweeny dynamics for the random-cluster model with small Q.

The Sweeny algorithm for the Q-state random-cluster model in two dimensions is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As Q decreases, the so-called critical speeding-up for nonlocal quantities becomes more and more pronounced. However, for some quantity of a specific local pattern, e.g., the number of half faces on the square lattice, we observe that, as Q→0, the integrated autocorrelation time τ diverges as Q^{-ζ}, with ζ≃1/2, leading to the nonergodicity of the Sweeny method for Q→0. Such Q-dependent critical slowing-down, attributed to the peculiar form of the critical bond weight v=sqrt[Q], can be eliminated by a combination of the Sweeny and the Kawasaki algorithm. Moreover, by classifying the occupied bonds into bridge bonds and backbone bonds, and the empty bonds into internal-perimeter bonds and external-perimeter bonds, one can formulate an improved version of the Sweeny-Kawasaki method such that the autocorrelation time for any quantity is of order O(1).

Magneto-structural characterization of different kinds of magnetic nanoparticles

Using well-established measurement techniques like transmission electron microscopy (TEM), dynamic light scattering (DLS), small and wide angle X-ray scattering (SAXS, WAXS), susceptometry, and magnetorelaxometry, the distribution of the physical and magnetic size (magnetic moments) and magnetic anisotropy of a variety of structurally different magnetic nanoparticle samples (MNPs) is analyzed and compared. A term which accounts for the presence of weak magnetic areas (WMAs) within the MNPs was introduced to the widespread analysis model for M(H) data, enabling a consistent interpretation of the data in most of the systems. A comparison of the size distributions as obtained for the physical and the magnetic diameter suggests a multidomain structure for three single core systems under investigation, in all probability evoked by the presence of a wustite phase, as identified by WAXS.Analyzing the relationship d < dm < dc between the average single core diameter d, the effective magnetic (domain) size dm and the cluster diameter dc quantitatively, two qualitatively different magnetic structures in multicore MNP (MCMNP) systems were identified: (i) The magnetic moments of single cores within the MCMNP of fluidMAG tend to build flux closure structures, driven by dipole–dipole interaction. (ii) The magnetic behavior of Resovist® was attributed to the presence of domain sizes of about 12 nm within MCMNP, exceeding the single core diameters of 5 nm. Thereby, WAXS revealed a bimodal crystallite size distribution suggesting a crystallite merging process within the MCMNP. The value of the effective magnetic moment of these MCMNP could be explained within the presented “random moment cluster model” (RMCM).We conclude that the combination of physical and magnetic structure parameters obtained from complementary measurement methods allows a reliable assessment of the magnetic structure of single and multicore MNPs.

Swendsen-Wang dynamics for the ferromagnetic Ising model with external fields

We study the sampling problem for the ferromagnetic Ising model with consistent external fields, and in particular, Swendsen-Wang dynamics on this model. We introduce a new grand model unifying two closely related models: the subgraph world and the random cluster model. Through this new viewpoint, we show:(1)polynomial mixing time bounds for Swendsen-Wang dynamics and (edge-flipping) Glauber dynamics of the random cluster model, generalising the bounds and simplifying the proofs for the no-field case by Guo and Jerrum (2018);(2)near linear mixing time for the two dynamics above if the maximum degree is bounded and all fields are (consistent and) bounded away from 1.

Finite-size scaling, phase coexistence, and algorithms for the random cluster model on random graphs

Pour Δ≥5 et q grand en fonction de Δ, nous donnons une description détaillée de la transition de phase du modèle de composantes connexes aléatoires (i.e., le modèle FK) sur des graphes Δ-réguliers aléatoires. En particulier, nous déterminons la distribution limite des poids des phases ordonnées et désordonnées au point critique et prouvons la décroissance exponentielle des corrélations et le comportement gaussien des fluctuations loin du point critique. Nos techniques sont basées sur l’utilisation de modèles de polymères et l’expansion en clusters pour contrôler les écarts par rapport aux états fondamentaux ordonnés et désordonnés. Ces techniques produisent également des algorithmes de comptage et d’échantillonnage efficaces pour les modèles de Potts et FK sur des graphes Δ-réguliers aléatoires à toutes les températures lorsque q est grand. Cela inclut la température critique à laquelle on sait que la dynamique de Glauber et de Swendsen–Wang pour le modèle de Potts mélangent lentement. Nous prouvons en outre de nouveaux résultats de mélange lent pour les chaînes de Markov, notamment que la dynamique de Swendsen–Wang mélange exponentiellement lentement tout au long d’un intervalle ouvert contenant la température critique. Ceci n’était auparavant connu qu’à la température critique. Beaucoup de nos résultats s’appliquent plus généralement aux graphes Δ-réguliers qui satisfont une borne inférieure sur le nombre d’arêtes quittant chaque « petit ensemble » de sommets dans le graphe.

Strict monotonicity, continuity, and bounds on the Kertész line for the random-cluster model on Zd

Ising and Potts models can be studied using the Fortuin–Kasteleyn representation through the Edwards–Sokal coupling. This adapts to the setting where the models are exposed to an external field of strength h &amp;gt; 0. In this representation, which is also known as the random-cluster model, the Kertész line is the curve that separates two regions of the parameter space defined according to the existence of an infinite cluster in Zd. This signifies a geometric phase transition between the ordered and disordered phases even in cases where a thermodynamic phase transition does not occur. In this article, we prove strict monotonicity and continuity of the Kertész line. Furthermore, we give new rigorous bounds that are asymptotically correct in the limit h → 0 complementing the bounds from the work of Ruiz and Wouts [J. Math. Phys. 49, 053303 (2008)], which were asymptotically correct for h → ∞. Finally, using a cluster expansion, we investigate the continuity of the Kertész line phase transition.

On the Two-Point Function of the Potts Model in the Saturation Regime

We consider the Random-Cluster model on {mathbb {Z}}^d with interactions of infinite range of the form J_x = psi (x){mathsf {scriptstyle e}}^{-rho (x)} with rho a norm on {mathbb {Z}}^d and psi a subexponential correction. We first provide an optimal criterion ensuring the existence of a nontrivial saturation regime (that is, the existence of beta _{textrm{sat}}(s)>0 such that the inverse correlation length in the direction s is constant on [0,beta _{textrm{sat}}(s))), thus removing a regularity assumption used in our previous work (Aoun et al. in Commun Math Phys 386:433–467, 2021). Then, under suitable assumptions, we derive sharp asymptotics (which are not of Ornstein–Zernike form) for the two-point function in the whole saturation regime (0,beta _{textrm{sat}}(s)). We also obtain a number of additional results for this class of models, including sharpness of the phase transition, mixing above the critical temperature and the strict monotonicity of the inverse correlation length in beta in the regime (beta _{textrm{sat}}(s), beta _{mathrm {scriptscriptstyle c}}).

Efficient sampling and counting algorithms for the Potts model on ℤd at all temperatures

AbstractFor and all we give an efficient algorithm to approximately sample from the ‐state ferromagnetic Potts and random cluster models on finite tori for any inverse temperature . This shows that the physical phase transition of the Potts model presents no algorithmic barrier to efficient sampling, and stands in contrast to Markov chain mixing time results: the Glauber dynamics mix slowly at and below the critical temperature, and the Swendsen–Wang dynamics mix slowly at the critical temperature. We also provide an efficient algorithm (an FPRAS) for approximating the partition functions of these models at all temperatures. Our algorithms are based on representing the random cluster model as a contour model using Pirogov–Sinai theory. The main innovation of our approach is an algorithmic treatment of unstable ground states, which is essential for our algorithms to apply to all inverse temperatures .

Multimodal data fusion based on IGERNNC algorithm for detecting pathogenic brain regions and genes in Alzheimer's disease.

At present, the study on the pathogenesis of Alzheimer's disease (AD) by multimodal data fusion analysis has been attracted wide attention. It often has the problems of small sample size and high dimension with the multimodal medical data. In view of the characteristics of multimodal medical data, the existing genetic evolution random neural network cluster (GERNNC) model combine genetic evolution algorithm and neural network for the classification of AD patients and the extraction of pathogenic factors. However, the model does not take into account the non-linear relationship between brain regions and genes and the problem that the genetic evolution algorithm can fall into local optimal solutions, which leads to the overall performance of the model is not satisfactory. In order to solve the above two problems, this paper made some improvements on the construction of fusion features and genetic evolution algorithm in GERNNC model, and proposed an improved genetic evolution random neural network cluster (IGERNNC) model. The IGERNNC model uses mutual information correlation analysis method to combine resting-state functional magnetic resonance imaging data with single nucleotide polymorphism data for the construction of fusion features. Based on the traditional genetic evolution algorithm, elite retention strategy and large variation genetic algorithm are added to avoid the model falling into the local optimal solution. Through multiple independent experimental comparisons, the IGERNNC model can more effectively identify AD patients and extract relevant pathogenic factors, which is expected to become an effective tool in the field of AD research.

Random Cluster Model on Regular Graphs

For a graph G=(V,E) with v(G) vertices the partition function of the random cluster model is defined by ZG(q,w)=∑A⊆E(G)qk(A)w|A|,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} Z_G(q,w)=\\sum _{A\\subseteq E(G)}q^{k(A)}w^{|A|}, \\end{aligned}$$\\end{document}where k(A) denotes the number of connected components of the graph (V, A). Furthermore, let g(G) denote the girth of the graph G, that is, the length of the shortest cycle. In this paper we show that if (G_n)_n is a sequence of d-regular graphs such that the girth g(G_n)rightarrow infty , then the limit limn→∞1v(Gn)lnZGn(q,w)=lnΦd,q,w\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\lim _{n\\rightarrow \\infty } \\frac{1}{v(G_n)}\\ln Z_{G_n}(q,w)=\\ln \\Phi _{d,q,w} \\end{aligned}$$\\end{document}exists if qge 2 and wge 0. The quantity Phi _{d,q,w} can be computed as follows. Let Φd,q,w(t):=1+wqcos(t)+(q-1)wqsin(t)d+(q-1)1+wqcos(t)-wq(q-1)sin(t)d,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Phi _{d,q,w}(t):= & {} \\left( \\sqrt{1+\\frac{w}{q}}\\cos (t)+\\sqrt{\\frac{(q-1)w}{q}}\\sin (t)\\right) ^{d}\\\\{} & {} +\\,(q-1)\\left( \\sqrt{1+\\frac{w}{q}}\\cos (t) -\\sqrt{\\frac{w}{q(q-1)}}\\sin (t)\\right) ^{d}, \\end{aligned}$$\\end{document}then Φd,q,w:=maxt∈[-π,π]Φd,q,w(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Phi _{d,q,w}:=\\max _{t\\in [-\\pi ,\\pi ]}\\Phi _{d,q,w}(t), \\end{aligned}$$\\end{document}The same conclusion holds true for a sequence of random d-regular graphs with probability one. Our result extends the work of Dembo, Montanari, Sly and Sun for the Potts model (integer q), and we prove a conjecture of Helmuth, Jenssen and Perkins about the phase transition of the random cluster model with fixed q.

Damage spreading in the random cluster model

We investigate the damage spreading effect in the Fortuin-Kasteleyn random cluster model for 2- and 3-dimensional grids with periodic boundary. For 2D the damage function has a global maximum at p=q/(1+q) for all q>0 and also local maxima at p=1/2 and p=q/(1+q) for q≲0.75. For 3D we observe a local maximum at p=q/(1+q) for q≲0.46 and a global maximum at p=1/2 for q≲4.5. The chaotic phase of the model's (p,q)-parameter space is where the coupling time is of exponential order and we locate points on its boundary. For 3-dimensional grids the lower bound of this phase may be equal to the corresponding critical point of the q-state Potts model for q≥3.

On the Six-Vertex Model’s Free Energy

In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime Delta <1. As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when a=b=1 and cge 1, and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.

The random cluster model on the complete graph via large deviations

We study the emergence of the giant component in the random cluster model on the complete graph, which was first studied by Bollobás et al. (1996). We give an alternative analysis using a thermodynamic/large deviations approach introduced by Biskup et al. (2007) for the case of percolation. In particular, we compute the rate function for large deviations of the size of the largest connected component of the random graph for q≥1.

Mass Scaling of the Near-Critical 2D Ising Model Using Random Currents

We examine the Ising model at its critical temperature with an external magnetic field \(h a^{\frac{15}{8}}\) on \(a\mathbb {Z}^2\) for \(a,h >0\). A new proof of exponential decay of the truncated two-point correlation functions is presented. It is proven that the mass (inverse correlation length) is of the order of \(h^\frac{8}{15}\) in the limit \(h \rightarrow 0\). This was previously proven with CLE-methods in Camia et al. in (Commun Pure Appl Math 73(7):1371–405, 2020). Our new proof uses instead the random current representation of the Ising model and its backbone exploration. The method further relies on recent couplings to the random cluster model (Aizenman et al. in Invent Math 216:661–743, 2018) as well as a near-critical RSW-result for the random cluster model (Duminil-Copin and Manolescu in Planar Random-Cluster Model: Scaling Relations, 2020).

The critical mean-field Chayes–Machta dynamics

AbstractThe random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes–Machta (CM) dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrised by an edge probability p and a cluster weight q. Our focus is on the critical regime: $p = p_c(q)$ and $q \in (1,2)$ , where $p_c(q)$ is the threshold corresponding to the order–disorder phase transition of the model. We show that the mixing time of the CM dynamics is $O({\log}\ n \cdot \log \log n)$ in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the $\log\log n$ factor) for the mixing time of the mean-field CM dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the CM dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.

Geometric algebra and algebraic geometry of loop and Potts models

We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.

Component‐based nearest neighbour subspace clustering

In this paper, the problem of clustering data points that lie near or on a union of independent low-dimensional subspaces is addressed. To this end, the popular spectral clustering-based algorithms usually follow a two-stage strategy that initially builds an affinity matrix and then applies spectral clustering. However, an inappropriate affinity matrix that does not sufficiently connect data points lying on the same subspace will easily lead to the issue of over-segmentation. To alleviate this issue, building the affinity matrix based on subspace hypotheses generated by an iterative sampling operation according to the Random Cluster Model under the framework of energy minimisation is proposed. Specifically, each hypothesis is generated from a large number of data points by sampling a component in a K-nearest neighbour graph. Extensive experiments on synthetic data and real-world datasets show that the proposed method can improve the connectivity of the affinity matrix and provide competitive results against state-of-the-art methods.