Related papers: The Random-Cluster Model
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point. By combining the replica-exchange algorithm with cluster updates…
Over the last few years it was pointed out that certain observables of time-evolving quantum systems may have singularities at certain moments in time, mimicking the singularities physical systems have when undergoing phase transitions.…
We consider the fractal dimensions of critical clusters occurring in configurations of a q-state Potts model coupled to the planar random graphs of the dynamical triangulations formulation of Euclidean quantum gravity in two dimensions. For…
A sharp-threshold theorem is proved for box-crossing probabilities on the square lattice. The models in question are the random-cluster model near the self-dual point $p_{\mathrm {sd}}(q)=\sqrt{q}/(1+\sqrt{q})$, the Ising model with…
We develop a fluctuation theory of connectivities for subcritical random cluster models. The theory is based on a comprehensive nonperturbative probabilistic description of long connected clusters in terms of essentially one-dimensional…
In critical percolation models, in a large cube there will typically be more than one cluster of comparable diameter. In 2D, the probability of $k>>1$ spanning clusters is of the order $e^{-\alpha k^{2}}$. In dimensions d>6, when $\eta = 0$…
The concept of random dynamical system is a comparatively recent development combining ideas and methods from the well developed areas of probability theory and dynamical systems. Due to our inaccurate knowledge of the particular physical…
The random-cluster model is a unifying framework for studying random graphs, spin systems in physics and random spanning trees. The model is closely related to, though much more general than the classical Ising and Potts models, but its…
An important paradigm in the understanding of mixing times of Glauber dynamics for spin systems is the correspondence between spatial mixing properties of the models and bounds on the mixing time of the dynamics. This includes, in…
The invaded cluster algorithm, a new method for simulating phase transitions, is described in detail. Theoretical, albeit nonrigorous, justification of the method is presented and the algorithm is applied to Potts models in two and three…
We report on single-cluster Monte Carlo simulations of the Ising, 4-state Potts and 10-state Potts models on quenched ensembles of planar, tri-valent random graphs. We confirm that the first-order phase transition of the 10-state Potts…
Mixture models are probabilistic models aimed at uncovering and representing latent subgroups within a population. In the realm of network data analysis, the latent subgroups of nodes are typically identified by their connectivity…
An approach to analyse the properties of a particle system is to compare it with different processes to understand when one of them is larger than other ones. The main technique for that is coupling, which may not be easy to construct. We…
We discuss the critical behaviour of 2D Ising and q-states Potts models coupled by their energy density. We found new tricritical points. The procedure employed is the renormalisation approach of the perturbations series around conformal…
As a problem in data science the inverse Ising (or Potts) problem is to infer the parameters of a Gibbs-Boltzmann distributions of an Ising (or Potts) model from samples drawn from that distribution. The algorithmic and computational…
Complex systems in nature and in society are often represented as networks, describing the rich set of interactions between objects of interest. Many deterministic and probabilistic clustering methods have been developed to analyze such…
The phase transitions of random-field q-state Potts models in d=3 dimensions are studied by renormalization-group theory by exact solution of a hierarchical lattice and, equivalently, approximate Migdal-Kadanoff solutions of a cubic…
In real life, networks are dynamic in nature; they grow over time and often exhibit power-law degree sequences. To model the evolving structure of the internet, Barab\'{a}si and Albert introduced a simple dynamic model with a power-law…
Motivated by recent progress on the scaling behavior of entanglement entropy, we study the scaling behavior of the number of clusters crossing the boundary between two subsystems for several classical statistical models in two dimension.…
We study gravitational clustering of mass points in three dimensions with random initial positions and periodic boundary conditions (no expansion) by numerical simulations. Correlation properties are well defined in the system and a sort of…