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Essential properties of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
is the partition function of the model. The Potts model represents a physical system in which particles have spins that may take one of q values, and neighboring particles like to have the same spin (the ferromagnetic Potts model). As so often in statistical physics, the partition function is a key quantity that encodes much of the behavior of the system. It may be calculated easily from the Tutte polynomial: ZP(G;β,q)=e−βe(G)qk(G)(eβ−1)r(G)TG;1+qeβ−1,eβ;
Magnetic Properties of Nanostructures in Non-Integer Dimensions
Published in Jean-Claude Levy, Magnetic Structures of 2D and 3D Nanoparticles, 2018
The Potts model generalizes the Ising one by taking into account the fact that spins can be in more than two discrete states. The Hamiltonian of the q-state Potts model reads H = −J ∑⟨i,j⟩δ(σi, σj), where σi and σj designate the spin states at the occupied sites i and j of the lattice and can take the integer values 1, 2, …, q. The sum runs over the nearest-neighbor spins pairs, δ(σi, σj) is equal to 1 if σi = σj, and 0 otherwise. J > 0 is the exchange ferromagnetic coupling uniform all over the lattice. For a given size L and a given spin configuration, the order parameter of the phase transition reads mL=qρL−1q−1 where ρL = max{N1/N, ⋯, Nq/N} and Nq0 is the number of spins whose state is q0. Since the number of states q is related to the symmetrical properties of the order parameter, the critical behavior of the Potts model, in particular the universality class (excepted for q = 2), depends upon the value of this additional variable. Furthermore, one of its most striking features is the effect of q on the order of the transition as a function of the space dimension d: In the case of translationally invariant lattices, the phase transition in the bidimensional case is first order for q > 4 and second order for q ≤ 4, whereas in the three-dimensional case, it is first order for q 3 and second order for q 2. This suggests the relevance of a phase diagram in the (d, q) plane [16], generalized to non-integer values of q and d, where some border should separate a first-order domain, where q is larger than a critical value qc(d), from a second-order one. The question of the physical meaning of the extrapolation to non-integer integer dimensions leads naturally to study phase transitions of the Potts model on fractal structures.
Mixture of Regression Models for Large Spatial Datasets
Published in Technometrics, 2019
For simplicity, it is often assumed that the interactions between sites are pairwise; that is, clique potentials are nonzero only for cliques containing one or two sites (e.g., sets (a) through (c) in Figure 1). In this case, the energy function can be written aswhere and represent clique potentials for a single site and a pair of neighboring sites, respectively. A simple model is the homogeneous, isotropic Potts model, which is obtained by setting , implying that a priori all classes are equally likely, and Here, the scalar is the only parameter of the MRF, and it controls the likelihood of neighboring sites belonging to the same component. Setting ψ = 0 corresponds to spatial independence, while larger values of ψ denote stronger spatial dependence. The joint distribution of the Potts model is then where denotes distinct pairs of neighboring sites. The Potts model is the K-state generalization of the popular two-state Ising model (Celeux, Forbes, and Peyrard 2003; Dang and Govaert 1998).
Data-driven subsurface modelling using a Markov random field model
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023
Takayuki Shuku, Kok-Kwang Phoon
The energy function E can be considered as a model for spatial structure of x, and the following models have been commonly used in image processing: where xk is a “state” or “category” at site k, θ is a hyperparameter called “inverse temperature”, and q is the number of states or categories. These two models are originally designed for describing physical phenomena such as magnetisms and phase transitions in thermodynamics. In the context of DDSC, they play a role of prior information of spatial patterns/distributions of soil types. While Ising model considers only binary states, Potts model can consider any numbers of states/soil types in principle. Therefore, it is natural to use Potts model for DDSC. In subsurface modelling, states correspond to “soil types”, and a classification system called Robertson chart, which is widely used in geotechnical practice, can be used. In Robertson chart, soil types (or soil behaviour types, SBTs) are classified based on soil behaviour type index Ic (Table 1). The notation {k, l} indicates that sites k and l are nearest neighbours (Figure 2). The Potts model with q = 2 is equivalent to the Ising model. The numerical algorithm to sample from an MRF is shown in Algorithm 1. This employs a standard Metropolis algorithm (e.g. Metropolis et al. 1953). Figure 3 shows changes of energy E(x) over iterations simulated using a four states Potts model with θv = θh = 1.0 and Algorithm 1. In the figure, simulated images that corresponds to the results of itr = 100, 1000, 50,000 and 50,000 are also shown. Simpler images are generated by increasing with iterations or with decreasing with E(x). Simulation usually runs until the energy converges.