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About Ising and Potts Models on Cayley Trees and Bayesian Networks ABOUT ISING AND POTTS MODELS ON CAYLEY TREES AND BAYESIAN NETWORKS GERARD COHEN TERVAERT, SUPERVISED BY A. VAN ENTER, AND D. VALESIN Master Project Mathematics March 2019 Student: G.D. Cohen Tervaert First supervisor: Prof. Dr. A. C. D. van Enter Second supervisor: Dr. D. Valesin 1 2 GERARD COHEN TERVAERT, SUPERVISED BY A. VAN ENTER, AND D. VALESIN Abstract. We perform numerical estimates for the q-state Potts model on a k-order Cayley tree. K¨ulske, Rozikov and Khakimov explicitly calculated up to 2q − 1 TISGMs (Translation Invariant Splitting Gibbs Measures) for the binary tree (k = 2), without an external field (α = 0). We extend these results numerically for k > 2 and α 6= 0. We conjecture that for α ≥ k − 1 the model has uniqueness. Additionally, decay of memory is proved for a Potts-type model on a Bayesian network with up to two parents and a counterexample is given for a more general case. Contents 1. Introduction 3 2. Cayley tree 4 2.1. Definitions 4 3. Ising model 4 3.1. Definitions 4 3.2. Phase transitions 5 3.3. Compatible measures 6 3.4. Numerical example 6 4. Potts model 7 4.1. Definition 7 4.2. Compatible measures 8 4.3. Uncountable set of Gibbs measures 9 4.4. Translation invariant Gibbs measures 9 4.5. Binary tree explicit solutions 12 4.6. Higher-order trees 13 4.7. External fields 16 4.8. Dobrushin's condition 18 5. Bayesian Networks 22 5.1. Definition 22 5.2. Excluding remote generations 23 6. Decay of memory on Bayesian networks 25 6.1. Ising-type model 25 6.2. Potts-type model 26 6.3. Counterexample for a general discrete model 35 7. Discussion and Conclusion 38 References 38 Internet 39 Applications 39 ABOUT ISING AND POTTS MODELS ON CAYLEY TREES AND BAYESIAN NETWORKS 3 1. Introduction In this thesis, several systems of random variables will be discussed. The theory of "interacting random variables" is a domain of probability theory that is gaining popularity. The goal of this theory is comprehending behaviour of large random systems. Applications are mainly in statistical mechanics, but also in other fields such as computer science or biology. In statistical mechanics, the concept 'phase transition' is often relevant. Phase transitions are sudden shifts of the state which can occur at certain temperatures. For example, matter changing from liquid to solid state. These phase transitions for the Ising model on a Cayley tree have been extensively studied [3]. For the Potts model on a Cayley tree there is still some ground to explore. The Ising model is used to describe ferromagnetism in statistical mechanics. The Potts model is a generalisation of the Ising model, in which the spins can take up to q different states whereas in the Ising model they can only take the values −1 and +1. These models consist of networks of interacting random variables. They will be described first as Gibbs measures on Cayley trees and later as Bayesian Networks to simulate inter-dependencies. In this thesis two topics will be investigated: First, we will describe the Ising and the Potts model on a Cayley tree. Results from the literature will be presented and extended with our own numerical results. Then, we will discuss decay of memory on Bayesian networks. We will present results from our efforts to set conditions for decay, when all variables have two parents or less. That is, earlier results from the Ising model are generalised to the Potts model and a counterexample for decay is given for a more general case. Throughout the thesis, to show the dynamics of these models, computer simu- lations will be added for both the Ising and the Potts model. For readability's sake, we try not to complicate formulas if not necessary and give some explanation where appropriate (E.g. We omit variables that are not relevant for the calculations and we expand extra on the transition from h~ 2 Rq to h 2 Rq−1 in the Potts model). This work extends the main theorem of my Bachelor thesis to a Potts-type model. 4 GERARD COHEN TERVAERT, SUPERVISED BY A. VAN ENTER, AND D. VALESIN 2. Cayley tree A tree in which each non-leaf graph vertex has a con- stant number of branches n is called a (n − 1)-Cayley tree. 1-Cayley trees are path graphs. The picture on the right shows a 2-Cayley tree. Both the Ising model as well as the Potts model will be defined as a collection of random variables on a Cayley tree. 2.1. Definitions. Let Γk = (V; L) be the Cayley tree of order k, which is a (k + 1) regular infinite tree. V is the set of vertices and L the set of edges. Two vertices x 2 V and y 2 V are called nearest neighbours if there exists an edge l 2 L : l = hx; yi: A path from x0 2 V to xn 2 V is a collection of nearest neighbour pairs hx0; x1i; hx1; x2i; :::; hxn−1; xni. The distance d(x0; xn) = n is the number of edges of the shortest path from x0 to xn. For a fixed arbitrary vertex x0 2 V called the root, we set n [ Wn = fx 2 V jd(x; x0) = ng;Vn = Wi i=0 The set of direct successors of x 2 Wn will be defined as: S(x) = fy 2 Wn+1 : d(x; y) = 1g Ln will be the subset of edges restricted to Vn: Ln = fhx; yi 2 L : x 2 Vn & y 2 Vng 3. Ising model The Ising model is named after the German physicist Ernst Ising. It describes ferromagnetism and consists of a collection of interdependent random variables, or spins, that can be in one of two states (+1 or −1). The model allows the identification of phase transitions. The one-dimensional Ising model has no phase transition and was solved by Ising himself in his 1924 thesis. The two-dimensional square-lattice Ising model is a simple statistical model that does show a phase transition. 3.1. Definitions. Let Γk = (V; L) be the infinite Cayley tree of order k, where V is the set of vertices and L is the set of edges. The state space X = {−1; 1g and the set of configurations is defined as Ω = XV . We say that for Λ ⊂ V and σ 2 Ω: σΛ = fσλ : λ 2 Λg. Let J > 0. The Hamiltonian of the ferromagnetic Ising model on the volume Vn is defined as: 0 X Hn(σ) = −J σxσy hx;yi2Ln where hx; yi is summed over all pairs of nearest neighbour vertices. Introducing the ABOUT ISING AND POTTS MODELS ON CAYLEY TREES AND BAYESIAN NETWORKS 5 boundary condition η = fηi 2 R : i 2 V g on the volume Vn yields: η X X Hn(σ) = −J σxσy − J σxηy hx;yi2Ln hx;yi2Ln+1 x2Wn y2Wn+1 We say that ηx = 1 for all x 2 V is the plus boundary condition, ηx = −1 for all x 2 V is the minus boundary condition and ηx = 0 for all x 2 V is the free boundary condition. ¯ Next, we add possibly spatially dependent external fields h = fhi 2 R : i 2 Ng: n η η X X Hn;h¯ (σ) = Hn(σ) − hkσx k=0 x2Wk If we add the inverse temperature β > 0, we can define the Gibbs measure for the Ising model on a finite Cayley tree with boundary condition η and with spatially dependent external fields h¯ by exp{−βHη (σ)g η n;h¯ (3.1) µn,β;h¯ (σ) = η Zn,β;h¯ η Zn,β;h¯ is the partition function: η X η Zn,β;h¯ = exp{−βHn;h¯ (σ)g σ2ΩVn The set of Gibbs measures Gβ;h¯ is defined as the convex hull of the set of all weak limits of the Gibbs measures on the volumes Vn. All measures that correspond to some sequence of boundary conditions belong to the set: ηmn Gβ;h¯ = conv µ : lim µ ¯ (σ) = µ; for all increasing (mn)n≥1; and (ηn)n≥1 n!+1 mn,β;h The model is said to undergo a phase transition if there exists a β for which jGβ;h¯ j > + − 1. This existence of a phase transition is equivalent to stating that µβ;h¯ 6= µβ;h¯ , + + ηn where µβ;h¯ = limn!1 µn,β;h¯ , the limit measure of the plus boundary condition − + − ηn (ηn;x = 1 : n 2 N; x 2 V ), and µβ;h¯ = limn!1 µn,β;h¯ , the limit measure of the − minus boundary condition (ηn;x = −1 : n 2 N; x 2 V ). The model is said to have uniqueness at inverse temperature β if jGβ;h¯ j = 1. k 3.2. Phase transitions. On Z , a homogeneous external field hn = h 2 Rnf0g for all n ≥ 0, implies uniqueness. Absence of an external field implies appearance of phase transitions for k > 1. However, on a Cayley tree, phase transitions can also appear if the non-zero external fields are weak enough. 6 GERARD COHEN TERVAERT, SUPERVISED BY A. VAN ENTER, AND D. VALESIN 3.3. Compatible measures. Probability measures, as defined in (3.1), are called compatible if for all n ≥ 1 and σ 2 ΩVn−1 : X (3.2) µn(σ _ !) = µn−1(σ) !2ΩWn Here σ _ ! represents the concatenation of configurations σ and !. For each sequence of compatible measures, Kolmogorov's theorem states that there exists a unique measure µ such that for all n and σVn 2 ΩVn : µ(σjVn = σVn ) = µn(σVn ) Such a measure µ is called a splitting Gibbs measure.
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