Continuous spin models on annealed generalized random graphs

Dommers, Sander; Külske, Christof; Philipp, Schriever,

doi:10.1016/j.spa.2017.03.009

Cited by 3 publications

(2 citation statements)

References 44 publications

(59 reference statements)

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“…In [7], continuous spin models on random graphs were studied in the annealed setting, also resulting in a mean-field approximation. It would be interesting to see if the central limit theorem for the sum of spins can also be proved using our techniques for that model.…”

Section: Discussionmentioning

confidence: 99%

Berry–Esseen bounds in the inhomogeneous Curie–Weiss model with external field

Dommers

Eichelsbacher

2020

Stochastic Processes and their Applications

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We study the inhomogeneous Curie-Weiss model with external field, where the inhomogeneity is introduced by adding a positive weight to every vertex and letting the interaction strength between two vertices be proportional to the product of their weights. In this model, the sum of the spins obeys a central limit theorem outside the critical line. We derive a Berry-Esseen rate of convergence for this limit theorem using Stein's method for exchangeable pairs. For this, we, amongst others, need to generalize this method to a multidimensional setting with unbounded random variables.

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Section: Discussionmentioning

confidence: 99%

Berry–Esseen bounds in the inhomogeneous Curie–Weiss model with external field

Dommers

Eichelsbacher

2020

Stochastic Processes and their Applications

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“…In these models the random graphs are often assumed (or proven in the largegraph limit) to be locally tree-like. To understand the behavior of these systems it is important to understand the situation on the tree first [4][5][6][7]. For the Ising model on an arbitrary infinite tree with constant interaction strength J the exact critical inverse temperature β c of phase transition is well-known and is given by Jβ c = k coth −1 (br T ), where k is Boltzmann's constant and br T is the branching number of the tree which captures the average number of edges per vertex [25].…”

Section: Introductionmentioning

confidence: 99%

Non-robust Phase Transitions in the Generalized Clock Model on Trees

Külske¹,

Philipp²

2017

J Stat Phys

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Abstract. Pemantle and Steif provided a sharp threshold for the existence of a RPT (robust phase transition) for the continuous rotator model and the Potts model in terms of the branching number and the second eigenvalue of the transfer operator, where a robust phase transition is said to occur if an arbitrarily weak coupling with symmetrybreaking boundary conditions suffices to induce symmetry breaking in the bulk. They further showed that for the Potts model RPT occurs at a different threshold than PT (phase transition in the sense of multiple Gibbs measures), and conjectured that RPT and PT should occur at the same threshold in the continuous rotator model.We consider the class of 4-and 5-state rotation-invariant spin models with reflection symmetry on general trees which contains the Potts model and the clock model with scalarproduct-interaction as limiting cases. The clock model can be viewed as a particular discretization which is obtained from the classical rotator model with state space S 1 . We analyze the transition between PT=RPT and PT =RPT, in terms of the eigenvalues of the transfer matrix of the model at the critical threshold value for the existence of RPT. The transition between the two regimes depends sensitively on the third largest eigenvalue.Mathematics Subject Classifications (2010). 82B26 (primary); 60K35 (secondary)

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Large Deviations for the Annealed Ising Model on Inhomogeneous Random Graphs: Spins and Degrees

et al. 2018

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We prove a large deviations principle for the total spin and the number of edges under the annealed Ising measure on generalized random graphs. We also give detailed results on how the annealing over the Ising model changes the degrees of the vertices in the graph and show how it gives rise to interesting correlated random graphs.where ℓ n = i∈[n] w i is the total weight of all vertices. Denote the law of GRG n (w) by P and its expectation by E. There are many related random graph models (also called rank-1 inhomogeneous random graphs [2]), such as the random graph with specified expected degrees or Chung-Lu model [6,7] and the Poisson random graph or Norros-Reittu model [24]. Janson [21] shows that many of these models are asymptotically equivalent. Even though his results do not apply to the large deviation properties of these random graphs, all our results also apply to these other models.We need to assume that the vertex weight sequences w = (w i ) i∈[n] are sufficiently nicely behaved. Let U n ∈ [n] denote a uniformly chosen vertex in GRG n (w) and W n = w Un its weight. Then, the following condition defines the asymptotic weight W and set the convergence properties of (W n ) n≥1 to W : Condition 1.1 (Weight regularity). There exists a random variable W such that, as n → ∞, (a) W n D −→ W , where D −→ denotes convergence in distribution;

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Continuous spin models on annealed generalized random graphs

Cited by 3 publications

References 44 publications

Berry–Esseen bounds in the inhomogeneous Curie–Weiss model with external field

Berry–Esseen bounds in the inhomogeneous Curie–Weiss model with external field

Non-robust Phase Transitions in the Generalized Clock Model on Trees

Large Deviations for the Annealed Ising Model on Inhomogeneous Random Graphs: Spins and Degrees

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