We analyze the high temperature fluctuations of the magnetization of the so-called Ising block model. This model was recently introduced by Berthet, Rigollet and Srivastavaz in [2]. We prove a Central Limit Theorems (CLT) for the magnetization in the high temperature regime. At the same time we show that this CLT breaks down at a line of critical temperatures. At this line we show the validity of a non-standard Central Limit Theorems for the magnetization.A closely related version of this model has been investigated in [14]. However, the couplings in [14] between the blocks have the same strength of interaction as the couplings within a block. We were informed that a more general version of the model will be studied in [15]. Similar to the Curie-Weiss model the Ising block model has a order parameter, which in this case is two-dimensional: the vector of block magnetizations, m := m N :=
We study a block spin mean-field Ising model. For the vector of block magnetizations we prove Large Deviation Principles and Central Limit Theorems under general assumptions for the block interaction matrix. Using the exchangeable pair approach of Stein's method we are also able to establish a speed of convergence for the Central Limit Theorem for the vector of block magnetizations in the high temperature regime.
We analyze Ising/Curie-Weiss models on the Erdős-Rényi graph with N vertices and edge probability p = p(N ) that were introduced by Bovier and Gayrard [J. Statist. Phys.,: 1993] and investigated in [20] and [21]. We prove Central Limit Theorems for the partition function of the model and -at other decay regimes of p(N ) -for the logarithmic partition function. We find critical regimes for p(N ) at which the behavior of the fluctuations of the partition function changes.
We continue our analysis of Ising models on the (directed) Erdős–Rényi random graph. This graph is constructed on N vertices and every edge has probability p to be present. These models were introduced and first studied by Bovier and Gayrard (1993 J. Stat. Phys.
72 643–64) and further analyzed by the authors in a previous note, in which we consider the case of p = p(N) satisfying p
3
N
2 → ∞ and β < 1. In the current note we prove a quenched central limit theorem for the magnetization for p satisfying pN → ∞ in the high-temperature regime β < 1. We also show a non-standard central limit theorem for p
4
N
3 → ∞ at the critical temperature β = 1. For p
4
N
3 → 0 we obtain a Gaussian limiting distribution for the magnetization. Finally, in the critical regime p
4
N
3 → c the limiting distribution for the magnetization contains a Gaussian component as well as a
e
−
x
4
-term. Hence, at β = 1 we observe a phase transition in p for the fluctuations of the magnetization.
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