Fluctuations for block spin Ising models

Abstract

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 :=

Fluctuation Results for General Block Spin Ising Models

“…Here we derive a standard CLT using the Hubbard-Stratonovich transform. This is in spirit similar to the third section in [26] and technically related to [20]. The result can also be derived from [15], where similar techniques are used.…”

“…which is exactly the covariance matrix in [26] (again up to a factor of 2). Note that similar results have been derived in [24].…”

“…formula (4.1) in [23]. Later, they were rediscovered as interesting models for statistical mechanics systems, see [18], [15], [8], [26], [24], as well as models for social interactions between several groups, e.g. in [17], [1], [28].…”

“…Our starting point is [26]. There, the fluctuations of an order parameter for a two-groups block model with equal block sizes were analyzed on the level of large deviations principles (LDPs, for short) and central limit theorems (CLTs).…”

Exact recovery in block spin Ising models at the critical line

“…Section 3 will generalize a result on the critical fluctuations of m 1 and m 1 − m 2 from [20]. In particular, we will treat the case of negative α which was omitted in [20]. These two ingredients will yield the proof of Theorem 1.3, which will be given in Section 4.…”