Limit theorems for sums of dependent random variables occurring in statistical mechanics

Ellis, Richard S.; Newman, Charles M.; Rosen, Jay

doi:10.1007/bf00536186

Cited by 97 publications

(130 citation statements)

References 4 publications

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“…Our interpretation is based on a series of rigorous mathematical works (Alberici et al, 2014a, b;Alberici and Contucci, 2014;Alberici et al, 2015Alberici et al, , 2016a where it has been shown that, in the presence of an imitative interaction among vertices (monomers), the model displays a phase transition: the dimer densities have a square-root growth by the critical point, where quartic exponential fluctuations are observed at the scale N 3/4 (rather than quadratic at the usual scale N 1/2 , i.e., the standard Gaussian scenario). This result is also typical of a large class of mean-field ferromagnetic spin models whose behaviour was understood in the works (Ellis and Newman, 1978a, b;Ellis et al, 1980;Ellis and Rosen, 1982). The imitative interaction, from a sociological modelistic point of view (Nowak, 2006;Hauert and Doebeli, 2004), is seen as the trustbond among two people.…”

Section: Introductionsupporting

confidence: 56%

Social interaction effects on immigrant integration

et al. 2018

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In recent years Italy has been involved in massive migration flows and, consequently, migrant integration is becoming a urgent political, economic and social issue. In this paper we apply quantitative methods, based on probability theory and statistical mechanics, to study the relative integration of migrants in Italy. In particular, we focus on the probability distribution of a classical quantifier that social scientists use to measure migrant integration, that is, the fraction of mixed (natives and immigrants) married couples, and we study, in particular, how it changes with respect to the migrant density. The analysed dataset collected yearly by ISTAT (Italian National Institute of Statistics), from 2002 to 2010, provides information on marriages and population compositions for all Italian municipalities. Our findings show that there are strong differences according to the size of the municipality. In fact, in large cities the occurrence of mixed marriages grows, on average, linearly with respect to the migrant density and its fluctuations are always Gaussian; conversely, in small cities, growth follows a square-root law and the fluctuations, which have a much larger scale, approach an exponential quartic distribution at very small densities. Following a quantitative approach, whose origins trace back to the probability theory of interacting systems, we argue that the difference depends on how connected the social tissue is in the two cases: large cities present a highly fragmented social network made of very small isolated components while villages behave as percolated systems with a rich tie structure where isolation is rare or completely absent. Our findings are potentially useful for policy makers; for instance, the incentives towards a smooth integration of migrants or the size of nativist movements should be predicted based on the size of the targeted population.

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Section: Introductionsupporting

confidence: 56%

Social interaction effects on immigrant integration

et al. 2018

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“…The case of the Curie-Weiss-Ising model (q = 2) was reported in [11] (see [12,13] for the proofs), while the Curie-Weiss Potts model was treated at zero external field in [14].…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Limit theorems and coexistence probabilities for the Curie–Weiss Potts model with an external field

Gandolfo¹,

Ruiz²,

Wouts³

2010

Stochastic Processes and their Applications

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The Curie-Weiss Potts model is a mean field version of the well-known Potts model. In this model, the critical line β = β c (h) is explicitly known and corresponds to a first-order transition when q > 2. In the present paper we describe the fluctuations of the density vector in the whole domain β 0 and h 0, including the conditional fluctuations on the critical line and the non-Gaussian fluctuations at the extremity of the critical line. The probabilities of each of the two thermodynamically stable states on the critical line are also computed. Similar results are inferred for the random-cluster model on the complete graph.

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“…This idea was successfully used in the context of the analysis of the Curie-Weiss model in e.g. [12] and already introduced to the analysis of the Hopfield model in [27]. It consists of convoluting our measures of interest with N (0, 1 βN I M(N) ), i.e.…”

Section: Corollary 22mentioning

confidence: 99%

Moderate Deviations for the overlap parameter in the Hopfield model

Eichelsbacher

Löwe

2004

Probab. Theory Relat. Fields

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We derive moderate deviation principles for the overlap parameter in the Hopfield model of spin glasses and neural networks. If the inverse temperature β is different from the critical inverse temperature β c = 1 and the number of patterns M(N) satisfies M(N )/N → 0, the overlap parameter multiplied by N γ , 1/2 < γ < 1, obeys a moderate deviation principle with speed N 1−2γ and a quadratic rate function (i.e. the Gaussian limit for γ = 1/2 remains visible on the moderate deviation scale). At the critical temperature we need to multiply the overlap parameter by N γ , 1/4 < γ < 1. If then M(N) satisfies (M(N ) 6 log N ∧M(N) 2 N 4γ log N)/N → 0, the rescaled overlap parameter obeys a moderate deviation principle with speed N 1−4γ and a rate function that is basically a fourth power. The random term occurring in the Central Limit theorem for the overlap at β c = 1 is no longer present on a moderate deviation scale. If the scaling is even closer to N 1/4 , e.g. if we multiply the overlap parameter by N 1/4 log log N the moderate deviation principle breaks down. The case of variable temperature converging to one is also considered. If β N converges to β c fast enough, i.e. faster than O(N −2γ ), the non-Gaussian rate function persists, whereas for β N converging to one slower than O(N −2γ ), the moderate deviations principle is given by the Gaussian rate. At the borderline the moderate deviation rate function is the one at criticality plus an additional Gaussian term.

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Limit theorems for sums of dependent random variables occurring in statistical mechanics

Cited by 97 publications

References 4 publications

Social interaction effects on immigrant integration

Social interaction effects on immigrant integration

Limit theorems and coexistence probabilities for the Curie–Weiss Potts model with an external field

Moderate Deviations for the overlap parameter in the Hopfield model

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