Continuous spin models on annealed generalized random graphs

Abstract

We study Gibbs distributions of spins taking values in a general compact Polish space, interacting via a pair potential along the edges of a generalized random graph with a given asymptotic weight distribution $P$, obtained by annealing over the random graph distribution. First we prove a variational formula for the corresponding annealed pressure and provide criteria for absence of phase transitions in the general case. We furthermore study classes of models with second order phase transitions which include rotation-invariant models on spheres and models on intervals, and classify their critical exponents. We find critical exponents which are modified relative to the corresponding mean-field values when $P$ becomes too heavy-tailed, in which case they move continuously with the tail-exponent of $P$. For large classes of models they are the same as for the Ising model treated in \cite{DomGiaGibHofPri16}. On the other hand, we provide conditions under which the model is in a different universality class, and construct an explicit example of such a model on the interval.

References in corpus (13)

• The structure and function of complex networks
• Critical phenomena in complex networks
• Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
• Ising models on locally tree-like graphs
• Ising-like dynamics in large-scale functional brain networks
• Critical Behaviour of the Randomly Spin--Diluted 2-d Ising Model --- A Grand Ensemble Approach
• Phase coexistence of gradient Gibbs states
• Ising models on power-law random graphs
• Ising critical behavior of inhomogeneous Curie-Weiss models and annealed random graphs
• Metastability of the Ising model on random regular graphs at zero temperature
• Comment on: Critical behavior on the spin diluted 2D Ising model: A grand ensemble approach, by R. Kuehn
• The Ising model on the random planar causal triangulation: bounds on the critical line and magnetization properties
• Sharp thresholds for Gibbs-non-Gibbs transition in the fuzzy Potts model with a Kac-type interaction

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