Critical phenomena on scale-free networks: logarithmic corrections and scaling functions

Abstract

In this paper, we address the logarithmic corrections to the leading power laws that govern thermodynamic quantities as a second-order phase transition point is approached. For phase transitions of spin systems on d-dimensional lattices, such corrections appear at some marginal values of the order parameter or space dimension. We present new scaling relations for these exponents. We also consider a spin system on a scale-free network which exhibits logarithmic corrections due to the specific network properties. To this end, we analyze the phase behavior of a model with coupled order parameters on a scale-free network and extract leading and logarithmic correction-to-scaling exponents that determine its field- and temperature behavior. Although both non-trivial sets of exponents emerge from the correlations in the network structure rather than from the spin fluctuations they fulfil the respective thermodynamic scaling relations. For the scale-free networks the logarithmic corrections appear at marginal values of the node degree distribution exponent. In addition we calculate scaling functions, which also exhibit nontrivial dependence on intrinsic network properties.

15 pages, 4 figures

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• The two-dimensional 4-state Potts model in a magnetic field
• The Diffusive Epidemic Process on Barabasi-Albert Networks
• Ising model with invisible states on scale-free networks
• Critical behavior of the 2D Ising model modulated by the Octonacci sequence
• The Fate of Ernst Ising and the Fate of his Model
• Scaling functions and amplitude ratios for the Potts model on an uncorrelated scale-free network