Abstract:We will present a brief overview of critical phenomena on periodic and scale free networks and ask if we can carry over field theoretic renormalization group methods from periodic to non-translationally invariant lattices with highly inhomogeneous connectivities. What is the analogue of taking the Fourier transform? How does one eliminate "short wavelength" fluctuations? What does rescaling mean in the absence of the notion of a distance? How are the critical exponents to be computed? Do new types of couplings arise which are not ordinarily present on periodic lattices? What takes the place of dimensionality? Is there an analogue of the upper critical dimensionality? What do we learn from the spectral density of the graph Laplacian? Do localized modes have a relevance to critical behaviour? Further questions bearing upon amorphous materials will be addressed.
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S.N. Dorogovtsev and A.V. Goltsev, "Critical phenomena in complex networks," Rev. Mod. Phys. 80, 1275 (2008).