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Renormalization group calculations are used to give exact solutions for rigidity percolation on hierarchical lattices. Algebraic scaling transformations for a simple example in two dimensions produce a transition of second order, with an unstable critical point and associated scaling laws. Values are provided for the order parameter exponent $beta = 0.0775$ associated with the spanning rigid cluster and also for $d u = 3.533$ which is associated with an anomalous lattice dimension $d$ and the divergence in the correlation length near the transition. In addition we argue that the number of floppy modes $F$ plays the role of a free energy and hence find the exponent $alpha$ and establish hyperscaling. The exact analytical procedures demonstrated on the chosen example readily generalize to wider classes of hierarchical lattice.
We develop a theoretical approach to ``spontaneous stochasticity in classical dynamical systems that are nearly singular and weakly perturbed by noise. This phenomenon is associated to a breakdown in uniqueness of solutions for fixed initial data and
The structure of the UV divergencies in higher dimensional nonrenormalizable theories is analysed. Based on renormalization operation and renormalization group theory it is shown that even in this case the leading divergencies (asymptotics) are gover
A nonconventional renormalization-group (RG) treatment close to and below four dimensions is used to explore, in a unified and systematic way, the low-temperature properties of a wide class of systems in the influence domain of their quantum critical
We study the evolution of the probability density of ensembles of iterates of the logistic map that advance towards and finally remain at attractors of representative dynamical regimes. We consider the mirror families of superstable attractors along
For classical lattice systems with finite (Ising) spins, we show that the implementation of momentum-space renormalization at the level of Hamiltonians runs into the same type of difficulties as found for real-space transformations: Renormalized Hami