Renormalization group for link percolation on planar hyperbolic manifolds

Kryven, Ivan; Ziff, Robert M.; Bianconi, Ginestra

doi:10.1103/physreve.100.022306

Cited by 25 publications

(41 citation statements)

References 57 publications

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“…The derivation of this result follows steps similar to the ones previously reported in Ref. [11]. Here we report these results for the self-consistency of this work.…”

Section: B Rg Theory At the Transcritical Bifurcation Pointsupporting

confidence: 81%

“…At the transcritical bifurcation point we observe a discontinuous percolation transition that is in the same universality class as percolation in the Farey graph [30] and in well-behaved 2d cell complexes [11]. The derivation of this result follows steps similar to the ones previously reported in Ref.…”

Section: B Rg Theory At the Transcritical Bifurcation Pointsupporting

confidence: 78%

“…In the following sections we will use the RG technique to predict that at the transcritical bifurcation point the percolation transition is discontinuous, similar to the Farey graph [30] and well-behaved generalized 2d hyperbolic manifolds [11]; at the saddle-node bifurcation point we predict a BKT transition and at the third-order pitchfork bifurcation point we predict a second-order critical behavior (see Fig. 8).…”

Section: A General Frameworkmentioning

confidence: 98%

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Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

2019

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Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in non-amenable two-dimensional branching simplicial and cell complexes. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with inter-layer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition. arXiv:1908.09392v1 [cond-mat.dis-nn]

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“…The derivation of this result follows steps similar to the ones previously reported in Ref. [11]. Here we report these results for the self-consistency of this work.…”

Section: B Rg Theory At the Transcritical Bifurcation Pointsupporting

confidence: 81%

Section: B Rg Theory At the Transcritical Bifurcation Pointsupporting

confidence: 78%

Section: A General Frameworkmentioning

confidence: 98%

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Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

2019

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“…By examining wrapping probabilities, Wang et al [16,17] simulated the bond and site percolation models on several threedimensional lattices, including simple cubic (SC), the diamond, body-centered cubic (BCC), and face-centered cubic (FCC) lattices. Other recent work on percolation includes [18][19][20][21][22][23][24][25][26][27].…”

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confidence: 99%

Precise bond percolation thresholds on several four-dimensional lattices

Xun

Ziff

2020

Phys. Rev. Research

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We study bond percolation on several four-dimensional (4D) lattices, including the simple (hyper) cubic (SC), the SC with combinations of nearest neighbors and second nearest neighbors (SC-NN+2NN), the body-centered cubic (BCC), and the face-centered cubic (FCC) lattices, using an efficient single-cluster growth algorithm. For the SC lattice, we find pc = 0.1601312(8), which confirms previous results (based on other methods), and find a new value pc = 0.035827(2) for the SC-NN+2NN lattice, which was not studied previously for bond percolation. For the 4D BCC and FCC lattices, we obtain pc = 0.074212(2) and 0.049517(2), which are substantially more precise than previous values. We also find critical exponents τ = 2.3135(5) and Ω = 0.40(3), consistent with previous results, including the recent four-loop series result of Gracey [Phys. Rev. D 92, 025012, (2015)], Ω = 0.4003.

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“…In other words, that the structure of a dynamic network may have memory of its past. Some examples of evolving network models include Price’s (1965) preferential attachment model 1 , the vertex copying model 2 , network optimisation models 3 , and branching simlicial complexes 4 , 5 . Preferential attachment, vertex copying models, and branching simplicial complexes all result in heavy-tailed degree distributions.…”

Section: Introductionmentioning

confidence: 99%

Coloured random graphs explain the structure and dynamics of cross-linked polymer networks

Schamboeck

Iedema

Kryven

2020

Sci Rep

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Step-growth and chain-growth are two major families of chemical reactions that result in polymer networks with drastically different physical properties, often referred to as hyper-branched and crosslinked networks. in contrast to step-growth polymerisation, chain-growth forms networks that are history-dependent. Such networks are defined not just by the degree distribution, but also by their entire formation history, which entails a modelling and conceptual challenges. We show that the structure of chain-growth polymer networks corresponds to an edge-coloured random graph with a defined multivariate degree distribution, where the colour labels represent the formation times of chemical bonds. The theory quantifies and explains the gelation in free-radical polymerisation of cross-linked polymers and predicts conditions when history dependance has the most significant effect on the global properties of a polymer network. As such, the edge colouring is identified as the key driver behind the difference in the physical properties of step-growth and chain-growth networks. We expect that this findings will stimulate usage of network science tools for discovery and design of cross-linked polymers. Already in the early days of network science it has been realised that in dynamic networks the entire time trajectory of network formation may reflect on the topological features of the structure that is formed at the end. In other words, that the structure of a dynamic network may have memory of its past. Some examples of evolving network models include Price's (1965) preferential attachment model 1 , the vertex copying model 2 , network optimisation models 3 , and branching simlicial complexes 4,5. Preferential attachment, vertex copying models, and branching simplicial complexes all result in heavy-tailed degree distributions. In this work, we introduce an evolving network model with an arbitrary degree distribution to show that the different effects induced by the two most common polymerisation processes on the resulting materials are provoked by the presence or absence of memory in the underlaying network structures. Step-growth and chain-growth polymerisation are two major families of chemical reactions that result in polymer networks. For step-growth, such polymers include polyethylene (PE), polyvinyl chloride (PVC), polypropylene (PP) and polyacrylates 6 , which have a variety of applications: including paints and adhesives 7,8 , food packaging 9 , biomaterials and medical devices 10,11. Well-known examples of chain-growth polymers are gels that find applications as absorbents for medical, chemical and agricultural purposes 12 , and coatings made by photopolymerisation 13,14. The reason for the difference in the physical and mechanical properties of the step-and chain-growth derived polymers lies in the distinct network structures. Moreover, the structure of chain-growth networks can be manipulated by adjusting polymerisation conditions and species concentrators in order to optimise physical and mechanical properties. ...

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Renormalization group for link percolation on planar hyperbolic manifolds

Cited by 25 publications

References 57 publications

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

Precise bond percolation thresholds on several four-dimensional lattices

Coloured random graphs explain the structure and dynamics of cross-linked polymer networks

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