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On near-critical and dynamical percolation in the tree case [article]

Olle Haggstrom, Robin Pemantle
2004 arXiv   pre-print
Here we relate this phenomenon to the near-critical behaviour of theta_Gamma(p) by showing that the root percolates in the dynamical percolation process if and only if int_p_c^1 (theta_Gamma(p))^-1dp<infty  ...  If theta_Gamma(p_c)=0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p_c, as demonstrated recently by Haggstrom, Peres and Steif.  ...  An ultimate goal might be to find some "if and only if" criterion in terms of the near-critical behaviour of θ G (p), determining whether there are percolating times for the critical dynamical percolation  ... 
arXiv:math/0404091v1 fatcat:khmyfs7zpja4lfryatu2hnvdvy

On near-critical and dynamical percolation in the tree case

Olle H�ggstr�m, Robin Pemantle
1999 Random structures & algorithms (Print)  
Here we relate this phenomenon to the near-critical behaviour of θ Γ (p) by showing that the root percolates in the dynamical percolation process if and only if 1 pc (θ Γ (p)) −1 dp < ∞.  ...  If θ Γ (p c ) = 0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p c , as demonstrated recently by Häggström, Peres and Steif.  ...  An ultimate goal might be to find some "if and only if" criterion in terms of the near-critical behaviour of θ G (p), determining whether there are percolating times for the critical dynamical percolation  ... 
doi:10.1002/(sici)1098-2418(199910/12)15:3/4<311::aid-rsa7>3.0.co;2-6 fatcat:uztixmpcv5gslm4xo7motkcpni

On near‐critical and dynamical percolation in the tree case

Olle Häggström, Robin Pemantle
1999 Random structures & algorithms (Print)  
Here we relate this phenomenon to the near-critical behaviour of θ Γ (p) by showing that the root percolates in the dynamical percolation process if and only if 1 pc (θ Γ (p)) −1 dp < ∞.  ...  If θ Γ (p c ) = 0, then the root may still percolate in the corresponding dynamical percolation process at the critical value p c , as demonstrated recently by Häggström, Peres and Steif.  ...  An ultimate goal might be to find some "if and only if" criterion in terms of the near-critical behaviour of θ G (p), determining whether there are percolating times for the critical dynamical percolation  ... 
doi:10.1002/(sici)1098-2418(199910/12)15:3/4<311::aid-rsa7>3.3.co;2-y fatcat:okskpgd7dbhmtp5s2h2sjroc4i

Page 4847 of Mathematical Reviews Vol. , Issue 90H [page]

1990 Mathematical Reviews  
In the same setting, the critical temperatures are derived for spin percolation, in both the full and diluted models. For the case of no interaction, percolation on tree-like graphs is studied.  ...  B. (4-OX-TP) Critical dynamics in Heisenberg ferromagnets and antiferromagnets near the percolation threshold. J. Phys. A 22 (1989), no. 18, 3959-3967.  ... 

The scaling limit of the Minimal Spanning Tree - a preliminary report [article]

Christophe Garban, Gábor Pete, Oded Schramm
2009 arXiv   pre-print
We describe how the recent proof of the existence and conformal covariance of the scaling limits of dynamical and near-critical planar percolation implies the existence and several topological properties  ...  of the scaling limit of the Minimal Spanning Tree, and that it is invariant under scalings, rotations and translations.  ...  Hence we get a well-defined Markov process in the scaling limit, both in the dynamical and the near-critical cases.  ... 
arXiv:0909.3138v1 fatcat:2wto6o3ktncf5aovlffstzvmfu

Invading interfaces and blocking surfaces in high-dimensional disordered systems

Omri Gat, Zeev Olami
1997 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
We find that the critical exponents in d=6 are very close to their values on the Cayley tree, and we conjecture on this basis the existence of a further dimension, where mean field behavior is obtained  ...  We present numerical simulations in which we measured the values of the critical exponents of the depinning transition, both in finite dimensional lattices with d≤6 and on the Cayley tree, which support  ...  There is however a subtle connection between the dynamics of a growing AP and the dynamics of a percolation cluster.  ... 
doi:10.1103/physreve.56.1714 fatcat:xc2djdy2gnajfk5s6gqyi4gjbi

Two-Dimensional Scaling Limits via Marked Nonsimple Loops [article]

F. Camia, L. R. G. Fontes, C. M. Newman
2006 arXiv   pre-print
We explain how these marked loops should yield continuum versions of near-critical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation.  ...  in 2D critical percolation.  ...  dynamical percolation (Section 5), near-critical exploration paths (Section 6), minimal spanning trees and related lattice-filling curves (Section 7), and invasion percolation (Section 8).  ... 
arXiv:cond-mat/0604390v1 fatcat:j53u7zb2trcl7fvavud4sna5ku

Dynamics of driven interfaces near isotropic percolation transition

M.-P. Kuittu, M. Haataja, N. Provatas, T. Ala-Nissila
1998 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
We consider the dynamics and kinetic roughening of interfaces embedded in uniformly random media near percolation treshold.  ...  Away from the percolation transition, the interface is self-affine with asymptotic dynamics consistent with the Kardar-Parisi-Zhang universality class.  ...  ACKNOWLEDGMENTS We wish to thank Tommi Karhela for help at the initial stages of this work and J. Krug for useful discussions. This work has been supported in part by the Academy of Finland.  ... 
doi:10.1103/physreve.58.1514 fatcat:2ikcxxwmlfebfcblv7mghrbwpu

Page 1386 of The Journal of Ecology Vol. 103, Issue 6 [page]

2014 The Journal of Ecology  
One of the earlier examples of spatial large-scale pattern formation in vegetation, in their case, the arrangement of vegetation in bands or belts along an altitudi- nal gradient as a consequence of physical  ...  with the existence of a critical percolation threshold for fire spread. 4.  ... 

A survey on dynamical percolation [article]

Jeffrey E. Steif
2009 arXiv   pre-print
Percolation is one of the simplest and nicest models in probability theory/statistical mechanics which exhibits critical phenomena.  ...  Our goal in thissurvey is to give an overview of the work in dynamical percolation that has been done (and some of which is in the process of being written up).  ...  Acknowledgments I would like to thank Erik Broman, Christophe Garban, Olle Häggström, Alan Hammond, Russell Lyons and Gábor Pete for very many helpful comments on earlier versions of the manuscript.  ... 
arXiv:0901.4760v2 fatcat:denxyge75nhstk7xsfsyat7taa

A Survey of Dynamical Percolation [chapter]

Jeffrey E. Steif
2009 Fractal Geometry and Stochastics IV  
Percolation is one of the simplest and nicest models in probability theory/statistical mechanics which exhibits critical phenomena.  ...  Our goal in this survey is to give an overview of the work in dynamical percolation that has been done (and some of which is in the process of being written up).  ...  Acknowledgments I would like to thank Erik Broman, Christophe Garban, Olle Häggström, Alan Hammond, Russell Lyons and Gábor Pete for very many helpful comments on earlier versions of the manuscript.  ... 
doi:10.1007/978-3-0346-0030-9_5 fatcat:63vtrm5r2vbynhwxtcvh5dej44

Page 7273 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews  
percolation in the tree case.  ...  Here we relate this phe- nomenon to the near-critical behavior of Or(p) by showing that the root percolates in the dynamical percolation process if and only if fe (@r(p))~'dp < co.  ... 

Simultaneous first and second order percolation transitions in interdependent networks [article]

Dong Zhou, Amir Bashan, Reuven Cohen, Yehiel Berezin, Nadav Shnerb,, Shlomo Havlin
2014 arXiv   pre-print
This sheds light on the origin of the plateau and on how its length scales with the size of the system.  ...  Understanding the critical nature of the dynamical process of cascading failures may be useful for designing strategies for preventing and mitigating catastrophic collapses.  ...  ACKNOWLEDGMENTS We thank DTRA, ONR, BSF, the LINC (No. 289447) and the Multiplex (No. 317532) EU projects, the DFG, and the Israel Science Foundation for support. We thank Michael M.  ... 
arXiv:1211.2330v3 fatcat:fbqq6wx2u5e7fe43mnrqxfutmq

Percolation in real wildfires

G Caldarelli, R Frondoni, A Gabrielli, M Montuori, R Retzlaff, C Ricotta
2001 Europhysics letters  
Results indicate that the burned clusters behave similarly to percolation clusters on boundaries and look more dense in their core.  ...  This paper focuses on the statistical properties of wild-land fires and, in particular, investigates if spread dynamics relates to simple invasion model.  ...  This would result in fires cluster as large as ξ, and then with the same features of ordinary clusters of percolation near criticality.  ... 
doi:10.1209/epl/i2001-00549-4 fatcat:34tqpkwztrhazc6zenxktidf3y

Explosive growth in biased dynamic percolation on two-dimensional regular lattice networks [article]

Robert M. Ziff
2009 arXiv   pre-print
The critical behavior of this growth model is shown to be of a different universality class than standard percolation.  ...  connects, is shown to go through an explosive, first-order kinetic phase transition with a sharp jump in the mass of the largest cluster as the number of bonds is increased.  ...  a closeup of the PR case, showing the connecting spanning-tree bonds.  ... 
arXiv:0903.4678v2 fatcat:ot52wuuo4ve4dhfhozr6q5w6xu
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