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## Empires and percolation: stochastic merging of adjacent regions

### Citations

159 |
The critical probability of bond percolation on the square lattice equals 1
- Kesten
- 1980
(Show Context)
Citation Context ...h the unit squares of the square grid, the areas of regions at time t are distributed as the numbers of edges of the connected components of bond percolation with p(t) = 1− e−t. The celebrated result =-=[8, 9]-=- that the critical value in bond percolation equals 1/2 implies that, in the particular empire process above, infinite regions appear at time log 2. 2. Hegemonic or not? Whether or not the qualitative... |

26 | Dust and self-similarity for the Smoluchowski coagulation equation
- Mischler
(Show Context)
Citation Context .... Scaling exponents. In the context of mean-field coalescence, a kernel such that K(cx, cy) = cγK(x, y) is said to have scaling exponent γ. It has long been understood, mostly non-rigorously (but see =-=[11]-=- for references to recent rigorous work) that the coalescence process should be non-gelling if γ ≤ 1 but gelling if γ > 1. We can define a scaling exponent γ analogously for the empire process: λ(cA, ... |

5 |
Random fragmentation and coagulation
- Bertoin
(Show Context)
Citation Context ... relationships, and the stochastic nature of mergers, matter in our model. Versions (now called stochastic coalescents) with finite total mass and stochastic merging but no geometry have been studied =-=[2, 4]-=-. And there is recent progress [5] in rigorous verification of the underlying presumption that in models of spatial diffusion and merging, the limiting (low density of massive particles) behaviour is ... |

4 |
Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5
- DJ
- 1999
(Show Context)
Citation Context ...-field” model. Write f(x, t)dx for the density per unit volume at time t of particles with mass in [x, x + dx]. The density function f(x, t) satisfies the well-known Smoluchowski coagulation equation =-=[1, 2, 3]-=- d dt f(x, t) = 1 2 ∫ x 0 K(y, x−y)f(y, t)f(x−y, t)dy−f(x, t) ∫ ∞ 0 K(x, y)f(y, t)dy.(1) In the empire process the conserved quantity is area, so there is an analogous density f(a, t) of area-a region... |

2 |
A and Rezakhanlou F 2007 Moment bounds for the Smoluchowski equation and their consequences Comm
- Hammond
(Show Context)
Citation Context ...ture of mergers, matter in our model. Versions (now called stochastic coalescents) with finite total mass and stochastic merging but no geometry have been studied [2, 4]. And there is recent progress =-=[5]-=- in rigorous verification of the underlying presumption that in models of spatial diffusion and merging, the limiting (low density of massive particles) behaviour is as predicted by the Smoluchowski c... |

2 |
Essam JW (1964) Exact critical percolation probabilities for site and bond problems in two dimensions
- MF
(Show Context)
Citation Context ...h the unit squares of the square grid, the areas of regions at time t are distributed as the numbers of edges of the connected components of bond percolation with p(t) = 1− e−t. The celebrated result =-=[8, 9]-=- that the critical value in bond percolation equals 1/2 implies that, in the particular empire process above, infinite regions appear at time log 2. 2. Hegemonic or not? Whether or not the qualitative... |

1 |
A general mathematical survey of the coagulation equation Topics
- RL
- 1972
(Show Context)
Citation Context ...-field” model. Write f(x, t)dx for the density per unit volume at time t of particles with mass in [x, x + dx]. The density function f(x, t) satisfies the well-known Smoluchowski coagulation equation =-=[1, 2, 3]-=- d dt f(x, t) = 1 2 ∫ x 0 K(y, x−y)f(y, t)f(x−y, t)dy−f(x, t) ∫ ∞ 0 K(x, y)f(y, t)dy.(1) In the empire process the conserved quantity is area, so there is an analogous density f(a, t) of area-a region... |

1 |
Smoluchowski’s coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent Ann
- JR
- 1999
(Show Context)
Citation Context ...-field” model. Write f(x, t)dx for the density per unit volume at time t of particles with mass in [x, x + dx]. The density function f(x, t) satisfies the well-known Smoluchowski coagulation equation =-=[1, 2, 3]-=- d dt f(x, t) = 1 2 ∫ x 0 K(y, x−y)f(y, t)f(x−y, t)dy−f(x, t) ∫ ∞ 0 K(x, y)f(y, t)dy.(1) In the empire process the conserved quantity is area, so there is an analogous density f(a, t) of area-a region... |

1 |
Percolation (2nd Ed
- GR
- 1999
(Show Context)
Citation Context ...e lattice assign a random time Te, with Exponential(1) distribution P (Te ≤ t) = 1− e−t, at which the edge becomes “open”. The configuration of open edges at time t is just the usual bond percolation =-=[6, 7]-=- process with p = 1− e−t. There is an associated spanning forest process in which is included only those edges which (upon becoming open) link two distinct open components. A connected component of th... |

1 |
Parviainen R 2004 Bounds for the connective constant of the hexagonal lattice J. Phys. A 37 pp 549–60
- SE
(Show Context)
Citation Context ...uits enclosing b of length 2n can be bounded as order 22n without taking into account their self-avoiding property; but taking this into account reduces the bound to order 2(2−δ)n for some δ > 0 (see =-=[10]-=- for discussion of the value of δ, which is not important for our calculation). So if we can Empires and percolation 7 derive an upper bound p2n(t) for the probability that any particular contour of l... |