• Corpus ID: 251564418

Multi-source invasion percolation on the complete graph

@inproceedings{AddarioBerry2022MultisourceIP,
  title={Multi-source invasion percolation on the complete graph},
  author={Louigi Addario-Berry and Jordan Barrett},
  year={2022}
}
. We consider invasion percolation on the randomly-weighted complete graph K n , started from some number k ( n ) of distinct source vertices. The outcome of the process is a forest consisting of k ( n ) trees, each containing exactly one source. Let M n be the size of the largest tree in this forest. Logan, Molloy and Pralat [23] proved that if k ( n ) =n 1 = 3 ! 0 then M n =n ! 1 in probability. In this paper we prove a complementary result: if k ( n ) =n 1 = 3 ! 1 then M n =n ! 0 in… 

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References

SHOWING 1-10 OF 36 REFERENCES

The local weak limit of the minimum spanning tree of the complete graph

Assign i.i.d. standard exponential edge weights to the edges of the complete graph K_n, and let M_n be the resulting minimum spanning tree. We show that M_n converges in the local weak sense (also

Invasion percolation on regular trees

It is proved that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster, and the two clusters have the same law locally, but not globally.

Component Behavior Near the Critical Point of the Random Graph Process

  • T. Luczak
  • Mathematics
    Random Struct. Algorithms
  • 1990
It is shown that, with probability 1 o(l), when M ( n ) = n12 + s, s 3 C 2 + -a, then during a random graph process in some step M’ > M a “new” largest component will emerge, and when s3n-*+m, the largest component of G(n, M) remains largest until the very end of the process.

Invasion percolation on the Poisson-weighted infinite tree

We study invasion percolation on Aldous' Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the $\sigma\to\infty$ limit of a

The continuum limit of critical random graphs

We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn−4/3, for some fixed $${\lambda \in \mathbb{R}}$$ . We prove that the sequence of connected

THE SIZE OF A POND IN 2 D INVASION PERCOLATION

We consider invasion percolation on the square lattice. In [3] it has been proved that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n

On finding a minimum spanning tree in a network with random weights

We investigate Prim's standard “tree-growing” method for finding a minimum spanning tree, when applied to a network in which all degrees are about d and the edges e have independent identically

The size of a pond in 2D invasion percolation

We consider invasion percolation on the square lattice. van den Berg, Peres, Sidoravicius and Vares have proved that the probability that the radius of a so-called pond is larger than n, differs at

Geometry of the minimal spanning tree of a random 3-regular graph

The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.

The fractal volume of the two-dimensional invasion percolation cluster

AbstractWe consider both invasion percolation and standard Bernoulli bond percolation on theZ2 lattice. Denote byV andC the invasion cluster and the occupied cluster of the origin, respectively. Let