# The phase transition in the uniformly grown random graph has infinite order

## Abstract

The aim of this paper is to study the emergence of the giant component in the uniformly grown random graph G n(c), 0 < c < 1, the graph on the set [n] = {1, 2,...,n} in which each possible edge ij is present with probability c/ max{i, j}, independently of all other edges. Equivalently, we may start with the random graph G n(1) with vertex set [n], where each vertex j is joined to each "earlier" vertex i < j with probability 1/j, independently of all other choices. The graph G n(c) is formed by the open bonds in the bond percolation on G n(1) in which a bond is open with probability c. The model G n(c) is the finite version of a model proposed by Dubins in 1984, and is also closely related to a random graph process defined by Callaway, Hopcroft, Kleinberg, Newman, and Strogatz [Phys Rev E 64 (2001), 041902]. Results of Kalikow and Weiss [Israel J Math 62 (1988), 257-268] and Shepp [Israel J Math 67 (1989), 23-33] imply that the percolation threshold is at c = 1/4. The main result of this paper is that for c = 1/4 + ε,ε > 0, the giant component in G n(c) has order exp (-⊖(1/√ε)) n. In particular, the phase transition in the bond percolation on G n(1) has infinite order. Using nonrigorous methods, Dorogovtsev, Mendes, and Samukhin [Phys Rev E 64 (2001), 066110] showed that an even more precise result is likely to hold. © 2004 Wiley Periodicals, Inc.

## Publication Title

Random Structures and Algorithms

## Recommended Citation

Bollobás, B., Janson, S., & Riordan, O.
(2005). The phase transition in the uniformly grown random graph has infinite order.* Random Structures and Algorithms**, 26* (1-2), 1-36.
https://doi.org/10.1002/rsa.20041