Lutz Warnke

Explosive Percolation Is Continuous

A mathematical proof shows that in many models of the growth of network connectivity, phase transitions are continuous. “Explosive percolation” is said to occur in an evolving network when a macroscopic connected component emerges in a number of steps that is much smaller than the system size. Recent predictions based on simulations suggested that certain Achlioptas processes (much-studied local modifications of the classical mean-field growth model of Erdős and Rényi) exhibit this phenomenon, undergoing a phase transition that is discontinuous in the scaling limit. We show that, in fact, all Achlioptas processes have continuous phase transitions, although related models in which the number of nodes sampled may grow with the network size can indeed exhibit explosive percolation.

When does the K4-free process stop?

The K4-free process starts with the empty graph on n vertices and at each step adds a new edge chosen uniformly at random from all remaining edges that do not complete a copy of K4. Let G be the random maximal K4-free graph obtained at the end of the process. We show that for some positive constant C, with high probability as , the maximum degree in G is at most . This resolves a conjecture of Bohman and Keevash for the K4-free process and improves on previous bounds obtained by Bollobas and Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash this shows that with high probability G has edges and is ‘nearly regular’, i.e., every vertex has degree . This answers a question of Erdős, Suen and Winkler for the K4-free process. We furthermore deduce an additional structural property: we show that whp the independence number of G is at least , which matches an upper bound obtained by Bohman up to a factor of . Our analysis of the K4-free process also yields a new result in Ramsey theory: for a special case of a well-studied function introduced by Erdős and Rogers we slightly improve the best known upper bound.Copyright

Upper tails for arithmetic progressions in random subsets

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,..., n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to Schur triples, and, more generally, to the number of edges in random induced subhypergraphs of ‘almost linear’ k-uniform hypergraphs.

The Janson inequalities for general up-sets

Janson and Janson, Łuczak and Rucinski proved several inequalities for the lower tail of the distribution of the number of events that hold, when all the events are up-sets increasing events of a special form-each event is the intersection of some subset of a single set of independent events i.e., a principal up-set. We show that these inequalities in fact hold for arbitrary up-sets, by modifying existing proofs to use only positive correlation, avoiding the need to assume positive correlation conditioned on one of the events.

Convergence of Achlioptas Processes via Differential Equations with Unique Solutions

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the largest component in such variations of the Erdős–Renyi random graph process has recently received considerable attention, in particular for Bollobass ‘product rule’. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the ‘giant’ component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes. Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.