Explosive Percolation: Unusual Transitions of a Simple Model

In this paper we review the recent advances on explosive percolation, a very sharp phase transition first observed by Achlioptas et al. (Science, 2009). There a simple model was proposed, which changed slightly the classical percolation process so that the emergence of the spanning cluster is delayed. This slight modification turns out to have a great impact on the percolation phase transition. The resulting transition is so sharp that it was termed explosive, and it was at first considered to be discontinuous. This surprising fact stimulated considerable interest in "Achlioptas processes". Later work, however, showed that the transition is continuous (at least for Achlioptas processes on Erdos networks), but with very unusual finite size scaling. We present a review of the field, indicate open "problems" and propose directions for future research.Comment: 27 pages, 4 figures, Review pape

Random k-SAT and the Power of Two Choices

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics (Achlioptas processes) to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k-SAT for k >= 3. We then propose a gap decision problem based upon this semi-random model. The aim of the problem is to investigate the hardness of the random k-SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes.Comment: 13 page

The augmented multiplicative coalescent and critical dynamic random graph models

Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of $n$ vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed $K\geq 1$, all components of size greater than $K$ are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous's standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is `nearly' Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from [8], on the size of the largest component in the barely subcritical regime.Comment: 49 page

Bounded-Size Rules: The Barely Subcritical Regime

Abstract. Bounded-size rules(BSR) are dynamic random graph processes which incorporate limited choice along with randomness in the evolution of the system. Typically one starts with the empty graph and at each stage two edges are chosen uniformly at random. One of the two edges is then placed into the system according to a decision rule based on the sizes of the components containing the four vertices. For bounded-size rules, all components of size greater than some fixed K ≥ 1 are accorded the same treatment. Writing BSR(t) for the state of the system with ⌊nt/2⌋ edges, Spencer and Wormald [24] proved that for such rules, there exists a (rule dependent) critical time tc such that when t tc, the size of the largest component is of order n. In this work we obtain upper bounds (that hold with high probability) of order n2 log4 n, on the size of the largest component, at time instants tn = tc − n− , where ∈ (0, 1/4). This result for the barely subcritical regime forms a key ingredient in the study undertaken in [4], of the asymptotic dynamic behavior of the process describing the vector of component sizes and associated complexity of the components for such random graph models in the critical scaling window. The proof uses a coupling of BSR processes with a certain family of inhomogeneous random graphs with vertices in the type space R+ × D([0,∞) : N0) where D([0,∞) : N0) is the Skorohod D-space of functions that are right continuous and have left limits, with values in the space of nonnegative integers N0, equipped with the usual Skorohod topology. The coupling construction also gives an alternative characterization (than the usual explosion time of the susceptibility function) of the critical time tc for the emergence of the giant component in terms of the operator norm of integral operators on certain L2 spaces