We consider a synchronous process of particles moving on the vertices of a graph G, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, M particles are placed on a vertex of G. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time. In this work we study the case where G is the complete graph on n vertices and the number of particles is M = n/2+α n^{1/2} + o(n^{1/2}), α ∈ ℝ. This choice of M corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by n^{-1/2}, converges in p-th mean, as n → ∞ and for any p ∈ ℝ, to a continuous and almost surely positive random variable T_α. We find that T_α is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that 𝔼[T₀] = π^{3/2}/√7, and furthermore we formulate explicit asymptotics when |α| gets large that quantify the transition into and out of the critical window. We also study the random variable counting the total number of jumps that are performed by the particles until the dispersion time is reached and prove that, if rescaled by nln(n), it converges to 2/7 in probability.

Abstract. We describe a robust methodology, based on the martingale argument of Nachmias and Peres and random walk estimates, to obtain simple upper and lower bounds on the size of a maximal component in several random graphs at criticality. Even though the main result is not new, we believe the material presented here is interesting because it unifies several proofs found in the literature into a common framework. More specifically, we give easy-to-check conditions that, when satisfied, allow an immediate derivation of the above-mentioned bounds.