The probability of unusually large components in the near-critical Erdős–Rényi graph

Abstract

The largest components of the critical Erdős-Rényi graph, G(n, p) with p = 1/n, have size of order n 2/3 with high probability. We give detailed asymptotics for the probability that there is an unusually large component, i.e. of size an 2/3 for large a. Our results, which extend work of Pittel, allow a to depend upon n and also hold for a range of values of p around 1/n. We also provide asymptotics for the distribution of the size of the component containing a particular vertex.

Exceptional times of the critical dynamical Erdős–Rényi graph

“…For the third question, it is natural to guess-in analogy with work on dynamical planar lattice percolation by Hammond, Pete and Schramm [14]-that the largest component at "typical" exceptional times looks like a static component conditioned to have size at least βn 2/3 log 1/3 n. Unfortunately, our combinatorial method for estimating the probability that such a component exists (using results from [21]) gives little insight into its structure. Analysis using Brownian excursions, after Aldous [2] and Addario-Berry, Broutin and Goldschmidt [1], might shed more light on this problem.…”

“…We begin by presenting a result that gives the tail behavior of the size of components. For a proof of Proposition 3.1, see [21]. Pittel ([20], Proposition 2) proved part (b) when λ is fixed and k = an 2/3 where a is large but does not depend on n. PROPOSITION 3.1.…”

Finite-pool queueing with heavy-tailed services

“…However, the approximation becomes less effective as α 2, and for α = 2, (15) is not theoretically justified. In fact, for this finite-variance case, Pittel [28] (see also [29] and [30]) showed that the tail asymptotically behaves as…”

“…The most difficult task in proving Theorem 1 is to deal with the complicated drift R n (•) in (29). We will prove the following result.…”

“…We now conclude the proof of Theorem 1 by collecting various results from the previous sections. First, we split the process N n (•) in its martingale and drift components as in (29) to obtain N n (t) = q 0 + n (t) + σ n (B n (t)) − R n (t).…”