A scaling limit for the length of the longest cycle in a sparse random graph

A scaling limit for the length of the longest cycle in a sparse random digraph

“…⃗ L c,n ≈ ⃗ f (c)n and compute the first few terms of ⃗ f (c) = 1 − ∑ ∞ k=1 p k (c)e −kc where p k (c) is a polynomial in c for k ≥ 1. That is, we prove a 1 Here we say A n ≈ B n if A n ∕B n → 1 as n → ∞.…”

“…This is essentially a repeat of Section 3.1.1 of [1]. The differences are minor, but we feel we need to include the argument.…”

“…Interchanging two elements in a permutation can only change 𝜈(H) by (log n) k 1 = n o (1) . We can therefore apply Azuma's inequality to show that…”

“…⃗ L c,n ≥ (1 − (c + 1 + 𝜀 c )e −c )n. This was improved by Krivelevich, Lubetzky, and Sudakov [13] who showed that w.h.p. ⃗ L c,n ≥ (1 − (2 + 𝜀 c )e −c )n. Recently, Anastos and Frieze [1] have shown that if c is sufficiently large then w.h.p. L c,n ≈ f (c)n as n → ∞, for some function f (c).…”

“…L c,n ≈ f (c)n as n → ∞, for some function f (c). 1 In this article, we use the ideas of [1] and show that w.h.p. ⃗ L c,n ≈ ⃗ f (c)n and compute the first few terms of ⃗ f (c) = 1 − ∑ ∞ k=1 p k (c)e −kc where p k (c) is a polynomial in c for k ≥ 1.…”