The distance on the slightly supercritical random series–parallel graph

We consider the random series–parallel graph introduced by Hambly and Jordan (2004 Adv. Appl. Probab. 36 , 824–838), which is a hierarchical graph with a parameter p∈ [0, \, 1] . The graph is built recursively: at each step, every edge in the graph is either replaced with probability p by a series of two edges, or with probability $1-p$ by two parallel edges, and the replacements are independent of each other and of everything up to then. At the n th step of the recursive procedure, the distance between the extremal points on the graph is denoted by Dₙ (p) . It is known that Dₙ(p) possesses a phase transition at p=pc \;:\!=\;1/2 ; more precisely, 1/nlog E[Dₙ(p)] → α(p) when n → ∞ , with α(p) >0 for p>pc and α(p)=0 for p≤ pc . We study the exponent α(p) in the slightly supercritical regime p=pc+ε . Our main result says that as ε→ 0⁺ , α(pc+ε) behaves like √ζ(2) \, ε , where ζ(2) \;:\!=\; π²/6 .