The distance on the slightly supercritical random series–parallel graph

Abstract 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 $p\in [0, \, 1]$ . 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 $D_n (p)$ . It is known that $D_n(p)$ possesses a phase transition at $p=p_c \;:\!=\;\frac{1}{2}$ ; more precisely, $\frac{1}{n}\log {{\mathbb{E}}}[D_n(p)] \to \alpha(p)$ when $n \to \infty$ , with $\alpha(p) >0$ for $p>p_c$ and $\alpha(p)=0$ for $p\le p_c$ . We study the exponent $\alpha(p)$ in the slightly supercritical regime $p=p_c+\varepsilon$ . Our main result says that as $\varepsilon\to 0^+$ , $\alpha(p_c+\varepsilon)$ behaves like $\sqrt{\zeta(2) \, \varepsilon}$ , where $\zeta(2) \;:\!=\; \frac{\pi^2}{6}$ .