Time constant in a recursive series-parallel first-passage model

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G_0$ be one oriented edge from a source to a sink. Replace each edge recursively by a copy of the three-edge pattern with a single edge from its source to an intermediate junction and two parallel edges from that junction to its sink, preserving the source-to-sink orientation; call the result $G_n$. Assign independent positive weights with common law $\xi$ to its edges and let $D_n$ be the minimum total weight of a source--sink path. Thus $D_0\stackrel d=\xi$ and \[D_n\stackrel d=D_{n-1}^{(1)}+\min\{D_{n-1}^{(2)},D_{n-1}^{(3)}\},\] where the three variables on the right are independent copies. For the comparison branching random walk, a binary tree starting with an edge has independent edge weights of the same law; let $M_m$ be its minimum root-to-level-$m$ weight. For laws for which the indicated finite limits exist, define the constants \[\gamma_{\rm rec}=\lim_{n\to\infty}2^{-n}\mathbb E D_n,\qquad\gamma_{\rm brw}=\lim_{m\to\infty}m^{-1}\mathbb E M_m.\]

Statement recorded in the supplied source.

Compute $\gamma_{\rm rec}$ in terms of the edge-weight law $\xi$. Determine the hypotheses on $\xi$ needed for these finite constants to exist. For nondegenerate laws under those hypotheses, is $\gamma_{\rm rec}>\gamma_{\rm brw}$? [S2]

Short English statement

Determine the first-passage time constant of a series-and-parallel hierarchy and whether it is strictly larger than the binary-tree constant.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000065 (AMR-099-0065). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Itai Benjamini, Euclidean vs. Graph Metric, version dated 16 July 2012; original author manuscript preserved in Arquivo.pt capture of 31 December 2020, Sections 9.1--9.2, equation (1) and Question 9.1, pp. 13--15. https://arquivo.pt/noFrame/replay/20201231041538id_/http://www.wisdom.weizmann.ac.il/~itai/erd100.pdf.