Average-weight percolation on the complete graph

TL;DR

The paper resolves sharp near-critical behaviour for the longest path with average edge weight constraint on the complete graph with Exp$(n)$-mean edges by connecting average-weight percolation to a branching random walk with selection. It introduces a BRW-inspired exploration process, proving upper bounds via tilted random-walk analyses and lower bounds via a carefully coupled exploration with a killed BRW and first/second-moment methods. The main findings show a phase transition near $\lambda_c$ with regimes: (i) for $\alpha<\pi^2/2$, ${\mathscr L}(n,\alpha/\ln^2 n)$ scales like $\Theta(\ln^3 n)$ with a prefactor $(\pi^2/2 - \alpha)^{-1}$; (ii) in the near-critical window, ${\mathscr L}(n,\beta_n)$ concentrates around $n\,e^{-\pi/(\sqrt{2\beta_n})}$; and (iii) for $\alpha>\pi^2/2$, the length is polynomial, of order $n^{1-\pi/\sqrt{2\alpha}}$. The methods blend a constructive BRW-with-selection exploration, tilted-random-walk arguments, and Mogulskii-type large deviations to obtain sharp asymptotics, providing deep insight into the phase transition and its finite-size behaviour.

Abstract

Attach to each edge of the complete graph on $n$ vertices, i.i.d. exponential random variables with mean $n$. Aldous [1] proved that the longest path with average weight below $p$ undergoes a phase transition at $p=\frac{1}{e}$: it is $o(n)$ when $p<\frac{1}{e}$ and of order $n$ if $p>\frac1e$. Later, Ding [4] revealed a finer phase transition around $\frac{1}{e}$: there exist $c'>c>0$ such that the length of the longest path is of order $\ln^3 n$ if $ p \le \frac{1}{e}+\frac{c}{\ln^2 n}$ and is polynomial if $p\ge \frac{1}{e}+\frac{c'}{\ln^2 n}$. We identify the location of this phase transition and obtain sharp asymptotics of the length near criticality. The proof uses an exploration mechanism mimicking a branching random walk with selection introduced by Brunet and Derrida [3].

Figures (2)

• Figure 1: Schematic illustration of the coupling between $(Z_j, G_j)$ and ${\mathcal{V}}^N$.
• Figure 2: Schematic illustration of the exploration process in the proof of the lower bound of Theorem \ref{['t:main2']}. The path $\Gamma_1$ is the unique path in the exploring process from $u^*_0$ to $u^*_{\tau_1-1}$. Note that $u^*_j$, for $1\le j < \tau_1-1$, do not necessarily lie on $\Gamma_1$. When a particle $u$ is selected in ${\mathcal{V}}^N$, we write interchangeably ${\mathcal{V}}(u)$ or ${\mathcal{V}}^N(u)$.

Theorems & Definitions (36)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof : Proof of the upper bound in Theorem \ref{['t:main']}
• Lemma 2.4
• proof : Proof of the upper bound in Theorem \ref{['t:main2']}