Edge-averaging dynamics on finite graphs: moment dependence

We study the edge-averaging process on a finite, connected graph $G = (V, E)$. Initially, the vertices in $V$ are endowed with i.i.d.\ real-valued opinions $(f_0(v))_{v \in V}$. Edges are activated according to i.i.d.\ Poisson clocks of rate $1$; when an edge is activated, the opinions at its endpoints are replaced by their average. Let $f_t(v)$ denote the opinion at $v$ at time $t$.Define the $\epsilon$-convergence time $\tau_\epsilon$ as the first time when the maximum and the minimum of $f_t$ differ by at most $\epsilon$. It is known that if the initial opinions $(f_0(v))_{v \in V}$ are bounded in $L^\infty$, then $\mathbb{E}(\tau_\epsilon)$ is at most $C_\epsilon \log^2 n$ for $\epsilon \in (0, 1]$. We assume instead that the $L^p$ norm of $f_0(v)$ is at most $1$ for every $v \in V$. For fixed $\epsilon \in (0, 1]$, and show that $\mathbb{E}(\tau_\epsilon) = \widetilde{O}(n^{\beta_p})$ up to logarithmic terms, where $\beta_p := \max(3 - p, 2/p)$. Moreover, this power law is tight on cycle graphs.