Convergence rate of $\ell^p$-relaxation on a graph to a $p$-harmonic function with given boundary values

We analyze the following dynamics on a connected graph $(V,E)$ with $n$ vertices. Let $V = I \bigcup B$, where the set of interior vertices $I \ne \emptyset$ is disjoint from the set of boundary vertices $B \neq \emptyset$. Given $p > 1$ and an initial opinion profile $f_0: V \to [0,1]$, at each integer step $t \ge 1$ a uniformly random vertex $v_t \in I$ is selected, and the opinion there is updated to the value $f_{t}(v_t)$ that minimizes the sum $\sum_{w \sim v_t} \lvert f_t(v_t)-f_{t-1}(w) \rvert^p$ over neighbours $w$ of $v_t$. The case $p=2$ yields linear averaging dynamics, but for all $p \ne 2$ the dynamics are nonlinear. It is well known that almost surely, $f_t$ converges to the $p$-harmonic extension $h$ of $f_0 \vert_{B}$. Denote the number of steps needed to obtain $\lVert f_t - h \rVert_{\infty} \le ε$ by $τ_p(ε).$ Recently, Amir, Nazarov, and Peres~\cite{noboundarycase} analyzed the same dynamics without boundary. For individual graphs, adding boundary values can slow down the convergence considerably; indeed, when $p = 2$ the approximation time is controlled by the hitting time of the boundary by random walk, and hitting times can be much larger than mixing times, which control the convergence when $B=\emptyset$. Nevertheless, we show that for all graphs with $n$ vertices, the mean approximation time $\E[τ_p(ε)]$ is at most $n^{β_p}$ (up to logarithmic factors in $\frac{n}ε$ for $p \in [2, \infty)$, and polynomial factors in $ε^{-1}$ for $p \in (1, 2)$), where $β_p=\max\big(\frac{2p}{p-1},3\big)$. This matches the definition of $β_p$ given in \cite{noboundarycase} and answers Question 6.2 in that paper. The exponent $β_p$ is optimal in both settings. We also prove sharp bounds for $n$-vertex graphs with given average degree, that are technically more challenging.