Consensus time for asynchronous $\ell^p$ relaxation: graph dependence

We study the asynchronous $\ell^p$ relaxation introduced by Amir, Nazarov, and Peres: at each step, a uniformly chosen vertex minimizes its incident $\ell^p$ energy. For the profile $f_t$ after $t$ updates, let $$ \mathsf T_p(G,1/2):= \sup_{\lVert f_0\rVert_\infty\le1} \mathbb{E}\bigl[\min\{t\ge0:\operatorname{osc}(f_t)\le1/2\}\bigr]. $$ For $1<p<\infty$, we obtain graph-dependent estimates for boxes, trees, and conductance expanders. On the nearest-neighbor box $[L]^d$, for $d,L\ge2$ and $n=L^d$, the answer is, up to logarithmic factors, $nd^{1/(p-1)}L^{p/(p-1)}$ for $1<p<2$ and $ndL^2$ for $p\ge2$. On bounded-degree trees, an explicit rerooting-invariant parameter $T_G$ determines the answer up to logarithmic factors; for arbitrary trees it gives upper and lower bounds that differ additionally by the maximum degree. If the volume conductance $h(G)\ge h_0>0$, then $\mathsf T_p(G,1/2)=Θ_{p,h_0}(n\log n)$ without a degree assumption. At $p=\infty$, every connected graph satisfies $\mathsf T_\infty(G,1/2)\ge c nD^2/Δ$, where $D$ and $Δ$ are its diameter and maximum degree.