Sharp Gromov Hyperbolicity Constants for Cartesian Products of Finitely Many Trees

Let $X=T_1\times \cdots \times T_k$, $k\ge 2$, be a Cartesian product of finite nontrivial trees with unit edges, and write $d_i=\text{diam}{T}_i$, $S={\sum }_i{d}_i$ and $M={\max }_i{d}_i$. We prove that the thin-triangle hyperbolicity constant of the geodesic realization of $X$ is $$\text{Rips}(X)=\min \{\frac{S}{2},S-M+\frac{1}{2}\},$$ and that the four-point hyperbolicity constant of its vertex metric is the value $$\mathcal{P}(d_1,\dots ,d_k)=\underset{E\subseteq \{1,\dots ,k\}}{\max }\min \{\sum _{i\in E}{d}_i,\sum _{i\mathrm{\notin }E}{d}_i\}\mathrm{.}$$ of the best two-way partition of the diameters. Both constants therefore depend only on the diameter vector, and their difference is governed by the partition imbalance $\eta$ of that vector: it equals $\eta \mathrm{/}2$ when $M\le S-M$ and $1\mathrm{/}2$ otherwise. In particular the two constants coincide if and only if the diameters admit a perfect partition, and their difference is unbounded. For $k\ge 3$ the first formula exhibits a many-factor balancing phenomenon: several short factors can collectively balance the longest one and force the universal bound $\frac{1}{2}\text{diam}X$ to be attained. The proofs are self-contained and rest on an exact distance formula for the geodesic realization of $X$. For the ${\mathrm{\ell }}_1$-product of the factors, i.e.\ the ${\mathrm{\ell }}_1$-metric on the associated cube complex, we obtain $\text{Rips}=\min \{S\mathrm{/}2,S-M\}$, which identifies the half-unit above as a pure $1$-skeleton effect, and we show that the four-point formula holds for ${\mathrm{\ell }}_1$-products of arbitrary $0$-hyperbolic metric spaces, in particular for weighted trees and $\mathbb{R}$-trees. We also treat products with unbounded factors and record algorithmic consequences.