KMS states on the Toeplitz algebras of higher-rank graphs

The Toeplitz algebra $\mathcal{T}C^{*}(\Lambda)$ for a finite $k$-graph $\Lambda$ is equipped with a continuous one-parameter group $\alpha^{r}$ for each $r\in \mathbb{R}^{k}$, obtained by composing the map $\mathbb{R} \ni t \to (e^{itr_{1}}, \dots , e^{itr_{k}}) \in \mathbb{T}^{k}$ with the gauge action on $\mathcal{T}C^{*}(\Lambda)$. In this paper we give a complete description of the $\beta$-KMS states for the $C^{*}$-dynamical system $(\mathcal{T}C^{*}(\Lambda), \alpha^{r})$ for all finite $k$-graphs $\Lambda$ and all values of $\beta \in \mathbb{R}$ and $r\in \mathbb{R}^{k}$.