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Let $(M, \omega, J)$ be a K\"ahler manifold, equipped with an effectiveHamiltonian torus action $\rho: T \rightarrow \mathrm{Diff}(M, \omega, J)$ byisometries with moment map $\mu: M \rightarrow \mathfrak{t}^{*}$. We firstconstruct a singular mixed polarization $\mathcal{P}_{\mathrm{mix}}$ on $M$.Second, we construct a one-parameter family of complex structures $J_{t}$ on$M$ which are compatible with $\omega$. Furthermore, the path of correspondingK\"ahler metrics $g_{t}$ is a complete geodesic ray in the space of K\"ahlermetrics of $M$, when $M$ is compact. Finally, we show that the correspondingfamily of K\"ahler polarizations $\mathcal{P}_{t}$ associated to $J_{t}$converges to $\mathcal{P}_{\mathrm{mix}}$ as $t \rightarrow \infty$.

Geodesic rays in space of K\"ahler metrics with $T$-symmetry