  We study the statistical properties of a single free quantum particle
evolving coherently on a discrete lattice in $d$ spatial dimensions where every
lattice site is additionally subject to continuous measurement of the
occupation number. Using perturbative renormalization group (RG) analysis, we
show that the systems undergoes a Measurement-induced Phase Transition (MiPT)
for $d>2$ from a $\textit{delocalized}$ to a $\textit{localized}$ phase as the
measurement strength $\gamma$ is increased beyond a critical value
$\gamma_{c}$. In the language of surface growth, the delocalized phase
corresponds to a $\textit{rough}$ phase while the localized phase corresponds
to a $\textit{smooth}$ phase. We support our analytical computations with
numerical analysis which are in qualitative and quantitative agreement with the
theory.
