$^{51}$V-NMR investigation of the spin-frustrated magnet Ni$_{3}$V$_{2}$O$_{8}$.

The Ni$^{2+}$ ions in the Kagom\'{e}-staircase lattice compound Ni$_{3}$V$_{2}$O$_{8}$ form an electron spin $S$~=~1 system with a geometrically frustrated magnetization and a rich variety of phases below 10 K. Here, we report a study of the local magnetic field and its fluctuations using $^{51}$V NMR measurements of the spectrum, the spin-phase memory time ($T_{2})$, and the spin-lattice relaxation rate (1/$T_{1})$ over a broad range of magnetic field and temperature. Above 10 K, the local field from the Ni$^{2+}$ ions follows a Curie-Weiss law. In the ordered phases below 10 K, the NMR spectrum has a very broad structure that changes according to the particular phase. Also, below 10 K, $1/T_1 \propto T^2$. These features suggest that 1/$T_{1}$ is dominated by Ni$^{2+}$ electron moment fluctuations whose dynamics are driven by two-dimensional antiferromagnetic coupling.