This article examines the spatial {dynamics of bed load particles} in water. We focus particularly on the fluctuations of particle activity, which is defined as the number of moving particles per unit bed {length}. Based on a stochastic model recently proposed by \citet{Ancey2013}, we derive the second moment of particle activity analytically; that is the spatial correlation functions of particle activity. From these expressions, we show that large moving particle clusters can develop spatially. Also, we provide evidence that fluctuations of particle activity are scale-dependent. Two characteristic lengths emerge from the model: a saturation length $\ell_{sat}$ describing the length needed for a perturbation in particle activity to relax to the homogeneous solution, and a correlation length $\ell_c$ describing the typical size of moving particle clusters. A dimensionless P'eclet number can also be defined according to the transport model. Three different experimental data sets are used to test the theoretical results. We show that the stochastic model describes spatial patterns of particle activity well at all scales. In particular, we show that $\ell_c$ and $\ell_{sat}$ may be relatively large compared to typical scales encountered in bed load experiments (grain diameter, water depth, bed form wavelength, flume length...) suggesting that the spatial fluctuations of particle activity have a non-negligible impact on the average transport process.