  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.
