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A novel stochastic technique combining a dilute source grid of $\mathbb{Z}_3$noise with iterative momentum-smearing is used to study the proton correlationfunction at rest and in boosted frames on two lattice volumes. The techniquemakes use of the baryonic version of the so-called one-end trick, and thedecomposition into signal and noise terms of the resulting stochastic protoncorrelation function is made explicit. The number and location of the sourcepoints in the dilute grid should be chosen so that the benefits of averagingover many locations overcomes the additional statistical error introduced bythe noise terms in the desired fitting region. At all nontrivial momentumvalues considered we find that the choice of $N=4$--$8$ maximally separatedsource locations is shown to be optimal, providing a reduced statistical errorwhen compared with a single point source. This enables us to successfully fitthe proton energy at momentum values as high as $|\vec{p}| \simeq 3.75$ GeV and$|\vec{p}| \simeq 2.82$ GeV on the small and large volume respectively.

Accessing high-momentum nucleons with dilute stochastic sources