In this third paper of a series, we discuss the physics of the population of accelerated particles in the precursor of an unmagnetized, relativistic collisionless pair shock. In particular, we provide a theoretical estimate of their scattering length $l_{scatt}(p)$ in the self-generated electromagnetic turbulence, as well as an estimate of their distribution function. We obtain $l_{scatt}(p) simeq (gamma_p /epsilon_B)(p/gamma_{infty} mc)^2 (c/omega_p)$, with p the particle momentum in the rest frame of the shock front, $epsilon_B$ the strength parameter of the microturbulence, $gamma_p$ the Lorentz factor of the background plasma relative to the shock front and $gamma_{infty}$ its asymptotic value outside the precursor. We compare this scattering length to large-scale PIC simulations and find good agreement for the various dependencies.
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