In this paper, we have examined the R$acute{e}$nyi holographic dark energy (RHDE) model in the framework of an isotropic and spatially homogeneous flat FLRW (Friedmann- Lema$hat i$tre-Robertson-Walker) Universe by considering different values of parameter $delta$, where the infrared cut-off is taken care by the Hubble horizon. We examined the RHDE model through the analysis of the growth rate of perturbations and the statefinder hierarchy. The evolutionary trajectories of the statefinder hierarchy $S_3^1$, $S_3^2$ $S_4^1$, $S_4^2$ versus redshift $z$, shows satisfactory behaviour throughout the Universe evolution. One of the favourable appliance for exploring the dark energy models is the CND (composite null diagnostic) ${ S_3^1 - epsilon}$ and ${ S_4^1 - epsilon}$, where the evolutionary trajectories of the ${ S_3^1 - epsilon}$ and ${ S_4^1 - epsilon}$ pair show remarkable characteristics and the departure from $Lambda$CDM could be very much assessed.
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