  Due to large scale flow inhomogeneities and the effects of temperature,
turbulence small-scale structure in thermal convection is still an active field
of investigation, especially considering sophisticated Lagrangian statistics.
Here we experimentally study Lagrangian pair dispersion (one of the canonical
problems of Lagrangian turbulence) in a Rayleigh-B\'enard convection cell. A
sufficiently high temperature difference is imposed on a horizontal layer of
fluid to observe a turbulent flow. We perform Lagrangian tracking of
sub-millimetric particles on a large measurement volume including part of the
Large Scale Circulation (LSC) revealing some large inhomogeneities. Our study
brings to light several new insights regarding our understanding of turbulent
thermal convection: (i) by decomposing particle Lagrangian dynamics into the
LSC contribution and the turbulent fluctuations, we highlight the relative
impact of both contributions on pair dispersion; (ii) using the same
decomposition, we estimate the Eulerian second-order velocity structure
functions from pair statistics and show that after removing the LSC
contribution, the remaining statistics recover usual homogeneous and isotropic
behaviours which are governed by a local energy dissipation rate to be
distinguished from the global dissipation rate classically used to characterise
turbulence in thermal convection; and (iii) we revisit the super-diffusive
Richardson-Obukhov regime of particle dispersion and propose a refined estimate
of the Richardson constant.
