Study of $\boldsymbol{J/\psi}$ production dependence with the charged particle multiplicity in p-Pb collisions at $\boldsymbol{\sqrt{s_{_{\mathrm{NN}}}}} =$ 5.02 TeV and pp collisions at $\boldsymbol{\sqrt{s}} =$ 8 TeV with the ALICE experiment at the LHC

A suppression of the $J/\psi$ production was found in Pb-Pb collisions at $\sqrt{s_{_{\mathrm{NN}}}} =$ 2.76 TeV, providing further evidence of the formation of a deconfined medium in ultra-relativistic heavy-ion collisions, the so-called Quark-Gluon Plasma. In addition, p-Pb collisions at $\sqrt{s_{_{\mathrm{NN}}}} =$ 5.02 TeV have been studied at the LHC to measure cold nuclear matter effects (e.g. gluon shadowing, energy loss, nuclear absorption). Understanding p-Pb collisions will help to disentangle hot and cold nuclear matter effects in Pb-Pb collisions. Surprisingly, some observables in p-Pb collisions behave as in heavy-ion collisions where it is understood as a result of a collective expansion of the medium. This thesis analyses inclusive $J/\psi$ production in p-Pb and pp collisions with the ALICE muon spectrometer. The $J/\psi$ production rate, and its mean transverse momentum, have been measured at forward and backward rapidities as a function of the charged particle multiplicity. Measurements of particle production as a function of the event multiplicity in small size systems provide a way to sign the presence of collective final state effects like those observed in Pb-Pb collisions. The observed increase of the $J/\psi$ production at backward rapidity with multiplicity, is consistent with that observed in pp collisions. However, a deviation from this behaviour in the $J/\psi$ production at forward rapidity at high multiplicity has been measured. A trend towards saturation has also been observed in the $J/\psi$ mean transverse momentum in p-Pb collisions. Whether these effects can be explained by cold nuclear matter effects or by the presence of further final state effects is still unresolved.