This thesis investigates the Random Euclidean Matching Problem for probability measures on $\mathbb{R}^d$ with exponentially or polynomially decaying densities. Given a sample of independent random variables distributed according to a probability measure $\mu$, the problem consists in determining the asymptotic behavior of the expected optimal transport cost between the empirical measure and $\mu$, measured through the $p$-Wasserstein distance. The problem is closely related to both random matching and the quantitative convergence of empirical measures. While its behavior is well understood for measures supported on bounded domains with uniformly positive densities, much less was known for measures on the whole space $\mathbb{R}^d$. Prior results were limited to partial estimates and conjectures, mainly in the Gaussian case. The thesis focuses on radial densities with exponential or polynomial tails and aims at deriving sharp asymptotic estimates for the expected matching cost. After introducing the necessary tools from optimal transport theory, including optimal transport maps and the Benamou--Brenier formulation of Wasserstein distance, the analysis develops concentration estimates and geometric decompositions adapted to the symmetry of the density. The main result identifies a critical exponent $p_{\mathrm{crit}}$, depending on the dimension and on the decay of the density, at which the asymptotic behavior of the matching cost changes. The precise rate of growth of the expected cost is determined for all dimensions and transport exponents. For $p\leq p_{\mathrm{crit}}$, more refined asymptotic estimates are obtained. These results significantly extend the theory of random matching to non-compact settings and reveal new phenomena induced by the decay of the underlying probability distribution.