  We construct deterministic particle solutions for linear and fast diffusion
equations using a nonlocal approximation. We exploit the $2$-Wasserstein
gradient flow structure of the equations in order to obtain the nonlocal
approximating PDEs by regularising the corresponding internal energy with
suitably chosen mollifying kernels, either compactly or globally supported.
Weak solutions are obtained by the JKO scheme. From the technical point of
view, we improve known commutator estimates, fundamental in the
nonlocal-to-local limit, to include globally supported kernels which, in
particular cases, allow us to justify the limit without any further
perturbation needed. Furthermore, we prove geodesic convexity of the nonlocal
energies in order to prove convergence of the particle solutions to the
nonlocal equations towards weak solutions of the local equations. We overcome
the crucial difficulty of dealing with the singularity of the first variation
of the free energies at the origin. As a byproduct, we provide convergence
rates expressed as a scaling relationship between the number of particles and
the localisation parameter. The analysis we perform leverages the fact that
globally supported kernels yield a better convergence rate compared to
compactly supported kernels. Our result is relevant in statistics, more
precisely in sampling Gibbs and heavy-tailed distributions.
