  The critical behaviour of statistical models with long-range interactions
exhibits distinct regimes as a function of $\rho$, the power of the interaction
strength decay. For $\rho$ large enough, $\rho>\rho_{\rm sr}$, the critical
behaviour is observed to coincide with that of the short-range model. However,
there are controversial aspects regarding this picture, one of which is the
value of the short-range threshold $\rho_{\rm sr}$ in the case of the
long-range XY model in two dimensions. We study the 2d XY model on the {\it
diluted} graph, a sparse graph obtained from the 2d lattice by rewiring links
with probability decaying with the Euclidean distance of the lattice as
$|r|^{-\rho}$, which is expected to feature the same critical behavior of the
long range model. Through Monte Carlo sampling and finite-size analysis of the
spontaneous magnetisation and of the Binder cumulant, we present numerical
evidence that $\rho_{\rm sr}=4$. According to such a result, one expects the
model to belong to the Berezinskii-Kosterlitz-Thouless (BKT) universality class
for $\rho\ge 4$, and to present a $2^{nd}$-order transition for $\rho<4$.
