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

The low-temperature phase in the two-dimensional long-range diluted XY model