Abstract
The critical behavior of the $O(2)$ model on dilute L\'evy graphs built on a two-dimensional square lattice is analyzed. Different qualitative cases are probed, varying the exponent $\ensuremath{\rho}$ governing the dependence on the distance of the connectivity probability distribution. The mean-field regime, as well as the long-range and short-range non-mean-field regimes, are investigated by means of high-performance parallel Monte Carlo numerical simulations running on GPUs. The relationship between the long-range $\ensuremath{\rho}$ exponent and the effective dimension of an equivalent short-range system with the same critical behavior is investigated. Evidence is provided for the effective short-range dimension to coincide with the spectral dimension of the L\'evy graph for the $\mathit{XY}$ model in the mean-field regime.
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