Scale invariance and universal probability distribution of an order parameter across a continuous phase transition David Clément - Institut d'Optique Scale invariance lies at the foundation of modern statistical physics and underpins the description of continuous phase transitions. Its most striking manifestation is the universal probability distribution function (PDF) of the order parameter, which encapsulates the complete statistical structure of critical fluctuations—beyond what traditional quantities such as averages or critical exponents can reveal. However, this universal distribution is exceptionally challenging to measure, as it reflects the non-Gaussian and scale-invariant nature of critical fluctuations. We will report on the experimental study of the statistics of the condensate order parameter across the superfluid–Mott transition in a gas of 3D lattice bosons, making use of singleatom-resolved detection in momentum space [1]. First, we observe non-Gaussian statistics of the order parameter near the transition, distinguished by non-zero and oscillating highorder cumulants [2]. We provide direct experimental evidence that these oscillations are universal. Second, crossing the Mott transition for various entropies and collapsing the cumulant oscillations, we obtain the non-universal coefficients required to reconstruct the universal PDF [3]. Finally, this universal scaling function determined experimentally is shown to yield algebraic scaling laws whose exponents are consistent with the critical exponents of the (expected) 3D XY universality class. [1] H. Cayla et al. Phys. Rev A 97, 061609(R) (2018); M. Allemand et al. Phys. Rev. X Quantum 5, 040324 (2024). [2] M. Allemand et al. arxiv:2508.21623 (2025). [3] In preparation (2025).