  Unconventional quasiparticles emerging in the fractional quantum Hall regime
present the challenge of observing their exotic properties unambiguously.
Although the fractional charge of quasiparticles has been demonstrated since
nearly three decades, the first convincing evidence of their anyonic quantum
statistics has only recently been obtained and, so far, the so-called scaling
dimension that determines the quasiparticles propagation dynamics remains
elusive. In particular, while the non-linearity of the tunneling quasiparticle
current should reveal their scaling dimension, the measurements fail to match
theory, arguably because this observable is not robust to non-universal
complications. Here we achieve an unambiguous measurement of the scaling
dimension from the thermal to shot noise cross-over, and observe a long-awaited
agreement with expectations. Measurements are fitted to the predicted finite
temperature expression involving both the quasiparticles scaling dimension and
their charge, in contrast to previous charge investigations focusing on the
high bias shot noise regime. A systematic analysis, repeated on multiple
constrictions and experimental conditions, consistently matches the theoretical
scaling dimensions for the fractional quasiparticles emerging at filling
factors 1/3, 2/5 and 2/3. This establishes a central property of fractional
quantum Hall anyons, and demonstrates a powerful and complementary window into
exotic quasiparticles.
