  The interplay between structural and electronic degrees of freedom in complex
materials is the subject of extensive debate in physics and materials science.
Particularly interesting questions pertain to the nature and extent of
pre-transitional short-range order in diverse systems ranging from shape-memory
alloys to unconventional superconductors, and how this microstructure affects
macroscopic properties. Here we use neutron and X-ray diffuse scattering to
uncover universal structural fluctuations in La$_{2-x}$Sr$_x$CuO$_4$ and
Tl$_2$Ba$_2$CuO$_{6+\delta}$, two cuprate superconductors with distinct point
disorder effects and optimal superconducting transition temperatures. The
fluctuations are present in wide doping and temperature ranges, including
compositions that maintain high average structural symmetry, and they exhibit
unusual, yet simple scaling behavior. The scaling regime is robust and
universal, similar to the well-known critical fluctuations close to
second-order phase transitions, but with a distinctly different physical
origin. We relate this behavior to pre-transitional phenomena in a broad class
of systems with structural and magnetic transitions, and propose an explanation
based on rare structural fluctuations caused by intrinsic nanoscale
inhomogeneity. We also uncover parallels with superconducting fluctuations,
which indicates that the underlying inhomogeneity plays an important role in
cuprate physics.
