  In this work, we report on hot carrier diffusion in graphene across large
enough length scales that the carriers are not thermalized across the crystal.
The carriers are injected into graphene at one site and their thermal transport
is studied as a function of applied power and distance from the heating source,
up to tens of micrometers away. Superconducting contacts prevent out-diffusion
of hot carriers to isolate the electron-phonon coupling as the sole channel for
thermal relaxation. As local thermometers, we use the amplitude of the
Universal Conductance Fluctuations, which varies monotonically as a function of
temperature. By measuring the electron temperature simultaneously along the
length we observe a thermal gradient which results from the competition between
electron-phonon cooling and lateral heat flow.
