  We present a broad study of charge and heat transport in a mesoscopic system
where one or several quantum Hall edge channels are strongly coupled to a
floating Ohmic contact (OC). It is well known that charge-current fluctuations
emanating from the OC along the edge channels are highly susceptible to the OC
charge capacitance in the heat Coulomb blockade regime (an impeded ability of
the OC to equilibrate edge channels). Here, we demonstrate how potential- and
temperature fluctuations due to finite OC charge and heat capacities impact the
heat-current fluctuations emitted from the OC. First, by assuming an infinite
OC heat capacity, we show that the output heat-current noise is strongly
dependent on the OC charge capacitance, following from a close relation between
one-dimensional charge- and heat currents. When also the OC heat capacity is
finite, an interplay of potential- and temperature fluctuations influences the
heat transport. Concretely, we find that the effect of the charge capacitance
on heat transport manifests in terms of a strongly increased energy relaxation
time in the heat Coulomb blockade regime. Furthermore, we find expressions for
a broad set of output observables, such as charge and heat auto- and cross
correlations, as functions of input and OC fluctuations, depending on the
relation between charge and energy relaxation times compared to the frequency
of fluctuations and inverse (local) temperatures as well as on the number of
edge channels attached to the OC. Finally, we show that a finite OC heat
capacity transforms the full counting statistics of the output charge from
Gaussian to non-Gaussian. Our findings provide novel opportunities to
experimentally probe and harness the quantum nature of heat transport in
strongly coupled electron circuits.
