Charge Relaxation in the Presence of Shot Noise in Coulomb Coupled Mesoscopic Systems

In the presence of shot noise the charge on a mesoscopic conductor fluctuates. We are interested in the charge fluctuations which arise if the conductor is in the proximity of a gate to which it is coupled by long range Coulomb forces only. Specifically we consider a gate coupled to the edge of a Hall bar subject to a quantizing magnetic field which contains a quantum point contact. The gate is located away from the quantum point contact. We evaluate the charge relaxation resistance for this geometry. The charge relaxation resistance determines the current fluctuations and potential fluctuations induced into the gate. If there is only one edge channel the charge relaxation resistance is determined by transmission and reflection probabilities alone, but in the presence of many channels the density of states of all edge states determines this resistance