Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices

Abstract

We consider the AC transport in a quantum RC circuit made of a coherent chaotic cavity with a top gate. Within a random matrix approach, we study the joint distribution for the mesoscopic capacitance $C_{\mu }=(1/C+1/C_q{)}^{-1}$ and the charge relaxation resistance $R_q$, where $C$ is the geometric capacitance and $C_q$ the quantum capacitance. We study the limit of a large number of conducting channels $N$ with a Coulomb gas method. We obtain $⟨R_q⟩\simeq h/(N{e}^2)=R_{\mathrm{dc}}$ and show that the relative fluctuations are of order $1/N$ both for $C_q$ and $R_q$, with strong correlations $⟨\delta C_q\delta R_q⟩/\sqrt{⟨\delta C_q^2⟩⟨\delta R_q^2⟩}\simeq +0.707$. The detailed analysis of large deviations involves a second order phase transition in the Coulomb gas. The two dimensional phase diagram is obtained.

arXiv commentLaTex, 6 pages, 3 pdf figures ; v2 : minor corrections & Refs. added

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• Full counting statistics of 1d short-range Riesz gases in confinement
• Large deviations of spread measures for Gaussian matrices
• Random matrix ensembles with singularities and a hierarchy of Painlevé III equations
• A Riemann-Hilbert problem for equations of Painlevé type in the one matrix model with semi-classical potential
• Truncated linear statistics in the one dimensional one-component plasma