Probability distributions of linear statistics in chaotic cavities and associated phase transitions

Abstract

We establish large deviation formulas for linear statistics on the $N$ transmission eigenvalues ${{T}_{i}}$ of a chaotic cavity, in the framework of random matrix theory. Given any linear statistics of interest $A={\ensuremath{\sum}}_{i=1}^{N}a({T}_{i})$, the probability distribution ${\mathcal{P}}_{A}(A,N)$ of $A$ generically satisfies the large deviation formula ${\text{lim}}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}[\ensuremath{-}2\text{ }\text{log}\text{ }{\mathcal{P}}_{A}(Nx,N)/\ensuremath{\beta}{N}^{2}]={\ensuremath{\Psi}}_{A}(x)$, where ${\ensuremath{\Psi}}_{A}(x)$ is a rate function that we compute explicitly in many cases (conductance, shot noise, and moments) and $\ensuremath{\beta}$ corresponds to different symmetry classes. Using these large deviation expressions, it is possible to recover easily known results and to produce new formulas, such as a closed form expression for $v(n)={\text{lim}}_{N\ensuremath{\rightarrow}\ensuremath{\infty}}\text{ }\text{var}({\mathcal{T}}_{n})$ (where ${\mathcal{T}}_{n}={\ensuremath{\sum}}_{i}{T}_{i}^{n}$) for arbitrary integer $n$. The universal limit ${v}^{\ensuremath{\star}}={\text{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}\text{ }v(n)=1/2\ensuremath{\pi}\ensuremath{\beta}$ is also computed exactly. The distributions display a central Gaussian region flanked on both sides by non-Gaussian tails. At the junction of the two regimes, weakly nonanalytical points appear, a direct consequence of phase transitions in an associated Coulomb gas problem. Numerical checks are also provided, which are in full agreement with our asymptotic results in both real and Laplace space even for moderately small $N$. Part of the results have been announced by Vivo et al. [Phys. Rev. Lett. 101, 216809 (2008)].