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

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