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 $A={\sum }_{i=1}^{N}a(T_i)$, the probability distribution ${\mathcal{P}}_A(A,N)$ of $A$ generically satisfies the large deviation formula ${\lim }_{N\to \infty }[-2\log {\mathcal{P}}_A(Nx,N)/\beta N^2]={\Psi }_A(x)$, where ${\Psi }_A(x)$ is a rate function that we compute explicitly in many cases (conductance, shot noise, moments) and $\beta$ 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 $v(n)={\lim }_{N\to \infty }\mathrm{var}({\mathcal{T}}_n)$ (where ${\mathcal{T}}_n={\sum }_i{T}_i^n$) for arbitrary integer $n$. The universal limit $v^{\star }={\lim }_{n\to \infty }v(n)=1/2\pi \beta$ 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 non-analytical 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 in [P. Vivo, S.N. Majumdar and O. Bohigas, {\it Phys. Rev. Lett.} {\bf 101}, 216809 (2008)].

31 pages, 16 figures. To appear in Phys. Rev. B. Added section IVD about comparison with other theories and numerical simulations

References in corpus (19)

• The large deviation approach to statistical mechanics
• Large Deviations of Extreme Eigenvalues of Random Matrices
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices
• Large Deviations of the Maximum Eigenvalue in Wishart Random Matrices
• Conditional statistics of electron transport in interacting nanoscale conductors
• The Index Distribution of Gaussian Random Matrices
• Distributions of Conductance and Shot Noise and Associated Phase Transitions
• Systematic approach to statistics of conductance and shot-noise in chaotic cavities
• Nonlinear statistics of quantum transport in chaotic cavities
• Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry
• Integrable theory of quantum transport in chaotic cavities
• Experimental Test of Universal Conductance Fluctuations by means of Wave-Chaotic Microwave Cavities
• Full counting statistics of chaotic cavities from classical action correlations
• Shot noise from action correlations
• Full counting statistics of chaotic cavities with many open channels
• Statistics of thermal to shot noise crossover in chaotic cavities
• The density of stationary points in a high-dimensional random energy landscape and the onset of glassy behaviour
• Full counting statistics of a chaotic cavity with asymmetric leads

Cited by in corpus (48)

• Extreme value statistics of correlated random variables: a pedagogical review
• Top eigenvalue of a random matrix: large deviations and third order phase transition
• Statistical distribution of quantum entanglement for a random bipartite state
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• Wigner time-delay distribution in chaotic cavities and freezing transition
• A First-Order Dynamical Transition in the displacement distribution of a Driven Run-and-Tumble Particle
• How many eigenvalues of a Gaussian random matrix are positive?
• Tau-Function Theory of Quantum Chaotic Transport with beta=1,2,4
• Symmetry-resolved Page curves
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• Entanglement distribution in the Quantum Symmetric Simple Exclusion Process
• Number statistics for $\beta$-ensembles of random matrices: applications to trapped fermions at zero temperature
• A shortcut through the Coulomb gas method for spectral linear statistics on random matrices
• Universality of the third-order phase transition in the constrained Coulomb gas
• Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities
• Full counting statistics for interacting trapped fermions
• Phase transitions in the distribution of the Andreev conductance of superconductor-metal junctions with multiple transverse modes
• Truncated linear statistics associated with the top eigenvalues of random matrices
• Capacitance and charge relaxation resistance of chaotic cavities - Joint distribution of two linear statistics in the Laguerre ensemble of random matrices
• Universality of the weak pushed-to-pulled transition in systems with repulsive interactions
• Index Distribution of Cauchy Random Matrices
• Edge fluctuations and third-order phase transition in harmonically confined long-range systems
• Turbulence Hierarchy and Multifractality in the Integer Quantum Hall Transition
• Distribution of spectral linear statistics on random matrices beyond the large deviation function -- Wigner time delay in multichannel disordered wires
• Universal covariance formula for linear statistics on random matrices
• Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel
• Purity distribution for generalized random Bures mixed states
• Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula
• Third-order phase transition: random matrices and screened Coulomb gas with hard walls
• Topological phase transitions in the 1D multichannel Dirac equation with random mass and a random matrix model
• Truncated linear statistics associated with the eigenvalues of random matrices II. Partial sums over proper time delays for chaotic quantum dots
• Joint statistics of quantum transport in chaotic cavities
• Largest Schmidt eigenvalue of entangled random pure states and conductance distribution in chaotic cavities
• Entanglement production in non-ideal cavities and optimal opacity
• Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• Statistics of quantum transport in weakly non-ideal chaotic cavities
• A singular-potential random matrix model arising in mean-field glassy systems
• Full counting statistics of 1d short-range Riesz gases in confinement
• General truncated linear statistics for the top eigenvalues of random matrices
• Spatio-temporal fluctuations in the passive and active Riesz gas on the circle
• Large deviations of spread measures for Gaussian matrices
• Klein Paradox in Chaotic Dirac Billiards
• Asymmetric scaling in large deviations for rare values bigger or smaller than the typical value
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"
• Global and Local Scaling Limits for Linear Eigenvalue Statistics of Jacobi $\beta$-Ensembles
• Entanglement transitions induced by large deviations