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

2009/09/30 by Pierpaolo Vivo, Satya N. Majumdar, O. Bohigas +1 · 71 citations
Mathematics · Physics and Astronomy · #Artificial intelligence #Chaotic #Computer science #Mathematics #Phase (matter) #Physics #Probability and statistics #Probability distribution #Quantum chaos and dynamical systems #Quantum mechanics #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.mes-hall #cond-mat.stat-mech #math-ph #math.MP

paper · pdf · open access · doi:10.1103/physrevb.81.104202

published in Physical Review B 81(10) (American Physical Society) · 31 pages, 16 figures. To appear in Phys. Rev. B. Added section IVD about comparison with other theories and numerical simulations

arxiv created 2010/02/07 · openalex publication_date 2010/03/12 · arxiv updated 2015/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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∑_i=1^Na(T_i), the probability distribution P_A(A,N) of A generically satisfies the large deviation formula lim_N\ensuremath→\ensuremath∞[\ensuremath-2 log P_A(Nx,N)/\ensuremathβN²]=\ensuremathΨ_A(x), where \ensuremathΨ_A(x) is a rate function that we compute explicitly in many cases (conductance, shot noise, and moments) and \ensuremathβ 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\ensuremath→\ensuremath∞ var(T_n) (where T_n=\ensuremath∑_iT_iⁿ) for arbitrary integer n. The universal limit v^\ensuremath⋆=lim_n\ensuremath→\ensuremath∞ v(n)=1/2\ensuremathπ\ensuremathβ 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)].

Citations

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