Invariant ensembles, linear statistics and Coulomb gas

An invariant ensemble of random matrices is defined by a probability measure over a set of matrices, which is invariant under unitary transformations. In this case, the eigenvectors and the eigenvalues are uncorrelated and the statistical properties of the matrices can be characterised by the joint probability distribution function of the eigenvalues, generically of the form

PN1,…,λN) = ANi<jij|βkφ(λk)

where the function φ(λ) depends on the ensemble. Examples are φ(λ)=exp(-λ2) for the Gaussian ensembles, where λ∈ℜ, φ(λ)=λα-1exp(-λ) for the Laguerre ensembles, where λ∈ℜ+, etc. β=1, 2 or 4 is the Dyson index corresponding to orthogonal, unitary or symplectic symmetry classes.

Linear statistics of eigenvalues are the form

L = (1/N)∑i=1N f(λi)

where f(λ) is a given function. They appear in many concrete applications of random matrix theory. We have considered several examples of linear statistics and have studied their statistical properties with the Coulomb gas technique. Interestingly, the analysis of the large deviations (atypical fluctuations) can be related in some cases to phase transitions in the underlying Coulomb gas.


Wigner time delay of chaotic cavities — Freezing transition in the Coulomb gas

We study the Wigner time delay of chaotic cavities within a random matrix approach (RMT). The analysis of Wigner time delay, partial time delays (derivatives of scattering phase shifts with respect to the energy) or proper time delays (eigenvalues of Wigner-Smith time delay matrix) have attracted a lot of attention at the end of the 90ies in the context of 1D disordered systems [Jayannavar,  Vijayagovindan & Kumar, Z. Phys. B – Condens. Matter 75 (1989) 77 ; Heinrichs, J. Phys. Cond. Matter 2 (1990) 1559 ; these first results were later corrected and the universality, suggested in Comtet & Texier J.Phys.A 30, 8017 (1997), explained in : C. Texier & A. Comtet, Phys. Rev. Lett. 82, 4220 (1999)] and chaotic cavities. An important step in this history was the discovery of the joint distribution for the proper time delays by Brouwer, Frahm & Beenakker [Phys. Rev. Lett. 78, 4737 (1997)], using the « Alternative Stochastic Approach » introduced slightly earlier by Brouwer & Büttiker [Europhys. Lett. 37, 441 (1997)]. Despite the joint distribution of proper times is known since 1997, little is known about the distribution of their sum, the Wigner time delay, despite the interest of this question in relation with charging effects in mesoscopic coherent conductors (cf. work by Markus Büttiker and coworkers). The distribution of the Wigner time delay is known for N=1 [Gopar, Mello & Büttiker, Phys. Rev. Lett. 77, 3005 (1996)] and N=2 conducting channels [Savin, Fyodorov & Sommers, Phys. Rev. E 63, 035202 (2001)]. The exponent of the power tail of the Wigner time delay distribution has been conjectured to coincide with the one of the marginal law for partial times, based on the heuristic picture of resonances [Fyodorov & Sommers, J. Math. Phys. 38, 1918 (1997)]. More recently, Mezzadri & Simm shown how to derive systematically the cumulants of the Wigner time delay by a generating function approach (math-ph arXiv:1206.4584), and gave explicitely the four first cumulants [note that the leading order of the variance has been derived for the first time (for β=1) some time ago in the pioneering work : Lehmann, Savin, Sokolov and Sommers, Physica 86D, 572 (1995) and is included in  Brouwer & Büttiker’s results [Europhys. Lett. 37, 441 (1997)].

In this article we derive the distribution of the Wigner time delay in the limit of large number of conducting channels by a Coulomb gas approach. We show that it presents a rich structure. Interestingly, the emergence of the power law tail is related to a second order freezing transition in the Coulomb gaz.

Statistical properties of the complex admittance in a one contact mesoscopic conductor


Distribution of the Wigner time delay (DoS) in semi-infinite multichannel disordered wires

Distribution of truncated linear statistics of eigenvalues

We have introduced a new type of question and considered truncated linear statistics of eigenvalues

L = (1/N)∑i=1K f(λi) for K<N

We have analysed this problem with the Coulomb gas technique. In a first article we have considered the case where an additional constraint is introduced and the sum runs over the largest (or smallest) eigenvalues λ12>…>λKK+1>…>λN.
In the second paper we have removed this addtional constraint.
The two cases lead to different phase diagrams.

Les commentaires sont fermés.