Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles

Forrester, Peter J.

doi:10.1088/1751-8113/45/14/145201

Cited by 17 publications

(21 citation statements)

References 66 publications

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“…Note that for w = 0 we recover the spectral density of the Jacobi ensemble, see e.g. [54,66,67]. As w varies from w * 1 to w * 2 , the charge density decreases at the center of the interval, eventually vanishing in λ = 1/2 at w = w * 2 .…”

Section: Figmentioning

confidence: 78%

Open Quantum Symmetric Simple Exclusion Process

Bernard

Jin

2019

Phys. Rev. Lett.

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We present the solution to a model of fermions hopping between neighbouring sites on a line with random Brownian amplitudes and open boundary conditions driving the system out of equilibrium. The average dynamics reduces to that of the symmetric simple exclusion process. However, the full distribution encodes for a richer behaviour entailing fluctuating quantum coherences which survive in the steady limit. We determine exactly the steady statistical distribution of the system state. We show that the out of equilibrium quantum coherence fluctuations satisfy a large deviation principle and we present a method to recursively compute exactly the large deviation function. As a byproduct, our approach gives a solution of the classical symmetric simple exclusion process based on fermion technology. Our results open the route towards the extension of the macroscopic fluctuation theory to many body quantum systems.Introduction.-Non-equilibrium phenomena are ubiquitous in Nature. Understanding the fluctuations of the flux of heat or particles through systems is a central question in non equilibrium statistical mechanics. Last decade has witnessed tremendous conceptual and technical progresses in this direction for classical systems, starting from the exact analysis of simple models [1-3], such as the Symmetric Simple Exclusion Process (SSEP) [4][5][6][7], via the understanding of fluctuation relations [8][9][10] and their interplay with time reversal [11,12], and culminating in the formulation of the macroscopic fluctuation theory (MFT) which is an effective theory adapted to describe transport and its fluctuations in diffusive classical systems [13]. Whether MFT may be extended to the quantum realm is yet unexplored.In parallel, the study of quantum systems out of equilibrium has received a large amount of attention in recent years [19][20][21][22]. Experimentally, unprecedented control of cold atom gases gave access to the observation of many body quantum systems in inhomogeneous and isolated setups [14][15][16][17][18]. Theoretically, results about closed, quantum systems have recently flourished, with a better perception of the roles of integrability, chaos or disorder [23][24][25][26][27][28][29][30][31][32][33][34]. In critical or integrable models, a good understanding has been obtained with a precise description of entanglement dynamics, quenched dynamics, as well as transport [35][36][37][38][39][40][41][42][43][44][45][46]. These efforts culminated in the development of a hydrodynamic picture adapted to integrable systems [47,48]. However, these understandings are restricted to closed, predominantly ballistic, systems.Many quantum transport processes are diffusive rather than ballistic [49] and, to some extends, physical systems are generically in contact with external environments. It is thus crucial to extend the previous studies by developing simple models for fluctuations in open, quantum many body, locally diffusive, out of equilibrium systems. Putting aside the quantum nature of the environments leads to co...

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Section: Figmentioning

confidence: 78%

Open Quantum Symmetric Simple Exclusion Process

Bernard

Jin

2019

Phys. Rev. Lett.

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“…[35] 4. If finite N corrections are taken into account, the density spreads over the full interval and presents an exponentionally small tail for x > b [28].…”

Section: 22mentioning

confidence: 99%

Distribution of spectral linear statistics on random matrices beyond the large deviation function—Wigner time delay in multichannel disordered wires

Grabsch¹,

Texier²

2016

J. Phys. A: Math. Theor.

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An invariant ensemble of N × N random matrices can be characterised by a joint distribution for eigenvalues P (λ 1 , · · · , λ N ). The study of the distribution of linear statistics, i.e. of quantities of the formis a given function, appears in many physical problems. In the N → ∞ limit, L scales as L ∼ N η , where the scaling exponent η depends on the ensemble and the function f (x). Its distribution can be written under the form Pis the Dyson index. The Coulomb gas technique naturally provides the large deviation function Φ(s), which can be efficiently obtained thanks to a "thermodynamic identity" introduced earlier. We conjecture the pre-exponential function A N,β (s). We check our conjecture on several well controlled cases within the Laguerre and the Jacobi ensembles. Then we apply our main result to a situation where the large deviation function has no minimum (and L has infinite moments) : this arises in the statistical analysis of the Wigner time delay for semi-infinite multichannel disordered wires (Laguerre ensemble). The statistical analysis of the Wigner time delay then crucially depends on the pre-exponential function A N,β (s), which ensures the decay of the distribution for large argument.

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“…In two recent works by the author [21,20], the spectral densities for the Gaussian, Laguerre and Jacobi β-ensembles have been similarly formulated, and the corresponding asymptotics computed by recognizing that such averages can be interpreted as the characteristic function for the linear statistic V (x) = N l=1 log |x−λ l |. The significance of this is that for ME β,N (e −2βN λ ), which is the scaled Laguerre ensemble with a = 0, eigenvalue density ρ (1),N (t) supported to leading order on (0, 1), it is a known theorem [3] that…”

Section: Strategymentioning

confidence: 99%

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

Forrester

2013

Random Matrices: Theory Appl.

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In a previous work [J. Math. Phys. 35 (1994) [2539][2540][2541][2542][2543][2544][2545][2546][2547][2548][2549][2550][2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large s asymptotic form of the general β > 0 gap probability E hard β (0; (0, s); βa/2), provided both βa/2 ∈ Z ≥0 and 2/β ∈ Z + . It is shown how the details of this method can be extended to remove the requirement that 2/β ∈ Z + . Furthermore, a large deviation formula for the gap probability E β (n; (0, x); ME β,N (λ aβ/2 e −2βNλ )) is deduced by writing it in terms of the characteristic function of a certain linear statistic. By scaling x = s/(4N ) 2 and taking N → ∞, this is shown to reproduce a recent conjectured formula for E hard β (n; (0, s); βa/2), βa/2 ∈ Z ≥0 , and moreover to give a prediction without the latter restriction. This extended formula, which for the constant term involves the Barnes double gamma function, is shown to satisfy an asymptotic functional equation relating the gap probability with parameters (β, n, a), to a gap probability with parameters (4/β, n , a ), where n = β(n + 1)/2 − 1, a = β(a − 2)/2 + 2.

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Large deviation eigenvalue density for the soft edge Laguerre and Jacobi β-ensembles

Cited by 17 publications

References 66 publications

Open Quantum Symmetric Simple Exclusion Process

Open Quantum Symmetric Simple Exclusion Process

Distribution of spectral linear statistics on random matrices beyond the large deviation function—Wigner time delay in multichannel disordered wires

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

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