New tools for the study of the aging behavior of disordered systems were developed in [G1] and successfully applied to the study of the arcsine aging regime of Bouchaud’s asymmetric trap model on the complete graph [B, BD, BRM].
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
Establish the optimal time scale and temperature domain where the model exhibits an arcsine aging regime, striving for statements that are valid in the strongest sense possible w.r.t. the law of the random environment.
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
Explain why and how the RHT dynamics of the REM and Bouchaud’s REM-like trap model exhibit the same arcsine aging regime.
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
We now specify the model and succinctly recall the basics of arcsine aging – a detailed exposition can be found in [G1].
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
Theorem 1.2 establishes that arcsine aging is not present on short scales, where ε = 0, and emerges on intermediate scales, as ε becomes positive.
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
These are the very last (and longest) scales where arcsine aging occurs before interruption.
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
Indeed, as t increases from 0 to ∞, the system moves out of its arcsine aging regime and crosses over to its stationarity regime.
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
The aging scheme based on the arcsine law for subordinators was proposed later in the landmark paper [BC] and applied to the study of intermediate scales, using potential theoretic tools.
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
Thus the ensemble of random density matrices constructed according to the procedure shown in Fig. 2 will be called arcsine ensemble.
Generating random density matrices
FIG. 2: To generate states from the arcsine ensemble described by (8) one has to i) construct a superposition of a maximally entangled state |Ψ+ AB i with another maximally enAB i, and ii) perform partial trace over tangled state (UA ⊗ an auxiliary subsystem B .
Generating random density matrices
In other words, one can take a family of random matrices parametrized by a real this expression in place of X into (3) we construct a family of ensembles of density matrices which gives the Dirac mass for a = 0 and a = 1, while the arcsine ensemble is obtained for a = 1/2.
Generating random density matrices
The arcsine ensemble introduced above can be obtained by superimposing k = 2 random maximally entangled states.
Generating random density matrices
The case k = 2 corresponds to the arcsine ensemble, while in the case k → ∞ the sum of k random unitaries has properties of a random Ginibre matrix, so the spectral distribution converges to the Marchenko-Pastur law.
Generating random density matrices
Ben Arous, G.; ˇCern´y, J. (2008) The arcsine law as a universal aging scheme for trap models, Comm.
On the dynamics of trap models in Z^d
V (x) = W (arccos(2x − 1)) is the Radon-Nikodym derivative of µ with respect to the arcsine matricial measure.
Large Deviations for Random Matricial Moment Problems
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