Ben Arous and J. ˇCern´y, The arcsine law as a universal aging scheme for trap models, Comm.
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
Emergence of the arcsine law in Bouchaud’s asymmetric trap model on the complete graph (2010).
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
Emergence of the arcsine law in the Metropolis dynamics of the REM (in preparation) (2010).
Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM
In Section III we use 2 independent random unitary operators to construct the arcsine ensemble obtained by superposition of two random maximally entangled states.
Generating random density matrices
The name ‘arcsine’ is due to the fact that the corresponding cumulative distribution is proportional to sin−1 px/2.
Generating random density matrices
Moreover, the last integrand in (10) is somewhat different from the integrand in the integral (6) deﬁning C (neglecting the constant in front of the integral in (6)) in that in (6) we subtract J0 (ψ) from the arcsine.
The defect variance of random spherical harmonics
To account for this difference, we need to subtract J0 (ψ) from the arcsine in the integrand (10), thus obtaining a conditionally convergent integral.
The defect variance of random spherical harmonics
As it was explained in Section 2, to extract the asymptotics of Il , we will expand the arcsine on the RHS of (15) into the Taylor series around the origin; we will encounter only the odd moments of Pl (cos θ), due to the arcsine being an odd function.
The defect variance of random spherical harmonics
The advantage of the latter representation (18) over (15) is that the only powers that will appear in the Taylor expansion of the arcsine on the righthand side of (18) are of order ≥ 3, so that the moments of Pl (cos θ) are all identical to the corresponding moments of the scaling limit.
The defect variance of random spherical harmonics
Note that all the Taylor coeﬃcients ak of arcsine are positive (see (20)).
The defect variance of random spherical harmonics
Let us keep in mind that Mδ sitting on the unstable stationary point ψ = arcsin (cid:0) 1 is close to M , which is a circle, so that also the dynamics on Mδ can be reduced to the dynamics of a phase (see Fig. 1).
Transitions in active rotator systems: invariant hyperbolic manifold approach
Keywords: hyperbolic trigonometry, arcsine law, continued fractions, Fibonacci numbers, non-linear transformations of random variables.
Angular processes related to Cauchy random walks
It is well-known that the random variable U1 possesses the arcsin law.
Angular processes related to Cauchy random walks
For a ﬁxed t > 0, the compound undershooting subordinator U −γ [Sγ (t/∆t)] has the generalized arcsine distribution rescaled by t/∆t.
Experimental Evidence of the Role of Compound Counting Processes in Random Walk Approaches to Fractional Dynamics
Ben Arous and J. ˇCern´y, The arcsine law as a universal aging scheme for trap models, Comm.
Biased random walk on critical Galton-Watson trees conditioned to survive
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