We give a direct alternative proof of an area law for the entanglement entropy of the ground state of disordered oscillator systems-a result due to Nachtergaele, Sims and Stolz [22]. Instead of studying the logarithmic negativity, we invoke the explicit formula for the entanglement entropy of Gaussian states to derive the upper bound. We also contrast this area law in the disordered case with divergent lower bounds on the entanglement entropy of the ground state of one-dimensional ordered oscillator chains. (2010): 82B44. Mathematics Subject Classification
We prove an averaging principle for interacting slow-fast systems driven by independent fractional Brownian motions. The mode of convergence is in Hölder norm in probability. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, partially improving a recent result of Panloup and Richard.
We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process ( y t ) t ⩾ 0 in the rough path topology. As an application, we establish weak convergence as ɛ → 0 of the solution of the random ordinary differential equation (ODE) d d t x t ε = 1 ε f ( x t ε , y t ε ) and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type Itô SDE dx t = G(x t , dt), where G(x, t) is a semi-martingale with spatial parameters.
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