Time-reversal of multiple-force-point chordal SLEκ(ρ‾)\mathrm{SLE}_\kappa(\underline{\rho})

Chordal SLE$_\kappa(\underline{\rho})$ is a natural variant of chordal SLE curve. It is a family of random non-crossing curves on the upper half plane from 0 to $\infty$, whose law is influenced by additional force points on $\mathbb R$. When there are force points away from the origin, the law of SLE$_\kappa(\underline{\rho})$ is not reversible as the ordinary chordal SLE$_\kappa$. Zhan (2019) give an explicit description of the law of the time reversal of SLE$_\kappa(\underline{\rho})$ when all force points lies on the same sides of the origin, and conjectured that a similar result holds in general. In this paper we prove his conjecture. In particular, based on Zhan's result, using the techniques from the Imaginary Geometry developed by Miller and Sheffield (2013), we show that when $\kappa\in(0,8)$, the law of the time reversal of non-boundary filling $\mathrm{SLE}_\kappa(\underline{\rho})$ process is absolutely continuous with respect to $\mathrm{SLE}_\kappa(\underline{\hat{\rho}})$ for some $\underline{\hat{\rho}}$ determined by $\underline{\rho}$, with the Radon-Nikodym derivative being a product of conformal derivatives.Comment: 13 pages, 6 figure

Dark information of black hole radiation raised by dark energy

The "lost" information of black hole through the Hawking radiation was discovered being stored in the correlation among the non-thermally radiated particles [Phys. Rev. Lett 85, 5042 (2000), Phys. Lett. B 675, 1 (2009)]. This correlation information, which has not yet been proved locally observable in principle, is named by dark information. In this paper, we systematically study the influences of dark energy on black hole radiation, especially on the dark information. Calculating the radiation spectrum in the existence of dark energy by the approach of canonical typicality, which is reconfirmed by the quantum tunneling method, we find that the dark energy will effectively lower the Hawking temperature, and thus makes the black hole has longer life time. It is also discovered that the non-thermal effect of the black hole radiation is enhanced by dark energy so that the dark information of the radiation is increased. Our observation shows that, besides the mechanical effect (e.g., gravitational lensing effect), the dark energy rises the the stored dark information, which could be probed by a non-local coincidence measurement similar to the coincidence counting of the Hanbury-Brown -Twiss experiment in quantum optics.Comment: 21 pages, 3 figures, complete journal-info of Ref.[4] is added, comments are welcome ([email protected]