Iterative Schwarz-Christoffel Transformations Driven by Random Walks and Fractal Curves

Stochastic Loewner evolution (SLE) is a differential equation driven by a one-dimensional Brownian motion (BM), whose solution gives a stochastic process of conformal transformation on the upper half complex-plane \H. As an evolutionary boundary of image of the transformation, a random curve (the SLE curve) is generated, which is starting from the origin and running in \H toward the infinity as time is going. The SLE curves provides a variety of statistical ensembles of important fractal curves, if we change the diffusion constant of the driving BM. In the present paper, we consider the Schwarz-Christoffel transformation (SCT), which is a conformal map from \H to the region \H with a slit starting from the origin. We prepare a binomial system of SCTs, one of which generates a slit in \H with an angle $\alpha \pi$ from the positive direction of the real axis, and the other of which with an angle $(1-\alpha) \pi$. One parameter $\kappa >0$ is introduced to control the value of $\alpha$ and the length of slit. Driven by a one-dimensional random walk, which is a binomial stochastic process, a random iteration of SCTs is performed. By interpolating tips of slits by straight lines, we have a random path in \H, which we call an Iterative SCT (ISCT) path. It is well-known that, as the number of steps $N$ of random walk goes infinity, each path of random walk divided by $\sqrt{N}$ converges to a Brownian curve. Then we expect that the ISCT paths divided by $\sqrt{N}$ (the rescaled ISCT paths) converge to the SLE curves in $N \to \infty$. Our numerical study implies that, for sufficiently large $N$, the rescaled ISCT paths will have the same statistical properties as the SLE curves have, supporting our expectation.Comment: RevTeX4, 21 pages, 12 figure

The context effect in the choice of earthquake insurance contracts in Japan

In this paper, we investigate the context effect in the choice of public and private earthquake insurance contracts using data from a questionnaire survey completed by an identical set of approximately 2,000 households in 2008 and again in 2009. According to the 2008 survey, the public earthquake insurance (PEI) was not popular among those who felt that the premiums were too high. On the other hand, the 2009 survey demonstrates that the choice of earthquake insurance changed substantially when a hypothetical private earthquake insurance contract, much more expensive than the PEI, was added to a choice menu. In particular, those who had initially felt that PEI was too expensive tended to find it less expensive relative to private insurance, and worth purchasing. A crucial advantage of the above choice architecture is that including additional options helps private insurance companies to develop market activities.earthquake insurance, public insurance, context effects

String scattering off the (2+1)-dimensional rotationg black hole

The $SL(2, R)/Z$ WZW orbifold describes the (2+1)-dimensional black hole which approaches anti-de~Sitter space asymptotically. We study the $1 \rightarrow 1$ tachyon scattering off the rotating black hole background and calculate the Hawking temperature using the Bogoliubov transformation.Comment: 10 pages, LaTe