No abstract

In order to understand characteristics common to distributions which have both fractal and non-fractal scale regions in a unified framework, we introduce a concept of typical scale. We employ a model of 2d gravity modified by the R 2 term as a tool to understand such distributions through the typical scale. This model is obtained by adding an interaction term with a typical scale to a scale invariant system. A distribution derived in the model provides power law one in the large scale region, but Weibull-like one in the small scale region. As examples of distributions which have both fractal and non-fractal regions, we take those of personal income and citation number of scientific papers. We show that these distributions are fitted fairly well by the distribution curves derived analytically in the R 2 2d gravity model. As a result, we consider that the typical scale is a useful concept to understand various distributions observed in the real world in a unified way. We also point out that the R 2 2d gravity model provides us with an effective tool to read the typical scales of various distributions in a systematic way.

No abstract

We investigate the correlators in unitary minimal conformal models coupled to two-dimensional gravity from the two-matrix model. We show that simple fusion rules for all of the scaling operators exist. We demonstrate the role played by the boundary operators and discuss its connection to how loops touch each other.

An explicit expression for continuum annulus amplitudes having boundary lengths 11 and l z is obtained from the two-matrix model for the case of the unitary series: (p, q) = ( m + l , m). In the limit of a vanishing cosmological constant, we find a n integral representation of these amplitudes which is reproduced, for the cases of m = 2 (c = 0) and m + cc (c = I ) , by a continuum approach consisting of quantum mechanics of loops and a matter system integrated over the modular parameter of the annulus. We comment on a possible relation t o the unconventional branch of Liouville gravity.PACS number(s): 11.25.Pm, 0 4 . 6 0 . K~