547 research outputs found
Boundary operators and touching of loops in 2d gravity
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.Comment: 19 pages, Latex, 3 Postscript figure
Macroscopic -Loop Amplitude for Minimal Models Coupled to Two-Dimensional Gravity: Fusion Rules and Interactions
We investigate the structure of the macroscopic -loop amplitude obtained
from the two-matrix model at the unitary minimal critical point . We
derive a general formula for the -resolvent correlator at the continuum
planar limit whose inverse Laplace transform provides the amplitude in terms of
the boundary lengths and the renormalized cosmological constant .
The amplitude is found to contain a term consisting of multiplied by the product of modified Bessel
functions summed over their degrees which conform to the fusion rules and the
crossing symmetry. This is found to be supplemented by an increasing number of
other terms with which represent residual interactions of loops. We reveal
the nature of these interactions by explicitly determining them as the
convolution of modified Bessel functions and their derivatives for the case
and the case . We derive a set of recursion relations which relate
the terms in the -resolvents to those in the -resolvents.Comment: 30 pages, Latex, figures: figures have been introduced to represent
our results on the resolvents. A better formula for the resolvents has been
put and the section on residual interactions has been expanded to a large
exten
Geometrical Construction of Heterogeneous Loop Amplitudes in 2D Gravity
We study a disk amplitude which has a complicated heterogeneous matter
configuration on the boundary in a system of the (3,4) conformal matter coupled
to two-dimensional gravity. It is analyzed using the two-matrix chain model in
the large N limit. We show that the disk amplitude calculated by
Schwinger-Dyson equations can completely be reproduced through purely
geometrical consideration. From this result, we speculate that all
heterogeneous loop amplitudes can be derived from the geometrical consideration
and the consistency among relevant amplitudes.Comment: 13 pages, 11 figure
Macroscopoic Three-Loop Amplitudes and the Fusion Rules from the Two-Matrix Model
From the computation of three-point singlet correlators in the two-matrix
model, we obtain an explicit expression for the macroscopic three-loop
amplitudes having boundary lengths in the case of
the unitary series coupled to two-dimensional gravity. The sum
appearing in this expression is found to conform to the structure of the CFT
fusion rules while the summand factorizes through a product of three modified
Bessel functions. We briefly discuss a possible generalization of these
features to macroscopic -loop amplitudes.Comment: 9 pages, no figure, late
Inequality in resource allocation and population dynamics models
The Hassell model has been widely used as a general discrete-time population
dynamics model that describes both contest and scramble intraspecific
competition through a tunable exponent. Since the two types of competition
generally lead to different degrees of inequality in the resource distribution
among individuals, the exponent is expected to be related to this inequality.
However, among various first-principles derivations of this model, none is
consistent with this expectation. This paper explores whether a Hassell model
with an exponent related to inequality in resource allocation can be derived
from first principles. Indeed, such a Hassell model can be derived by assuming
random competition for resources among the individuals wherein each individual
can obtain only a fixed amount of resources at a time. Changing the size of the
resource unit alters the degree of inequality, and the exponent changes
accordingly. The Beverton-Holt and Ricker models can be regarded as special
cases of the derived Hassell model. Two additional Hassell models are derived
under some modified assumptions.Comment: 13 pages, 5 figure
Interaction of boundaries with heterogeneous matter states in matrix models
We study disk amplitudes whose boundary conditions on matter configurations
are not restricted to homogeneous ones. They are examined in the two-matrix
model as well as in the three-matrix model for the case of the tricritical
Ising model. Comparing these amplitudes, we demonstrate relations between
degrees of freedom of matter states in the two models. We also show that they
have a simple geometrical interpretation in terms of interactions of the
boundaries. It plays an important role that two parts of a boundary with
different matter states stick each other. We also find two closed sets of
Schwinger-Dyson equations which determine disk amplitudes in the three-matrix
model.Comment: 20 pages, LaTex, 2 eps figures, comments added, introduction
replaced, version to appear in Nuclear Physics
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