59,671 research outputs found
Cyclic Critical Groups of Graphs
In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the group exactly
Critical percolation of free product of groups
In this article we study percolation on the Cayley graph of a free product of
groups.
The critical probability of a free product of groups
is found as a solution of an equation involving only the expected subcritical
cluster size of factor groups . For finite groups these
equations are polynomial and can be explicitly written down. The expected
subcritical cluster size of the free product is also found in terms of the
subcritical cluster sizes of the factors. In particular, we prove that
for the Cayley graph of the modular group (with the
standard generators) is , the unique root of the polynomial
in the interval .
In the case when groups can be "well approximated" by a sequence of
quotient groups, we show that the critical probabilities of the free product of
these approximations converge to the critical probability of
and the speed of convergence is exponential. Thus for residually finite groups,
for example, one can restrict oneself to the case when each free factor is
finite.
We show that the critical point, introduced by Schonmann,
of the free product is just the minimum of for the factors
Feynman Diagrams via Graphical Calculus
This paper is an introduction to the language of Feynman Diagrams. We use
Reshetikhin-Turaev graphical calculus to define Feynman diagrams and prove that
asymptotic expansions of Gaussian integrals can be written as a sum over a
suitable family of graphs. We discuss how different kind of interactions give
rise to different families of graphs. In particular, we show how symmetric and
cyclic interactions lead to ``ordinary'' and ``ribbon'' graphs respectively. As
an example, the 't Hooft-Kontsevich model for 2D quantum gravity is treated in
some detail.Comment: 30 pages, AMS-LaTeX, 19 EPS figures + several in-text XY-Pic,
PostScript \specials, corrected attributions, 'PROP's instead of 'operads
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