2,539 research outputs found
Entropy of random coverings and 4D quantum gravity
We discuss the counting of minimal geodesic ball coverings of -dimensional
riemannian manifolds of bounded geometry, fixed Euler characteristic and
Reidemeister torsion in a given representation of the fundamental group. This
counting bears relevance to the analysis of the continuum limit of discrete
models of quantum gravity. We establish the conditions under which the number
of coverings grows exponentially with the volume, thus allowing for the search
of a continuum limit of the corresponding discretized models. The resulting
entropy estimates depend on representations of the fundamental group of the
manifold through the corresponding Reidemeister torsion. We discuss the sum
over inequivalent representations both in the two-dimensional and in the
four-dimensional case. Explicit entropy functions as well as significant bounds
on the associated critical exponents are obtained in both cases.Comment: 54 pages, latex, no figure
The geometry of dynamical triangulations
We discuss the geometry of dynamical triangulations associated with
3-dimensional and 4-dimensional simplicial quantum gravity. We provide
analytical expressions for the canonical partition function in both cases, and
study its large volume behavior. In the space of the coupling constants of the
theory, we characterize the infinite volume line and the associated critical
points. The results of this analysis are found to be in excellent agreement
with the MonteCarlo simulations of simplicial quantum gravity. In particular,
we provide an analytical proof that simply-connected dynamically triangulated
4-manifolds undergo a higher order phase transition at a value of the inverse
gravitational coupling given by 1.387, and that the nature of this transition
can be concealed by a bystable behavior. A similar analysis in the
3-dimensional case characterizes a value of the critical coupling (3.845) at
which hysteresis effects are present.Comment: 166 pages, Revtex (latex) fil
Ample canonical heights for endomorphisms on projective varieties
We define an "ample canonical height" for an endomorphism on a projective
variety, which is essentially a generalization of the canonical heights for
polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical
analogue of the Northcott finiteness theorem for ample canonical heights as a
conjecture, and prove it for endomorphisms on varieties of small Picard
numbers, abelian varieties, and surfaces. As applications, for the
endomorphisms which satisfy the conjecture, we show the non-density of the set
of preperiodic points over a fixed number field, and obtain a dynamical
Mordell--Lang type result on the intersection of two Zariski dense orbits of
two endomorphisms on a common variety.Comment: 41 pages. The previous version has a serious mistake on the proof of
the main conjecture for simple abelian varieties, but the present version
gives a renewed proof that works for arbitrary abelian varietie
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