97 research outputs found
Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space
Associated to any finite simple graph is the chromatic polynomial
whose complex zeroes are called the chromatic zeros of .
A hierarchical lattice is a sequence of finite simple graphs
built recursively using a substitution rule
expressed in terms of a generating graph. For each , let denote the
probability measure that assigns a Dirac measure to each chromatic zero of
. Under a mild hypothesis on the generating graph, we prove that the
sequence converges to some measure as tends to infinity. We
call the limiting measure of chromatic zeros associated to
. In the case of the Diamond Hierarchical Lattice we
prove that the support of has Hausdorff dimension two.
The main techniques used come from holomorphic dynamics and more specifically
the theories of activity/bifurcation currents and arithmetic dynamics. We prove
a new equidistribution theorem that can be used to relate the chromatic zeros
of a hierarchical lattice to the activity current of a particular marked point.
We expect that this equidistribution theorem will have several other
applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added
considerably more background on activity currents and especially on the
Dujardin-Favre classification of the passive locus. Exposition in the proof
of the main theorem was improved. Comments welcome
Compact hyperbolic tetrahedra with non-obtuse dihedral angles
Given a combinatorial description of a polyhedron having edges, the
space of dihedral angles of all compact hyperbolic polyhedra that realize
is generally not a convex subset of \cite{DIAZ}. If has five
or more faces, Andreev's Theorem states that the corresponding space of
dihedral angles obtained by restricting to {\em non-obtuse} angles is a
convex polytope. In this paper we explain why Andreev did not consider
tetrahedra, the only polyhedra having fewer than five faces, by demonstrating
that the space of dihedral angles of compact hyperbolic tetrahedra, after
restricting to non-obtuse angles, is non-convex. Our proof provides a simple
example of the ``method of continuity'', the technique used in classification
theorems on polyhedra by Alexandrow \cite{ALEX}, Andreev \cite{AND}, and
Rivin-Hodgson \cite{RH}.Comment: 19 page
Some neglected issues in food demand analysis: retail-level demand, health information and product quality
Food demand analysis is dominated by the econometric estimation of demand systems based on aggregate market data and steady progress has been made in analytical techniques. Yet some issues have been neglected in food demand analysis which are crucial for understanding recent consumption trends in industrialised countries. Three of these issues are dealt with here: analysis of food demand at the retail level; influence of health information on food demand; and importance of product quality for food demand. It is shown that answers to important questions in these areas can be given when large and unconventional data sets are used.Agribusiness, Demand and Price Analysis, Health Economics and Policy,
Lee-Yang-Fisher zeros for DHL and 2D rational dynamics, II. Global Pluripotential Interpretation
In a classical work of the 1950's, Lee and Yang proved that for fixed
nonnegative temperature, the zeros of the partition functions of a
ferromagnetic Ising model always lie on the unit circle in the complex magnetic
field. Zeros of the partition function in the complex temperature were then
considered by Fisher, when the magnetic field is set to zero. Limiting
distributions of Lee-Yang and of Fisher zeros are physically important as they
control phase transitions in the model. One can also consider the zeros of the
partition function simultaneously in both complex magnetic field and complex
temperature. They form an algebraic curve called the Lee-Yang-Fisher (LYF)
zeros. In this paper we continue studying their limiting distribution for the
Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms
of the dynamics of an explicit rational function R in two variables (the
Migdal-Kadanoff renormalization transformation). We study properties of the
Fatou and Julia sets of this transformation and then we prove that the
Lee-Yang-Fisher zeros are equidistributed with respect to a dynamical
(1,1)-current in the projective space. The free energy of the lattice gets
interpreted as the pluripotential of this current. We also prove a more general
equidistribution theorem which applies to rational mappings having
indeterminate points, including the Migdal-Kadanoff renormalization
transformation of various other hierarchical lattices.Comment: Continues arXiv:1009.4691. Final version. To appear in The Journal of
Geometric Analysis. (This is a tiny update to correct an error in the Latex
file from previous version.
Typical dynamics of plane rational maps with equal degrees
Let be a rational map with
algebraic and topological degrees both equal to . Little is known in
general about the ergodic properties of such maps. We show here, however, that
for an open set of automorphisms , the
perturbed map admits exactly two ergodic measures of maximal entropy
, one of saddle and one of repelling type. Neither measure is supported
in an algebraic curve, and is `fully two dimensional' in the sense
that it does not preserve any singular holomorphic foliation. Absence of an
invariant foliation extends to all outside a countable union of algebraic
subsets. Finally, we illustrate all of our results in a more concrete
particular instance connected with a two dimensional version of the well-known
quadratic Chebyshev map.Comment: Many small changes in accord with referee comments and suggestion
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