1,497 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
Phylogenetic toric varieties on graphs
We define phylogenetic projective toric model of a trivalent graph as a
generalization of a binary symmetric model of a trivalent phylogenetic tree.
Generators of the pro- jective coordinate ring of the models of graphs with one
cycle are explicitly described. The phylogenetic models of graphs with the same
topological invariants are deforma- tion equivalent and share the same Hilbert
function. We also provide an algorithm to compute the Hilbert function.Comment: 36 pages, improved expositio
Singular structure of Toda lattices and cohomology of certain compact Lie groups
We study the singularities (blow-ups) of the Toda lattice associated with a
real split semisimple Lie algebra . It turns out that the total
number of blow-up points along trajectories of the Toda lattice is given by the
number of points of a Chevalley group related to the maximal
compact subgroup of the group with over the finite field . Here is the Langlands dual of . The blow-ups of the Toda lattice
are given by the zero set of the -functions. For example, the blow-ups of
the Toda lattice of A-type are determined by the zeros of the Schur polynomials
associated with rectangular Young diagrams. Those Schur polynomials are the
-functions for the nilpotent Toda lattices. Then we conjecture that the
number of blow-ups is also given by the number of real roots of those Schur
polynomials for a specific variable. We also discuss the case of periodic Toda
lattice in connection with the real cohomology of the flag manifold associated
to an affine Kac-Moody algebra.Comment: 23 pages, 12 figures, To appear in the proceedings "Topics in
Integrable Systems, Special Functions, Orthogonal Polynomials and Random
Matrices: Special Volume, Journal of Computational and Applied Mathematics
Toda lattice, cohomology of compact Lie groups and finite Chevalley groups
In this paper, we describe a connection that exists among (a) the number of
singular points along the trajectory of Toda flow, (b) the cohomology of a
compact subgroup , and (c) the number of points of a Chevalley group
related to over a finite field . The Toda
lattice is defined for a real split semisimple Lie algebra , and
is a maximal compact Lie subgroup of associated to .
Relations are also obtained between the singularities of the Toda flow and the
integral cohomology of the real flag manifold with the Borel subgroup
of (here we have with a finite group ). We also compute the
maximal number of singularities of the Toda flow for any real split semisimple
algebra, and find that this number gives the multiplicity of the singularity at
the intersection of the varieties defined by the zero set of Schur polynomials.Comment: 28 pages, 5 figure
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