142 research outputs found
A proof of the rooted tree alternative conjecture
Bonato and Tardif conjectured that the number of isomorphism classes of trees
mutually embeddable with a given tree T is either 1 or infinite. We prove the
analogue of their conjecture for rooted trees. We also discuss the original
conjecture for locally finite trees and state some new conjectures
Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank
We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) \u3e 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, Tc(f)T(f)
Dynamics of surface diffeomorphisms relative to homoclinic and heteroclinic orbits
The Nielsen-Thurston theory of surface diffeomorphisms shows that useful
dynamical information can be obtained about a surface diffeomorphism from a
finite collection of periodic orbits.In this paper, we extend these results to
homoclinic and heteroclinic orbits of saddle points. These orbits are most
readily computed and studied as intersections of unstable and stable manifolds
comprising homoclinic or heteroclinic tangles in the surface. We show how to
compute a map of a one-dimensional space similar to a train-track which
represents the isotopy-stable dynamics of the surface diffeomorphism relative
to a tangle. All orbits of this one-dimensional representative are globally
shadowed by orbits of the surface diffeomorphism, and periodic, homoclinic and
heteroclinic orbits of the one-dimensional representative are shadowed by
similar orbits in the surface.By constructing suitable surface diffeomorphisms,
we prove that these results are optimal in the sense that the topological
entropy of the one-dimensional representative is the greatest lower bound for
the entropies of diffeomorphisms in the isotopy class.Comment: Version submitted to "Dynamical Systems: An International Journal"
Section 7 has been further revised; the method for pA maps is new. Notation
has been standardised throughou
A computational approach for finding 6-List-critical graphs on the Torus
La coloració de grafs dibuixats a superfÃcies és un à rea antiga i molt estudiada de la teoria de grafs. Thomassen va demostrar que hi ha un nombre finit de grafs 6-crÃtics a qualsevol superfÃcie fixa i va proporcionar el conjunt explÃcit dels grafs 6-crÃtics al torus. Després, Postle va demostrar que hi ha un nombre finit de grafs 6-llista-crÃtics a qualsevol superfÃcie fixa. Amb l'objectiu de trobar el conjunt de grafs 6-llista-crÃtics al torus, desenvolupem i implementem tècniques algorÃtmiques per la cerca per ordinador de grafs crÃtics en diferents situacions de coloració per llistes.La coloración de grafos dibujados en superficies es un área antigua y muy estudiada de la teorÃa de grafos. Thomassen demostró que hay un número finito de grafos 6-crÃticos en cualquier superficie fija y proporcionó el conjunto explÃcito de los grafos 6-crÃticos en el toro. Después, Postle demostró que hay un número finito de grafos 6-lista-crÃticos en cualquier superficie fija. Con el objetivo de encontrar el conjunto de grafos 6-lista-crÃticos en el toro, desarrollamos e implementamos técnicas algorÃtmicas para la búsqueda por ordenador de grafos crÃticos en diferentes situaciones de coloración por listas.Coloring graphs embedded on surfaces is an old and well-studied area of graph theory. Thomassen proved that there are finitely many 6-critical graphs on any fixed surface and provided the explicit set of 6-critical graphs on the torus. Later, Postle proved that there are finitely many 6-list-critical graphs on any fixed surface. With the goal of finding the set of 6-list-critical graphs on the torus, we develop and implement algorithmic techniques for computer search of critical graphs in different list-coloring settings.Outgoin
Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank
We consider pairs of few-body Ising models where each spin enters a bounded
number of interaction terms (bonds), such that each model can be obtained from
the dual of the other after freezing spins on large-degree sites. Such a
pair of Ising models can be interpreted as a two-chain complex with being
the rank of the first homology group. Our focus is on the case where is
extensive, that is, scales linearly with the number of bonds . Flipping any
of these additional spins introduces a homologically non-trivial defect
(generalized domain wall). In the presence of bond disorder, we prove the
existence of a low-temperature weak-disorder region where additional summation
over the defects have no effect on the free energy density in the
thermodynamical limit, and of a high-temperature region where in the
ferromagnetic case an extensive homological defect does not affect . We
also discuss the convergence of the high- and low-temperature series for the
free energy density, prove the analyticity of limiting at high and low
temperatures, and construct inequalities for the critical point(s) where
analyticity is lost. As an application, we prove multiplicity of the
conventionally defined critical points for Ising models on all
tilings of the hyperbolic plane, where . Namely, for these infinite
graphs, we show that critical temperatures with free and wired boundary
conditions differ, .Comment: 18 pages, 6 figure
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