Ray representation for k-trees

k-trees have established themselves as useful data structures in pattern recognition. A fundamental operation regarding k-trees is the construction of a k-tree. We present a method to store an object as a set of rays and an algorithm to convert such a set into a k-tree. The algorithm is conceptually simple, works for any k, and builds a k-tree from the rays very fast. It produces a minimal k-tree and does not lead to intermediate storage swell. © 1989

Unit Grid Intersection Graphs: Recognition and Properties

It has been known since 1991 that the problem of recognizing grid intersection graphs is NP-complete. Here we use a modified argument of the above result to show that even if we restrict to the class of unit grid intersection graphs (UGIGs), the recognition remains hard, as well as for all graph classes contained inbetween. The result holds even when considering only graphs with arbitrarily large girth. Furthermore, we ask the question of representing UGIGs on grids of minimal size. We show that the UGIGs that can be represented in a square of side length 1+epsilon, for a positive epsilon no greater than 1, are exactly the orthogonal ray graphs, and that there exist families of trees that need an arbitrarily large grid

Character varieties and harmonic maps to R-trees

We show that the Korevaar-Schoen limit of the sequence of equivariant harmonic maps corresponding to a sequence of irreducible $SL_2({\mathbb C})$ representations of the fundamental group of a compact Riemannian manifold is an equivariant harmonic map to an ${\mathbb R}$-tree which is minimal and whose length function is projectively equivalent to the Morgan-Shalen limit of the sequence of representations. We then examine the implications of the existence of a harmonic map when the action on the tree fixes an end.Comment: 12 pages. Latex. to appear in Math. Res. Let

Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary

We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien (2015) by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense.Comment: 29 pages, 13 figures. Added reference and figur

Harmonic analysis on Cayley Trees II: the Bose Einstein condensation

We investigate the Bose-Einstein Condensation on non homogeneous non amenable networks for the model describing arrays of Josephson junctions on perturbed Cayley Trees. The resulting topological model has also a mathematical interest in itself. The present paper is then the application to the Bose-Einstein Condensation phenomena, of the harmonic analysis aspects arising from additive and density zero perturbations, previously investigated by the author in a separate work. Concerning the appearance of the Bose-Einstein Condensation, the results are surprisingly in accordance with the previous ones, despite the lack of amenability. We indeed first show the following fact. Even when the critical density is finite (which is implied in all the models under consideration, thanks to the appearance of the hidden spectrum), if the adjacency operator of the graph is recurrent, it is impossible to exhibit temperature locally normal states (i.e. states for which the local particle density is finite) describing the condensation at all. The same occurs in the transient cases for which it is impossible to exhibit locally normal states describing the Bose--Einstein Condensation at mean particle density strictly greater than the critical density . In addition, for the transient cases, in order to construct locally normal temperature states through infinite volume limits of finite volume Gibbs states, a careful choice of the the sequence of the finite volume chemical potential should be done. For all such states, the condensate is essentially allocated on the base--point supporting the perturbation. This leads that the particle density always coincide with the critical one. It is shown that all such temperature states are Kubo-Martin-Schwinger states for a natural dynamics. The construction of such a dynamics, which is a very delicate issue, is also done.Comment: 28 pages, 6 figures, 1 tabl