2,798 research outputs found
Planar maps as labeled mobiles
We extend Schaeffer's bijection between rooted quadrangulations and
well-labeled trees to the general case of Eulerian planar maps with prescribed
face valences, to obtain a bijection with a new class of labeled trees, which
we call mobiles. Our bijection covers all the classes of maps previously
enumerated by either the two-matrix model used by physicists or by the
bijection with blossom trees used by combinatorists. Our bijection reduces the
enumeration of maps to that, much simpler, of mobiles and moreover keeps track
of the geodesic distance within the initial maps via the mobiles' labels.
Generating functions for mobiles are shown to obey systems of algebraic
recursion relations.Comment: 31 pages, 17 figures, tex, lanlmac, epsf; improved tex
Shapes of topological RNA structures
A topological RNA structure is derived from a diagram and its shape is
obtained by collapsing the stacks of the structure into single arcs and by
removing any arcs of length one. Shapes contain key topological, information
and for fixed topological genus there exist only finitely many such shapes. We
shall express topological RNA structures as unicellular maps, i.e. graphs
together with a cyclic ordering of their half-edges. In this paper we prove a
bijection of shapes of topological RNA structures. We furthermore derive a
linear time algorithm generating shapes of fixed topological genus. We derive
explicit expressions for the coefficients of the generating polynomial of these
shapes and the generating function of RNA structures of genus . Furthermore
we outline how shapes can be used in order to extract essential information of
RNA structure databases.Comment: 27 pages, 11 figures, 2 tables. arXiv admin note: text overlap with
arXiv:1304.739
Matrix Models on Large Graphs
We consider the spherical limit of multi-matrix models on regular target
graphs, for instance single or multiple Potts models, or lattices of arbitrary
dimension. We show, to all orders in the low temperature expansion, that when
the degree of the target graph , the free energy becomes
independent of the target graph, up to simple transformations of the matter
coupling constant. Furthermore, this universal free energy contains
contributions only from those surfaces which are made up of ``baby universes''
glued together into trees, all non-universal and non-tree contributions being
suppressed by inverse powers of . Each order of the free energy is put
into a simple, algebraic form.Comment: 19pp. (uses harvmac and epsf), PUPT-139
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