5,269 research outputs found
The arctic curve of the domain-wall six-vertex model
The problem of the form of the `arctic' curve of the six-vertex model with
domain wall boundary conditions in its disordered regime is addressed. It is
well-known that in the scaling limit the model exhibits phase-separation, with
regions of order and disorder sharply separated by a smooth curve, called the
arctic curve. To find this curve, we study a multiple integral representation
for the emptiness formation probability, a correlation function devised to
detect spatial transition from order to disorder. We conjecture that the arctic
curve, for arbitrary choice of the vertex weights, can be characterized by the
condition of condensation of almost all roots of the corresponding saddle-point
equations at the same, known, value. In explicit calculations we restrict to
the disordered regime for which we have been able to compute the scaling limit
of certain generating function entering the saddle-point equations. The arctic
curve is obtained in parametric form and appears to be a non-algebraic curve in
general; it turns into an algebraic one in the so-called root-of-unity cases.
The arctic curve is also discussed in application to the limit shape of
-enumerated (with ) large alternating sign matrices. In
particular, as the limit shape tends to a nontrivial limiting curve,
given by a relatively simple equation.Comment: 39 pages, 2 figures; minor correction
Integrability of graph combinatorics via random walks and heaps of dimers
We investigate the integrability of the discrete non-linear equation
governing the dependence on geodesic distance of planar graphs with inner
vertices of even valences. This equation follows from a bijection between
graphs and blossom trees and is expressed in terms of generating functions for
random walks. We construct explicitly an infinite set of conserved quantities
for this equation, also involving suitable combinations of random walk
generating functions. The proof of their conservation, i.e. their eventual
independence on the geodesic distance, relies on the connection between random
walks and heaps of dimers. The values of the conserved quantities are
identified with generating functions for graphs with fixed numbers of external
legs. Alternative equivalent choices for the set of conserved quantities are
also discussed and some applications are presented.Comment: 38 pages, 15 figures, uses epsf, lanlmac and hyperbasic
Asymptotic formulas for stacks and unimodal sequences
We study enumeration functions for unimodal sequences of positive integers,
where the size of a sequence is the sum of its terms. We survey known results
for a number of natural variants of unimodal sequences, including Auluck's
generalized Ferrer diagrams, Wright's stacks, and Andrews' convex compositions.
These results describe combinatorial properties, generating functions, and
asymptotic formulas for the enumeration functions. We also prove several new
asymptotic results that fill in the notable missing cases from the literature,
including an open problem in statistical mechanics due to Temperley.
Furthermore, we explain the combinatorial and asymptotic relationship between
partitions, Andrews' Frobenius symbols, and stacks with summits.Comment: 19 pages, 4 figure
Generic method for bijections between blossoming trees and planar maps
This article presents a unified bijective scheme between planar maps and
blossoming trees, where a blossoming tree is defined as a spanning tree of the
map decorated with some dangling half-edges that enable to reconstruct its
faces. Our method generalizes a previous construction of Bernardi by loosening
its conditions of applications so as to include annular maps, that is maps
embedded in the plane with a root face different from the outer face.
The bijective construction presented here relies deeply on the theory of
\alpha-orientations introduced by Felsner, and in particular on the existence
of minimal and accessible orientations. Since most of the families of maps can
be characterized by such orientations, our generic bijective method is proved
to capture as special cases all previously known bijections involving
blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable
maps and simple triangulations and quadrangulations of a k-gon. Moreover, it
also permits to obtain new bijective constructions for bipolar orientations and
d-angulations of girth d of a k-gon.
As for applications, each specialization of the construction translates into
enumerative by-products, either via a closed formula or via a recursive
computational scheme. Besides, for every family of maps described in the paper,
the construction can be implemented in linear time. It yields thus an effective
way to encode and generate planar maps.
In a recent work, Bernardi and Fusy introduced another unified bijective
scheme, we adopt here a different strategy which allows us to capture different
bijections. These two approaches should be seen as two complementary ways of
unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom
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
Asymptotic enumeration and limit laws for graphs of fixed genus
It is shown that the number of labelled graphs with n vertices that can be
embedded in the orientable surface S_g of genus g grows asymptotically like
where , and is the exponential growth rate of planar graphs. This generalizes the
result for the planar case g=0, obtained by Gimenez and Noy.
An analogous result for non-orientable surfaces is obtained. In addition, it
is proved that several parameters of interest behave asymptotically as in the
planar case. It follows, in particular, that a random graph embeddable in S_g
has a unique 2-connected component of linear size with high probability
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