4,457 research outputs found
Random trees between two walls: Exact partition function
We derive the exact partition function for a discrete model of random trees
embedded in a one-dimensional space. These trees have vertices labeled by
integers representing their position in the target space, with the SOS
constraint that adjacent vertices have labels differing by +1 or -1. A
non-trivial partition function is obtained whenever the target space is bounded
by walls. We concentrate on the two cases where the target space is (i) the
half-line bounded by a wall at the origin or (ii) a segment bounded by two
walls at a finite distance. The general solution has a soliton-like structure
involving elliptic functions. We derive the corresponding continuum scaling
limit which takes the remarkable form of the Weierstrass p-function with
constrained periods. These results are used to analyze the probability for an
evolving population spreading in one dimension to attain the boundary of a
given domain with the geometry of the target (i) or (ii). They also translate,
via suitable bijections, into generating functions for bounded planar graphs.Comment: 25 pages, 7 figures, tex, harvmac, epsf; accepted version; main
modifications in Sect. 5-6 and conclusio
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
Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop
We consider quadrangulations with a boundary and derive explicit expressions
for the generating functions of these maps with either a marked vertex at a
prescribed distance from the boundary, or two boundary vertices at a prescribed
mutual distance in the map. For large maps, this yields explicit formulas for
the bulk-boundary and boundary-boundary correlators in the various encountered
scaling regimes: a small boundary, a dense boundary and a critical boundary
regime. The critical boundary regime is characterized by a one-parameter family
of scaling functions interpolating between the Brownian map and the Brownian
Continuum Random Tree. We discuss the cases of both generic and self-avoiding
boundaries, which are shown to share the same universal scaling limit. We
finally address the question of the bulk-loop distance statistics in the
context of planar quadrangulations equipped with a self-avoiding loop. Here
again, a new family of scaling functions describing critical loops is
discovered.Comment: 55 pages, 14 figures, final version with minor correction
Combinatorics of bicubic maps with hard particles
We present a purely combinatorial solution of the problem of enumerating
planar bicubic maps with hard particles. This is done by use of a bijection
with a particular class of blossom trees with particles, obtained by an
appropriate cutting of the maps. Although these trees have no simple local
characterization, we prove that their enumeration may be performed upon
introducing a larger class of "admissible" trees with possibly doubly-occupied
edges and summing them with appropriate signed weights. The proof relies on an
extension of the cutting procedure allowing for the presence on the maps of
special non-sectile edges. The admissible trees are characterized by simple
local rules, allowing eventually for an exact enumeration of planar bicubic
maps with hard particles. We also discuss generalizations for maps with
particles subject to more general exclusion rules and show how to re-derive the
enumeration of quartic maps with Ising spins in the present framework of
admissible trees. We finally comment on a possible interpretation in terms of
branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction
and discussion/conclusion extended, minor corrections, references adde
Multicritical continuous random trees
We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.Comment: 34 pages, 12 figures, uses lanlmac, hyperbasics, eps
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