73 research outputs found
Unified bijections for maps with prescribed degrees and girth
This article presents unified bijective constructions for planar maps, with
control on the face degrees and on the girth. Recall that the girth is the
length of the smallest cycle, so that maps of girth at least are
respectively the general, loopless, and simple maps. For each positive integer
, we obtain a bijection for the class of plane maps (maps with one
distinguished root-face) of girth having a root-face of degree . We then
obtain more general bijective constructions for annular maps (maps with two
distinguished root-faces) of girth at least . Our bijections associate to
each map a decorated plane tree, and non-root faces of degree of the map
correspond to vertices of degree of the tree. As special cases we recover
several known bijections for bipartite maps, loopless triangulations, simple
triangulations, simple quadrangulations, etc. Our work unifies and greatly
extends these bijective constructions. In terms of counting, we obtain for each
integer an expression for the generating function
of plane maps of girth with root-face of
degree , where the variable counts the non-root faces of degree .
The expression for was already obtained bijectively by Bouttier, Di
Francesco and Guitter, but for the expression of is new. We
also obtain an expression for the generating function
\G_{p,q}^{(d,e)}(x_d,x_{d+1},...) of annular maps with root-faces of degrees
and , such that cycles separating the two root-faces have length at
least while other cycles have length at least . Our strategy is to
obtain all the bijections as specializations of a single "master bijection"
introduced by the authors in a previous article. In order to use this approach,
we exhibit certain "canonical orientations" characterizing maps with prescribed
girth constraints
Cutting down trees with a Markov chainsaw
We provide simplified proofs for the asymptotic distribution of the number of
cuts required to cut down a Galton-Watson tree with critical, finite-variance
offspring distribution, conditioned to have total progeny . Our proof is
based on a coupling which yields a precise, nonasymptotic distributional result
for the case of uniformly random rooted labeled trees (or, equivalently,
Poisson Galton-Watson trees conditioned on their size). Our approach also
provides a new, random reversible transformation between Brownian excursion and
Brownian bridge.Comment: Published in at http://dx.doi.org/10.1214/13-AAP978 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Enumeration of m-ary cacti
The purpose of this paper is to enumerate various classes of cyclically
colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is
motivated by the topological classification of complex polynomials having at
most m critical values, studied by Zvonkin and others. We obtain explicit
formulae for both labelled and unlabelled m-ary cacti, according to i) the
number of polygons, ii) the vertex-color distribution, iii) the vertex-degree
distribution of each color. We also enumerate m-ary cacti according to the
order of their automorphism group. Using a generalization of Otter's formula,
we express the species of m-ary cacti in terms of rooted and of pointed cacti.
A variant of the m-dimensional Lagrange inversion is then used to enumerate
these structures. The method of Liskovets for the enumeration of unrooted
planar maps can also be adapted to m-ary cacti.Comment: LaTeX2e, 28 pages, 9 figures (eps), 3 table
Hopf-algebraic deformations of products and Wick polynomials
We present an approach to classical definitions and results on
cumulant--moment relations and Wick polynomials based on extensive use of
convolution products of linear functionals on a coalgebra. This allows, in
particular, to understand the construction of Wick polynomials as the result of
a Hopf algebra deformation under the action of linear automorphisms induced by
multivariate moments associated to an arbitrary family of random variables with
moments of all orders. We also generalise the notion of deformed product in
order to discuss how these ideas appear in the recent theory of regularity
structures.Comment: Revised and improved Section 9. 29 page
A priori bounds for the φ⁴ equation in the full sub-critical regime
We derive a priori bounds for the Φ4 equation in the full sub-critical regime using Hairer’s theory of regularity structures. The equation is formally given by (∂t − )φ = −φ3 + ∞φ + ξ , where the term +∞ϕ represents infinite terms that have to be removed in a renormalisation procedure. We emulate fractional dimensions d<4 by adjusting the regularity of the noise term ξ, choosing ξ∈C−3+δ. Our main result states that if ϕ satisfies this equation on a space–time cylinder D=(0,1)×{|x|⩽1} , then away from the boundary ∂D the solution ϕ can be bounded in terms of a finite number of explicit polynomial expressions in ξ . The bound holds uniformly over all possible choices of boundary data for ϕ and thus relies crucially on the super-linear damping effect of the non-linear term −ϕ3 . A key part of our analysis consists of an appropriate re-formulation of the theory of regularity structures in the specific context of (*), which allows us to couple the small scale control one obtains from this theory with a suitable large scale argument. Along the way we make several new observations and simplifications: we reduce the number of objects required with respect to Hairer’s work. Instead of a model (Πx)x and the family of translation operators (Γx,y)x,y we work with just a single object (Xx,y) which acts on itself for translations, very much in the spirit of Gubinelli’s theory of branched rough paths. Furthermore, we show that in the specific context of (*) the hierarchy of continuity conditions which constitute Hairer’s definition of a modelled distribution can be reduced to the single continuity condition on the “coefficient on the constant level”
Sparse Probabilistic Models:Phase Transitions and Solutions via Spatial Coupling
This thesis is concerned with a number of novel uses of spatial coupling, applied to a class of probabilistic graphical models. These models include error correcting codes, random constraint satisfaction problems (CSPs) and statistical physics models called diluted spin systems. Spatial coupling is a technique initially developed for channel coding, which provides a recipe to transform a class of sparse linear codes into codes that are longer but more robust at high noise level. In fact it was observed that for coupled codes there are efficient algorithms whose decoding threshold is the optimal one, a phenomenon called threshold saturation. The main aim of this thesis is to explore alternative applications of spatial coupling. The goal is to study properties of uncoupled probabilistic models (not just coding) through the use of the corresponding spatially coupled models. The methods employed are ranging from the mathematically rigorous to the purely experimental. We first explore spatial coupling as a proof technique in the realm of LDPC codes. The Maxwell conjecture states that for arbitrary BMS channels the optimal (MAP) threshold of the standard (uncoupled) LDPC codes is given by the Maxwell construction. We are able to prove the Maxwell Conjecture for any smooth family of BMS channels by using (i) the fact that coupled codes perform optimally (which was already proved) and (ii) that the optimal thresholds of the coupled and uncoupled LDPC codes coincide. The method is used to derive two more results, namely the equality of GEXIT curves above the MAP threshold and the exactness of the averaged Bethe free energy formula derived under the RS cavity method from statistical physics. As a second application of spatial coupling we show how to derive novel bounds on the phase transitions in random constraint satisfaction problems, and possibly a general class of diluted spin systems. In the case of coloring, we investigate what happens to the dynamic and freezing thresholds. The phenomenon of threshold saturation is present also in this case, with the dynamic threshold moving to the condensation threshold, and the freezing moving to colorability. These claims are supported by experimental evidence, but in some cases, such as the saturation of the freezing threshold it is possible to make part of this claim more rigorous. This allows in principle for the computation of thresholds by use of spatial coupling. The proof is in the spirit of the potential method introduced by Kumar, Young, Macris and Pfister for LDPC codes. Finally, we explore how to find solutions in (uncoupled) probabilistic models. To test this, we start with a typical instance of random K-SAT (the base problem), and we build a spatially coupled structure that locally inherits the structure of the base problem. The goal is to run an algorithm for finding a suitable solution in the coupled structure and then "project" this solution to obtain a solution for the base problem. Experimental evidence points to the fact it is indeed possible to use a form of unit-clause propagation (UCP), a simple algorithm, to achieve this goal. This approach works also in regimes where the standard UCP fails on the base problem
Do algorithms and barriers for sparse principal component analysis extend to other structured settings?
We study a principal component analysis problem under the spiked Wishart
model in which the structure in the signal is captured by a class of
union-of-subspace models. This general class includes vanilla sparse PCA as
well as its variants with graph sparsity. With the goal of studying these
problems under a unified statistical and computational lens, we establish
fundamental limits that depend on the geometry of the problem instance, and
show that a natural projected power method exhibits local convergence to the
statistically near-optimal neighborhood of the solution. We complement these
results with end-to-end analyses of two important special cases given by path
and tree sparsity in a general basis, showing initialization methods and
matching evidence of computational hardness. Overall, our results indicate that
several of the phenomena observed for vanilla sparse PCA extend in a natural
fashion to its structured counterparts
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