12 research outputs found
Weighted enumeration of spanning subgraphs with degree constraints
The Heilmann-Lieb Theorem on (univariate) matching polynomials states that
the polynomial has only real nonpositive zeros, in which
is the number of -edge matchings of a graph . There is a
stronger multivariate version of this theorem. We provide a general method by
which ``theorems of Heilmann-Lieb type'' can be proved for a wide variety of
polynomials attached to the graph . These polynomials are multivariate
generating functions for spanning subgraphs of with certain weights and
constraints imposed, and the theorems specify regions in which these
polynomials are nonvanishing. Such theorems have consequences for the absence
of phase transitions in certain probabilistic models for spanning subgraphs of
.Comment: complete re-write of arXiv:math/0412059 with some new result
Some applications of Wagner's weighted subgraph counting polynomial
We use Wagner's weighted subgraph counting polynomial to prove that the
partition function of the anti-ferromagnetic Ising model on line graphs is real
rooted and to prove that roots of the edge cover polynomial have length at most
. We moreover discuss how our results relate to efficient algorithms for
approximately computing evaluations of these polynomials
The Lee-Yang and P\'olya-Schur Programs. II. Theory of Stable Polynomials and Applications
In the first part of this series we characterized all linear operators on
spaces of multivariate polynomials preserving the property of being
non-vanishing in products of open circular domains. For such sets this
completes the multivariate generalization of the classification program
initiated by P\'olya-Schur for univariate real polynomials. We build on these
classification theorems to develop here a theory of multivariate stable
polynomials. Applications and examples show that this theory provides a natural
framework for dealing in a uniform way with Lee-Yang type problems in
statistical mechanics, combinatorics, and geometric function theory in one or
several variables. In particular, we answer a question of Hinkkanen on
multivariate apolarity.Comment: 32 page
Weighted enumeration of spanning subgraphs in locally treeâlike graphs
International audienceUsing the theory of negative association for measures and the notion of unimodularity for random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically treeâlike graphs. Specifically, the normalized logarithm of the associated partition function (free energy) is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a GaltonâWatson tree, the latter simplifies into a recursive distributional equation which can be solved explicitly. As an illustration, we provide a new asymptotic formula for the maximum size of a bâmatching in the ErdĆsâRĂ©nyi random graph with fixed average degree and diverging size, for any equation image. To the best of our knowledge, this is the first time that correlation inequalities and unimodularity are combined together to yield a general proof of uniqueness of Gibbs measures on infinite trees. We believe that a similar argument is applicable to other Gibbs measures than those over spanning subgraphs considered here
The Lee-Yang and P\'olya-Schur Programs. I. Linear Operators Preserving Stability
In 1952 Lee and Yang proposed the program of analyzing phase transitions in
terms of zeros of partition functions. Linear operators preserving
non-vanishing properties are essential in this program and various contexts in
complex analysis, probability theory, combinatorics, and matrix theory. We
characterize all linear operators on finite or infinite-dimensional spaces of
multivariate polynomials preserving the property of being non-vanishing
whenever the variables are in prescribed open circular domains. In particular,
this solves the higher dimensional counterpart of a long-standing
classification problem originating from classical works of Hermite, Laguerre,
Hurwitz and P\'olya-Schur on univariate polynomials with such properties.Comment: Final version, to appear in Inventiones Mathematicae; 27 pages, no
figures, LaTeX2
Topics in random graphs, combinatorial optimization, and statistical inference
The manuscript is made of three chapters presenting three differenttopics on which I worked with Ph.D. students. Each chapter can be read independently of the others andshould be relatively self-contained. Chapter 1 is a gentle introduction to the theory of random graphswith an emphasis on contagions on such networks. In Chapter 2, I explain the main ideas of the objectivemethod developed by Aldous and Steele applied to the spectral measure of random graphs and themonomer-dimer problem. This topic is dear to me and I hope that this chapter will convince the readerthat it is an exciting field of research. Chapter 3 deals with problems in high-dimensional statistics whichnow occupy a large proportion of my time. Unlike Chapters 1 and 2 which could be easily extended inlecture notes, I felt that the material in Chapter 3 was not ready for such a treatment. This field ofresearch is currently very active and I decided to present two of my recent contributions