567 research outputs found
Matchings on infinite graphs
Elek and Lippner (2010) showed that the convergence of a sequence of
bounded-degree graphs implies the existence of a limit for the proportion of
vertices covered by a maximum matching. We provide a characterization of the
limiting parameter via a local recursion defined directly on the limit of the
graph sequence. Interestingly, the recursion may admit multiple solutions,
implying non-trivial long-range dependencies between the covered vertices. We
overcome this lack of correlation decay by introducing a perturbative parameter
(temperature), which we let progressively go to zero. This allows us to
uniquely identify the correct solution. In the important case where the graph
limit is a unimodular Galton-Watson tree, the recursion simplifies into a
distributional equation that can be solved explicitly, leading to a new
asymptotic formula that considerably extends the well-known one by Karp and
Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page
Sublinear-Time Algorithms for Monomer-Dimer Systems on Bounded Degree Graphs
For a graph , let be the partition function of the
monomer-dimer system defined by , where is the
number of matchings of size in . We consider graphs of bounded degree
and develop a sublinear-time algorithm for estimating at an
arbitrary value within additive error with high
probability. The query complexity of our algorithm does not depend on the size
of and is polynomial in , and we also provide a lower bound
quadratic in for this problem. This is the first analysis of a
sublinear-time approximation algorithm for a # P-complete problem. Our
approach is based on the correlation decay of the Gibbs distribution associated
with . We show that our algorithm approximates the probability
for a vertex to be covered by a matching, sampled according to this Gibbs
distribution, in a near-optimal sublinear time. We extend our results to
approximate the average size and the entropy of such a matching within an
additive error with high probability, where again the query complexity is
polynomial in and the lower bound is quadratic in .
Our algorithms are simple to implement and of practical use when dealing with
massive datasets. Our results extend to other systems where the correlation
decay is known to hold as for the independent set problem up to the critical
activity
Spectrum of non-Hermitian heavy tailed random matrices
Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}|
is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our
main result is a heavy tailed counterpart of Girko's circular law. Namely,
under some additional smoothness assumptions on the law of X_{jk}, we prove
that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability
measure mu_alpha on C depending only on alpha such that with probability one,
the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1}
(X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our
approach combines Aldous & Steele's objective method with Girko's Hermitization
using logarithmic potentials. The underlying limiting object is defined on a
bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive
relations on the tree provide some properties of mu_alpha. In contrast with the
Hermitian case, we find that mu_alpha is not heavy tailed.Comment: Expanded version of a paper published in Communications in
Mathematical Physics 307, 513-560 (2011
Parentage of grapevine rootstock âFercalâ finally elucidated
Using a set of 20 microsatellite markers, âB.C. n°1Bâ (mother) and â31 Richterâ (father) were demonstrated to be the true parents of âFercalâ rootstock. â333 Ecole de Montpellierâ was definitively excluded as the putative father. âB.C. n°1Aâ and âB.C. n°1Bâ were shown to be distinct genotypes. âUgni blancâ, and not âColombardâ, was discovered to be the Vitis vinifera father of âB.C. n°1Bâ.
Spectral density of random graphs with topological constraints
The spectral density of random graphs with topological constraints is
analysed using the replica method. We consider graph ensembles featuring
generalised degree-degree correlations, as well as those with a community
structure. In each case an exact solution is found for the spectral density in
the form of consistency equations depending on the statistical properties of
the graph ensemble in question. We highlight the effect of these topological
constraints on the resulting spectral density.Comment: 24 pages, 6 figure
Stability Analysis of Frame Slotted Aloha Protocol
Frame Slotted Aloha (FSA) protocol has been widely applied in Radio Frequency
Identification (RFID) systems as the de facto standard in tag identification.
However, very limited work has been done on the stability of FSA despite its
fundamental importance both on the theoretical characterisation of FSA
performance and its effective operation in practical systems. In order to
bridge this gap, we devote this paper to investigating the stability properties
of FSA by focusing on two physical layer models of practical importance, the
models with single packet reception and multipacket reception capabilities.
Technically, we model the FSA system backlog as a Markov chain with its states
being backlog size at the beginning of each frame. The objective is to analyze
the ergodicity of the Markov chain and demonstrate its properties in different
regions, particularly the instability region. By employing drift analysis, we
obtain the closed-form conditions for the stability of FSA and show that the
stability region is maximised when the frame length equals the backlog size in
the single packet reception model and when the ratio of the backlog size to
frame length equals in order of magnitude the maximum multipacket reception
capacity in the multipacket reception model. Furthermore, to characterise
system behavior in the instability region, we mathematically demonstrate the
existence of transience of the backlog Markov chain.Comment: 14 pages, submitted to IEEE Transaction on Information Theor
CUTOFF AT THE " ENTROPIC TIME " FOR SPARSE MARKOV CHAINS
International audienceWe study convergence to equilibrium for a large class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix P the mass is essentially concentrated on few entries. Moreover, the random environment is such that rows of P are independent and such that the entries are exchangeable within each row. This includes various models of random walks on sparse random directed graphs. The models are generally non reversible and the equilibrium distribution is itself unknown. In this general setting we establish the cutoff phenomenon for the total variation distance to equilibrium, with mixing time given by the logarithm of the number of states times the inverse of the average row entropy of P. As an application, we consider the case where the rows of P are i.i.d. random vectors in the domain of attraction of a Poisson-Dirichlet law with index α â (0, 1). Our main results are based on a detailed analysis of the weight of the trajectory followed by the walker. This approach offers an interpretation of cutoff as an instance of the concentration of measure phenomenon
A real quaternion spherical ensemble of random matrices
One can identify a tripartite classification of random matrix ensembles into
geometrical universality classes corresponding to the plane, the sphere and the
anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the
anti-sphere with truncations of unitary matrices. This paper focusses on an
ensemble corresponding to the sphere: matrices of the form \bY= \bA^{-1} \bB,
where \bA and \bB are independent matrices with iid standard
Gaussian real quaternion entries. By applying techniques similar to those used
for the analogous complex and real spherical ensembles, the eigenvalue jpdf and
correlation functions are calculated. This completes the exploration of
spherical matrices using the traditional Dyson indices .
We find that the eigenvalue density (after stereographic projection onto the
sphere) has a depletion of eigenvalues along a ring corresponding to the real
axis, with reflective symmetry about this ring. However, in the limit of large
matrix dimension, this eigenvalue density approaches that of the corresponding
complex ensemble, a density which is uniform on the sphere. This result is in
keeping with the spherical law (analogous to the circular law for iid
matrices), which states that for matrices having the spherical structure \bY=
\bA^{-1} \bB, where \bA and \bB are independent, iid matrices the
(stereographically projected) eigenvalue density tends to uniformity on the
sphere.Comment: 25 pages, 3 figures. Added another citation in version
Spectral Theory of Sparse Non-Hermitian Random Matrices
Sparse non-Hermitian random matrices arise in the study of disordered
physical systems with asymmetric local interactions, and have applications
ranging from neural networks to ecosystem dynamics. The spectral
characteristics of these matrices provide crucial information on system
stability and susceptibility, however, their study is greatly complicated by
the twin challenges of a lack of symmetry and a sparse interaction structure.
In this review we provide a concise and systematic introduction to the main
tools and results in this field. We show how the spectra of sparse
non-Hermitian matrices can be computed via an analogy with infinite dimensional
operators obeying certain recursion relations. With reference to three
illustrative examples --- adjacency matrices of regular oriented graphs,
adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency
matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the
use of these methods to obtain both analytic and numerical results for the
spectrum, the spectral distribution, the location of outlier eigenvalues, and
the statistical properties of eigenvectors.Comment: 60 pages, 10 figure
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