2,443 research outputs found
Asymptotic Behavior of Partition Functions with Graph Laplacian
We introduce the matrix sums that represent a discrete analog of the matrix
models with quartic potential. The probability space is given by the set of all
simple n-vertex graphs with the Gibbs weight determined by the graph Laplacian.
We study the large-n limit of the free energy per site and show that it is
determined by the number of connected acyclic diagrams on the set of two-valent
vertices.Comment: 18 pages, 3 figures; misprints corrected, minor improvements of the
text, one reference adde
On Connected Diagrams and Cumulants of Erdos-Renyi Matrix Models
Regarding the adjacency matrices of n-vertex graphs and related graph
Laplacian, we introduce two families of discrete matrix models constructed both
with the help of the Erdos-Renyi ensemble of random graphs. Corresponding
matrix sums represent the characteristic functions of the average number of
walks and closed walks over the random graph. These sums can be considered as
discrete analogs of the matrix integrals of random matrix theory.
We study the diagram structure of the cumulant expansions of logarithms of
these matrix sums and analyze the limiting expressions in the cases of constant
and vanishing edge probabilities as n tends to infinity.Comment: 34 pages, 8 figure
The densest subgraph problem in sparse random graphs
We determine the asymptotic behavior of the maximum subgraph density of large
random graphs with a prescribed degree sequence. The result applies in
particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of
Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398-1414]. Our proof consists in
extending the notion of balanced loads from finite graphs to their local weak
limits, using unimodularity. This is a new illustration of the objective method
described by Aldous and Steele [In Probability on Discrete Structures (2004)
1-72 Springer].Comment: Published at http://dx.doi.org/10.1214/14-AAP1091 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Exchangeable pairs, switchings, and random regular graphs
We consider the distribution of cycle counts in a random regular graph, which
is closely linked to the graph's spectral properties. We broaden the asymptotic
regime in which the cycle counts are known to be approximately Poisson, and we
give an explicit bound in total variation distance for the approximation. Using
this result, we calculate limiting distributions of linear eigenvalue
functionals for random regular graphs.
Previous results on the distribution of cycle counts by McKay, Wormald, and
Wysocka (2004) used the method of switchings, a combinatorial technique for
asymptotic enumeration. Our proof uses Stein's method of exchangeable pairs and
demonstrates an interesting connection between the two techniques.Comment: Very minor changes; 23 page
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