352 research outputs found
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
Local statistics of lattice dimers
We show how to compute the probability of any given local configuration in a
random tiling of the plane with dominos. That is, we explicitly compute the
measures of cylinder sets for the measure of maximal entropy on the space
of tilings of the plane with dominos.
We construct a measure on the set of lozenge tilings of the plane, show
that its entropy is the topological entropy, and compute explicitly the
-measures of cylinder sets.
As applications of these results, we prove that the translation action is
strongly mixing for and , and compute the rate of convergence to
mixing (the correlation between distant events). For the measure we
compute the variance of the height function.Comment: 27 pages, 6 figure
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
The asymptotic determinant of the discrete Laplacian
We compute the asymptotic determinant of the discrete Laplacian on a
simply-connected rectilinear region in R^2. As an application of this result,
we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.Comment: 36 pages, 4 figures, to appear in Acta Mathematic
Polynomial-time perfect matchings in dense hypergraphs
Let be a -graph on vertices, with minimum codegree at least for some fixed . In this paper we construct a polynomial-time
algorithm which finds either a perfect matching in or a certificate that
none exists. This essentially solves a problem of Karpi\'nski, Ruci\'nski and
Szyma\'nska; Szyma\'nska previously showed that this problem is NP-hard for a
minimum codegree of . Our algorithm relies on a theoretical result of
independent interest, in which we characterise any such hypergraph with no
perfect matching using a family of lattice-based constructions.Comment: 64 pages. Update includes minor revisions. To appear in Advances in
Mathematic
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