66 research outputs found
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
On product, generic and random generic quantum satisfiability
We report a cluster of results on k-QSAT, the problem of quantum
satisfiability for k-qubit projectors which generalizes classical
satisfiability with k-bit clauses to the quantum setting. First we define the
NP-complete problem of product satisfiability and give a geometrical criterion
for deciding when a QSAT interaction graph is product satisfiable with positive
probability. We show that the same criterion suffices to establish quantum
satisfiability for all projectors. Second, we apply these results to the random
graph ensemble with generic projectors and obtain improved lower bounds on the
location of the SAT--unSAT transition. Third, we present numerical results on
random, generic satisfiability which provide estimates for the location of the
transition for k=3 and k=4 and mild evidence for the existence of a phase which
is satisfiable by entangled states alone.Comment: 9 pages, 5 figures, 1 table. Updated to more closely match published
version. New proof in appendi
Graph Zeta Function and Gauge Theories
Along the recently trodden path of studying certain number theoretic
properties of gauge theories, especially supersymmetric theories whose vacuum
manifolds are non-trivial, we investigate Ihara's Graph Zeta Function for large
classes of quiver theories and periodic tilings by bi-partite graphs. In
particular, we examine issues such as the spectra of the adjacency and whether
the gauge theory satisfies the strong and weak versions of the graph
theoretical analogue of the Riemann Hypothesis.Comment: 35 pages, 7 Figure
Crossings and nestings in colored set partitions
Chen, Deng, Du, Stanley, and Yan introduced the notion of -crossings and
-nestings for set partitions, and proved that the sizes of the largest
-crossings and -nestings in the partitions of an -set possess a
symmetric joint distribution. This work considers a generalization of these
results to set partitions whose arcs are labeled by an -element set (which
we call \emph{-colored set partitions}). In this context, a -crossing or
-nesting is a sequence of arcs, all with the same color, which form a
-crossing or -nesting in the usual sense. After showing that the sizes of
the largest crossings and nestings in colored set partitions likewise have a
symmetric joint distribution, we consider several related enumeration problems.
We prove that -colored set partitions with no crossing arcs of the same
color are in bijection with certain paths in \NN^r, generalizing the
correspondence between noncrossing (uncolored) set partitions and 2-Motzkin
paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a
proof that the sequence counting noncrossing 2-colored set partitions is
P-recursive. We also discuss how our methods extend to several variations of
colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further
revised, additional section adde
Recommended from our members
Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combinatorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Strings from Feynman Graph counting : without large N
A well-known connection between n strings winding around a circle and
permutations of n objects plays a fundamental role in the string theory of
large N two dimensional Yang Mills theory and elsewhere in topological and
physical string theories. Basic questions in the enumeration of Feynman graphs
can be expressed elegantly in terms of permutation groups. We show that these
permutation techniques for Feynman graph enumeration, along with the Burnside
counting lemma, lead to equalities between counting problems of Feynman graphs
in scalar field theories and Quantum Electrodynamics with the counting of
amplitudes in a string theory with torus or cylinder target space. This string
theory arises in the large N expansion of two dimensional Yang Mills and is
closely related to lattice gauge theory with S_n gauge group. We collect and
extend results on generating functions for Feynman graph counting, which
connect directly with the string picture. We propose that the connection
between string combinatorics and permutations has implications for QFT-string
dualities, beyond the framework of large N gauge theory.Comment: 55 pages + 10 pages Appendices, 23 figures ; version 2 - typos
correcte
Extremal Graph Theory and Dimension Theory for Partial Orders
This dissertation analyses several problems in extremal combinatorics.In Part I, we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph H and an integer t, what is the minimum number of coloured edges in a t-edge-coloured graph G on n vertices such that G does not contain a rainbow copy of H, but adding a new edge to G in any colour creates a rainbow copy of H? We determine the growth rates of these numbers for almost all graphs H and all t e(H).In Part II, we study dimension theory for finite partial orders. In Chapter 1, we introduce and define the concepts we use in the succeeding chapters.In Chapter 2, we determine the dimension of the divisibility order on [n] up to a factor of (log log n).In Chapter 3, we answer a question of Kim, Martin, Masak, Shull, Smith, Uzzell, and Wang on the local bipartite covering numbers of difference graphs.In Chapter 4, we prove some bounds on the local dimension of any pair of layers of the Boolean lattice. In particular, we show that the local dimension of the first and middle layers is asymptotically n / log n.In Chapter 5, we introduce a new poset parameter called local t-dimension. We also discuss the fractional variants of this and other dimension-like parameters.All of Part I is joint work with Antnio Giro of the University of Cambridge and Kamil Popielarz of the University of Memphis.Chapter 2 of Part II is joint work with Victor Souza of IMPA (Instituto de Matemtica Pura e Aplicada, Rio de Janeiro).Chapter 3 of Part II is joint work with Antnio Giro
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