Equality of bond percolation critical exponents for pairs of dual lattices

For a certain class of two-dimensional lattices, lattice-dual pairs are shown to have the same bond percolation critical exponents. A computational proof is given for the martini lattice and its dual to illustrate the method. The result is generalized to a class of lattices that allows the equality of bond percolation critical exponents for lattice-dual pairs to be concluded without performing the computations. The proof uses the substitution method, which involves stochastic ordering of probability measures on partially ordered sets. As a consequence, there is an infinite collection of infinite sets of two-dimensional lattices, such that all lattices in a set have the same critical exponents.Comment: 10 pages, 7 figure

The Poset of Hypergraph Quasirandomness

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's.Comment: 43 pages, 1 figur

Embeddings and Ramsey numbers of sparse k-uniform hypergraphs

Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In previous work, we proved the same result for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random' hypergraphs.Comment: 24 pages, 2 figures. To appear in Combinatoric

Phase transitions in Delaunay Potts models

We establish phase transitions for classes of continuum Delaunay multi-type particle systems (continuum Potts models) with infinite range repulsive interaction between particles of different type. In one class of the Delaunay Potts models studied the repulsive interaction is a triangle (multi-body) interaction whereas in the second class the interaction is between pairs (edges) of the Delaunay graph. The result for the edge model is an extension of finite range results in \cite{BBD04} for the Delaunay graph and in \cite{GH96} for continuum Potts models to an infinite range repulsion decaying with the edge length. This is a proof of an old conjecture of Lebowitz and Lieb. The repulsive triangle interactions have infinite range as well and depend on the underlying geometry and thus are a first step towards studying phase transitions for geometry-dependent multi-body systems. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transitions manifest themselves in the percolation of the corresponding random-cluster model. Our proofs rely on recent studies \cite{DDG12} of Gibbs measures for geometry-dependent interactions

On the Public Communication Needed to Achieve SK Capacity in the Multiterminal Source Model

The focus of this paper is on the public communication required for generating a maximal-rate secret key (SK) within the multiterminal source model of Csisz{\'a}r and Narayan. Building on the prior work of Tyagi for the two-terminal scenario, we derive a lower bound on the communication complexity, $R_{\text{SK}}$, defined to be the minimum rate of public communication needed to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by $R_{\text{CO}}$, is an upper bound on $R_{\text{SK}}$. For the class of pairwise independent network (PIN) models defined on uniform hypergraphs, we show that a certain "Type $\mathcal{S}$" condition, which is verifiable in polynomial time, guarantees that our lower bound on $R_{\text{SK}}$ meets the $R_{\text{CO}}$ upper bound. Thus, PIN models satisfying our condition are $R_{\text{SK}}$-maximal, meaning that the upper bound $R_{\text{SK}} \le R_{\text{CO}}$ holds with equality. This allows us to explicitly evaluate $R_{\text{SK}}$ for such PIN models. We also give several examples of PIN models that satisfy our Type $\mathcal S$ condition. Finally, we prove that for an arbitrary multiterminal source model, a stricter version of our Type $\mathcal S$ condition implies that communication from \emph{all} terminals ("omnivocality") is needed for establishing a SK of maximum rate. For three-terminal source models, the converse is also true: omnivocality is needed for generating a maximal-rate SK only if the strict Type $\mathcal S$ condition is satisfied. Counterexamples exist that show that the converse is not true in general for source models with four or more terminals.Comment: Submitted to the IEEE Transactions on Information Theory. arXiv admin note: text overlap with arXiv:1504.0062

Grassmann Integral Representation for Spanning Hyperforests

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.Comment: 50 pages, it uses some latex macros. Accepted for publication on J. Phys.