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Register a new userProbabilistic existence of regular combinatorial structures
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Ron Peled
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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We show the existence of regular combinatorial objects which previously were not known to exist. Specifically, for a wide range of the underlying parameters, we show the existence of non-trivial orthogonal arrays, t-designs, and t-wise permutations. In all cases, the sizes of the objects are optimal up to polynomial overhead. The proof of existence is probabilistic. We show that a randomly chosen structure has the required properties with positive yet tiny probability. Our method allows also to give rather precise estimates on the number of objects of a given size and this is applied to count the number of orthogonal arrays, t-designs and regular hypergraphs. The main technical ingredient is a special local central limit theorem for suitable lattice random walks with finitely many steps.
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Let the random variable $X, :=, e(mathcal{H}[B])$ count the number of edges of a hypergraph $mathcal{H}$ induced by a random $m$ element subset $B$ of its vertex set. Focussing on the case that $mathcal{H}$ satisfies some regularity condition we prove bounds on the probability that $X$ is far from its mean. It is possible to apply these results to discrete structures such as the set of $k$-term arithmetic progressions in the cyclic group $mathbb{Z}_N$. Furthermore, we show that our main theorem is essentially best possible and we deduce results for the case $Bsim B_p$ is generated by including each vertex independently with probability $p$.
For any positive integers $n, s, t, l$ such that $n geq 10$, $s, t geq 2$, $l geq 1$ and $n geq s+t+l$, a new infinite family of regular 3-hypertopes with type $(2^s, 2^t, 2^l)$ and automorphism group of order $2^n$ is constructed.
We derive combinatorial necessary conditions for discrete-time quantum walks defined by regular mixed graphs to be periodic. If the quantum walk is periodic, all the eigenvalues of the time evolution matrices must be algebraic integers. Focusing on this, we explore which ring the coefficients of the characteristic polynomials should belong to. On the other hand, the coefficients of the characteristic polynomials of $eta$-Hermitian adjacency matrices have combinatorial implications. From these, we can find combinatorial implications in the coefficients of the characteristic polynomials of the time evolution matrices, and thus derive combinatorial necessary conditions for mixed graphs to be periodic. For example, if a $k$-regular mixed graph with $n$ vertices is periodic, then $2n/k$ must be an integer. As an application of this work, we determine periodicity of mixed complete graphs and mixed graphs with a prime number of vertices.
A (q,k,t)-design matrix is an m x n matrix whose pattern of zeros/non-zeros satisfies the following design-like condition: each row has at most q non-zeros, each column has at least k non-zeros and the supports of every two columns intersect in at most t rows. We prove that the rank of any (q,k,t)-design matrix over a field of characteristic zero (or sufficiently large finite characteristic) is at least n - (qtn/2k)^2 . Using this result we derive the following applications: (1) Impossibility results for 2-query LCCs over the complex numbers: A 2-query locally correctable code (LCC) is an error correcting code in which every codeword coordinate can be recovered, probabilistically, by reading at most two other code positions. Such codes have numerous applications and constructions (with exponential encoding length) are known over finite fields of small characteristic. We show that infinite families of such linear 2-query LCCs do not exist over the complex numbers. (2) Generalization of results in combinatorial geometry: We prove a quantitative analog of the Sylvester-Gallai theorem: Let $v_1,...,v_m$ be a set of points in $C^d$ such that for every $i in [m]$ there exists at least $delta m$ values of $j in [m]$ such that the line through $v_i,v_j$ contains a third point in the set. We show that the dimension of ${v_1,...,v_m }$ is at most $O(1/delta^2)$. Our results generalize to the high dimensional case (replacing lines with planes, etc.) and to the case where the points are colored (as in the Motzkin-Rabin Theorem).
The Ising antiferromagnet is an important statistical physics model with close connections to the {sc Max Cut} problem. Combining spatial mixing arguments with the method of moments and the interpolation method, we pinpoint the replica symmetry breaking phase transition predicted by physicists. Additionally, we rigorously establish upper bounds on the {sc Max Cut} of random regular graphs predicted by Zdeborova and Boettcher [Journal of Statistical Mechanics 2010]. As an application we prove that the information-theoretic threshold of the disassortative stochastic block model on random regular graphs coincides with the Kesten-Stigum bound.
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