pinecone/core-2020-05-10-deduplication huggingface dataset:
On randomly generated intersecting hypergraphs II

Let $c$ be a positive constant. Suppose that $r=o(n^{5/12})$ and the members
of $\binom{[n]}{r}$ are chosen sequentially at random to form an intersecting
hypergraph $\mathcal{H}$. We show that whp $\mathcal{H}$ consists of a simple
hypergraph $\mathcal{S}$ of size $\Theta(r/n^{1/3})$, a distinguished vertex
$v$ and all $r$-sets which contain $v$ and meet every edge of $\mathcal{S}$.
This is a continuation of the study of such random intersecting systems started
in [Electron. J. Combin, (2003) R29] where the case $r=O(n^{1/3})$ was
considered. To obtain the stated result we continue to investigate this
question in the range $\omega(n^{1/3})\le r \le o(n^{5/12})$.Comment: 20 page
The phase transition in inhomogeneous random graphs

We introduce a very general model of an inhomogenous random graph with
independence between the edges, which scales so that the number of edges is
linear in the number of vertices. This scaling corresponds to the p=c/n scaling
for G(n,p) used to study the phase transition; also, it seems to be a property
of many large real-world graphs. Our model includes as special cases many
models previously studied.
  We show that under one very weak assumption (that the expected number of
edges is `what it should be'), many properties of the model can be determined,
in particular the critical point of the phase transition, and the size of the
giant component above the transition. We do this by relating our random graphs
to branching processes, which are much easier to analyze.
  We also consider other properties of the model, showing, for example, that
when there is a giant component, it is `stable': for a typical random graph, no
matter how we add or delete o(n) edges, the size of the giant component does
not change by more than o(n).Comment: 135 pages; revised and expanded slightly. To appear in Random
  Structures and Algorithm
Large-deviations/thermodynamic approach to percolation on the complete
  graph

We present a large-deviations/thermodynamic approach to the classic problem
of percolation on the complete graph. Specifically, we determine the
large-deviation rate function for the probability that the giant component
occupies a fixed fraction of the graph while all other components are
``small.'' One consequence is an immediate derivation of the ``cavity'' formula
for the fraction of vertices in the giant component. As a by-product of our
analysis we compute the large-deviation rate functions for the probability of
the event that the random graph is connected, the event that it contains no
cycles and the event that it contains only ``small'' components.Comment: 16 pages, 1 figure; revised version accomodating literature remarks
  of the referee
Spread-out percolation in R^d

Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of
intensity 1. Given parameters $r$ and $p$, join each pair of points of $X$
within distance $r$ independently with probability $p$. This is the simplest
case of a `spread-out' percolation model studied by Penrose, who showed that,
as $r\to\infty$, the average degree of the corresponding random graph at the
percolation threshold tends to 1, i.e., the percolation threshold and the
threshold for criticality of the naturally associated branching process
approach one another. Here we show that this result follows immediately from of
a general result of the authors on inhomogeneous random graphs.Comment: 9 pages. Title changed. Minor changes to text, including updated
  references to [3]. To appear in Random Structures and Algorithm
Percolation on dual lattices with k-fold symmetry

Zhang found a simple, elegant argument deducing the non-existence of an
infinite open cluster in certain lattice percolation models (for example, p=1/2
bond percolation on the square lattice) from general results on the uniqueness
of an infinite open cluster when it exists; this argument requires some
symmetry. Here we show that a simple modification of Zhang's argument requires
only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for
percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's
argument, our extension applies in many contexts; in particular, it enables us
to answer a question of Grimmett concerning the anisotropic random cluster
model on the triangular lattice.Comment: 11 pages, 1 figure. Revised with applications added; to appear in
  Random Structures and Algorithm
The simple random walk and max-degree walk on a directed graph

We show bounds on total variation and $L^{\infty}$ mixing times, spectral gap
and magnitudes of the complex valued eigenvalues of a general (non-reversible
non-lazy) Markov chain with a minor expansion property. This leads to the first
known bounds for the non-lazy simple and max-degree walks on a (directed)
graph, and even in the lazy case they are the first bounds of the optimal
order. In particular, it is found that within a factor of two or four, the
worst case of each of these mixing time and eigenvalue quantities is a walk on
a cycle with clockwise drift
Words Maps and Spectra of Random Graph Lifts

We begin with a new analysis of formal words. Let w be a formal word in
letters g_1,...,g_k. The word map associated with w maps the permutations
s_1,...,s_k in S_n to the permutation obtained by replacing for each i, every
occurrence of g_i in w by s_i. We investigate the random variable X_w^n that
counts the fixed points in this permutation when the s_i are selected uniformly
at random. A major ingredient of our work is a new categorization of words
which considerably extends the dichotomy of primitive vs. imprimitive words. We
establish some results and make a few conjectures about the relation between
the expectation E(X_w^n) and this new categorization.
  This analysis contributes deeply to our study of the spectra of random lifts
of graphs. Let G be a connected graph, and let the infinite tree T be its
universal cover space. If L and R are the spectral radii of G and T
respectively, then, as shown by J. Friedman, for almost every n-lift H of G,
all "new" eigenvalues of H are < O(L^(1/2)R^(1/2)). We improve this upper bound
to O(L^(1/3)R^(2/3)), and our aforementioned conjectures suggest a possible
approach to proving an upper bound of O(R). This is a generalization of the
problem of bounding the second eigenvalue in a random 2d-regular graph.
  As an aside, we obtain a new conceptual and relatively simple proof of a
theorem of A. Nica, which determines, for every fixed w, the limit distribution
(as n \to \infty) of X_w^n. A surprising aspect of this theorem is that the
answer depends only on the largest integer d so that w=u^d for some word u.Comment: 46 pages, 10 figures. For associated pdf file, see
  http://www.cs.huji.ac.il/~nati/PAPERS/puder.pdf . Accepted for publication in
  Random Structures & Algorithms. Main change upon former version: We refine
  the main theorem and state the exact bound our method yields for the spectra
  of random lifts of graph (the former version had a mere O()
Deterministic Random Walks on Regular Trees

Jim Propp's rotor router model is a deterministic analogue of a random walk
on a graph. Instead of distributing chips randomly, each vertex serves its
neighbors in a fixed order.
  Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable
similarity of both models. If an (almost) arbitrary population of chips is
placed on the vertices of a grid $\Z^d$ and does a simultaneous walk in the
Propp model, then at all times and on each vertex, the number of chips on this
vertex deviates from the expected number the random walk would have gotten
there by at most a constant. This constant is independent of the starting
configuration and the order in which each vertex serves its neighbors.
  This result raises the question if all graphs do have this property. With
quite some effort, we are now able to answer this question negatively. For the
graph being an infinite $k$-ary tree ($k \ge 3$), we show that for any
deviation $D$ there is an initial configuration of chips such that after
running the Propp model for a certain time there is a vertex with at least $D$
more chips than expected in the random walk model. However, to achieve a
deviation of $D$ it is necessary that at least $\exp(\Omega(D^2))$ vertices
contribute by being occupied by a number of chips not divisible by $k$ at a
certain time.Comment: 15 pages, to appear in Random Structures and Algorithm
Weighted Random Popular Matchings

For a set A of n applicants and a set I of m items, we consider a problem of
computing a matching of applicants to items, i.e., a function M mapping A to I;
here we assume that each applicant $x \in A$ provides a preference list on
items in I. We say that an applicant $x \in A$ prefers an item p than an item q
if p is located at a higher position than q in its preference list, and we say
that x prefers a matching M over a matching M' if x prefers M(x) over M'(x).
For a given matching problem A, I, and preference lists, we say that M is more
popular than M' if the number of applicants preferring M over M' is larger than
that of applicants preferring M' over M, and M is called a popular matching if
there is no other matching that is more popular than M. Here we consider the
situation that A is partitioned into $A_{1},A_{2},...,A_{k}$, and that each
$A_{i}$ is assigned a weight $w_{i}>0$ such that w_{1}>w_{2}>...>w_{k}>0$. For
such a matching problem, we say that M is more popular than M' if the total
weight of applicants preferring M over M' is larger than that of applicants
preferring M' over M, and we call M an k-weighted popular matching if there is
no other matching that is more popular than M. In this paper, we analyze the
2-weighted matching problem, and we show that (lower bound) if
$m/n^{4/3}=o(1)$, then a random instance of the 2-weighted matching problem
with $w_{1} \geq 2w_{2}$ has a 2-weighted popular matching with probability
o(1); and (upper bound) if $n^{4/3}/m = o(1)$, then a random instance of the
2-weighted matching problem with $w_{1} \geq 2w_{2}$ has a 2-weighted popular
matching with probability 1-o(1).Comment: 13 pages, 2 figure
Dependent Random Choice

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them.National Science Foundation (U.S.). Graduate Research Fellowship ProgramPrinceton University (Centennial Fellowship)National Science Foundation (U.S.). (CAREER award DMS-0812005)U.S.-Israel Binational Science Foundation (BSF
On covering by translates of a set

In this paper we study the minimal number of translates of an arbitrary
subset $S$ of a group $G$ needed to cover the group, and related notions of the
efficiency of such coverings. We focus mainly on finite subsets in discrete
groups, reviewing the classical results in this area, and generalizing them to
a much broader context. For example, we show that while the worst-case
efficiency when $S$ has $k$ elements is of order $1/\log k$, for $k$ fixed and
$n$ large, almost every $k$-subset of any given $n$-element group covers $G$
with close to optimal efficiency.Comment: 41 pages; minor corrections; to appear in Random Structures and
  Algorithm
Graph diameter in long-range percolation

We study the asymptotic growth of the diameter of a graph obtained by adding
sparse "long" edges to a square box in $\Z^d$. We focus on the cases when an
edge between $x$ and $y$ is added with probability decaying with the Euclidean
distance as $|x-y|^{-s+o(1)}$ when $|x-y|\to\infty$. For $s\in(d,2d)$ we show
that the graph diameter for the graph reduced to a box of side $L$ scales like
$(\log L)^{\Delta+o(1)}$ where $\Delta^{-1}:=\log_2(2d/s)$. In particular, the
diameter grows about as fast as the typical graph distance between two vertices
at distance $L$. We also show that a ball of radius $r$ in the intrinsic metric
on the (infinite) graph will roughly coincide with a ball of radius
$\exp\{r^{1/\Delta+o(1)}\}$ in the Euclidean metric.Comment: 17 pages, extends the results of arXiv:math.PR/0304418 to graph
  diameter, substantially revised and corrected, added a result on volume
  growth asymptoti
Generalization of a theorem of Erdős and Rényi on Sidon sequences

10 páginas.Erdős and Rényi claimed and Vu proved that for all h ≥ 2 and for all ϵ > 0, there exists g = gh(ϵ) and a sequence of integers A such that the number of ordered representations of any number as a sum of h elements of A is bounded by g, and such that |A ∩ [1,x]| ≫ x1/h-ϵ.\ud
\ud
We give two new proofs of this result. The first one consists of an explicit construction of such a sequence. The second one is probabilistic and shows the existence of such a g that satisfies gh(ϵ) ≪ ϵ−1, improving the bound gh(ϵ) ≪ ϵ−h+1 obtained by Vu. Finally we use the “alteration method” to get a better bound for g3(ϵ), obtaining a more precise estimate for the growth of B3[g] sequencesSupported by MICINN (MTM 2008-03880); ERC–AdG (228005); Hungarian National Foundation for Scientific\ud
Research (OTKA) (K 61908, K 72731).Peer reviewe
Hitting time results for Maker-Breaker games

We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property $\mP$. We focus on three natural
properties for Maker's graph, namely being $k$-vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the $k$-vertex connectivity game
exactly at the time the random graph process first reaches minimum degree $2k$;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree $2$; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree $4$. The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page
The distribution of height and diameter in random non-plane binary trees

This study is dedicated to precise distributional analyses of the height of
non-plane unlabelled binary trees ("Otter trees"), when trees of a given size
are taken with equal likelihood. The height of a rooted tree of size $n$ is
proved to admit a limiting theta distribution, both in a central and local
sense, as well as obey moderate as well as large deviations estimates. The
approximations obtained for height also yield the limiting distribution of the
diameter of unrooted trees. The proofs rely on a precise analysis, in the
complex plane and near singularities, of generating functions associated with
trees of bounded height
Tur\'annical hypergraphs

This paper is motivated by the question of how global and dense restriction
sets in results from extremal combinatorics can be replaced by less global and
sparser ones. The result we consider here as an example is Turan's theorem,
which deals with graphs G=([n],E) such that no member of the restriction set
consisting of all r-tuples on [n] induces a copy of K_r.
  Firstly, we examine what happens when this restriction set is replaced just
by all r-tuples touching a given m-element set. That is, we determine the
maximal number of edges in an n-vertex such that no K_r hits a given vertex
set.
  Secondly, we consider sparse random restriction sets. An r-uniform hypergraph
R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if
for any graph G on [n] with more edges than the Turan number ex(n,K_r)
(respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G.
We determine the thresholds for random r-uniform hypergraphs to be Turannical
and to epsilon-Turannical.
  Thirdly, we transfer this result to sparse random graphs, using techniques
recently developed by Schacht [Extremal results for random discrete structures]
to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random
graphs.Comment: 33 pages, minor improvements thanks to two referee
Characterizing optimal sampling of binary contingency tables via the configuration model

A binary contingency table is an m × n array of binary entries with row sums r = (r1,..., rm) and column sums c = (c1,..., cn). The configuration model generates a contingency table by considering ri tokens of type 1 for each row i and cj tokens of type 2 for each column j, and then taking a uniformly random pairing between type-1 and type-2 tokens. We give a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as goes to ∞. Our finding shows surprising differences from recent results for binary symmetric contingency tables
Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k -tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so-called graph-based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph-based distribution, while they are exponentially generic in the word-based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group.Peer ReviewedPostprint (author’s final draft
The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on  n  labeled vertices with prescribed degrees  D  = ( d  1 ,…, d   n  ). For a wide class of  tame  degree sequences  D  we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as  n  grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence  D  is well described by a certain  maximum entropy matrix  computed from  D . We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 201
A new approach to nonrepetitive sequences

A sequence is nonrepetitive if it does not contain two adjacent identical
blocks. The remarkable construction of Thue asserts that 3 symbols are enough
to build an arbitrarily long nonrepetitive sequence. It is still not settled
whether the following extension holds: for every sequence of 3-element sets
$L_1,..., L_n$ there exists a nonrepetitive sequence $s_1, ..., s_n$ with
$s_i\in L_i$. Applying the probabilistic method one can prove that this is true
for sufficiently large sets $L_i$. We present an elementary proof that sets of
size 4 suffice (confirming the best known bound). The argument is a simple
counting with Catalan numbers involved. Our approach is inspired by a new
algorithmic proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos and its
interpretations by Fortnow and Tao. The presented method has further
applications to nonrepetitive games and nonrepetitive colorings of graphs.Comment: 5 pages, no figures.arXiv admin note: substantial text overlap with
  arXiv:1103.381
On Extractors and Exposure-Resilient Functions for Sublogarithmic Entropy

We study resilient functions and exposure-resilient functions in the low-entropy regime. A resilient function (a.k.a. deterministic extractor for oblivious bit-fixing sources) maps any distribution on n -bit strings in which k bits are uniformly random and the rest are fixed into an output distribution that is close to uniform. With exposure-resilient functions, all the input bits are random, but we ask that the output be close to uniform conditioned on any subset of n - k input bits. In this paper, we focus on the case that k is sublogarithmic in n.
We simplify and improve an explicit construction of resilient functions for k sublogarithmic in n due to Kamp and Zuckerman (SICOMP 2006), achieving error exponentially small in k rather than polynomially small in k. Our main result is that when k is sublogarithmic in n, the short output length of this construction (O(log k) output bits) is optimal for extractors computable by a large class of space-bounded streaming algorithms.
Next, we show that a random function is a resilient function with high probability if and only if k is superlogarithmic in n, suggesting that our main result may apply more generally. In contrast, we show that a random function is a static (resp. adaptive) exposure-resilient function with high probability even if k is as small as a constant (resp. loglog n). No explicit exposure-resilient functions achieving these parameters are known.Engineering and Applied SciencesMathematic
Invertibility of symmetric random matrices

We study    n  ×  n    symmetric random matrices  H , possibly discrete, with iid above‐diagonal entries. We show that  H  is singular with probability at most    exp  (  −   n  c   )   , and    |  |   H   −  1    |  |  =  O  (   n   )   . Furthermore, the spectrum of  H  is delocalized on the optimal scale    o  (   n   −  1  /  2    )   . These results improve upon a polynomial singularity bound due to Costello, Tao and Vu, and they generalize, up to constant factors, results of Tao and Vu, and Erdös, Schlein and Yau.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 135‐182, 201
When does the K_4-free process stop?

The K_4-free process starts with the empty graph on n vertices and at each
step adds a new edge chosen uniformly at random from all remaining edges that
do not complete a copy of K_4. Let G be the random maximal K_4-free graph
obtained at the end of the process. We show that for some positive constant C,
with high probability as $n \to \infty$, the maximum degree in G is at most $C
n^{3/5}\sqrt[5]{\log n}$. This resolves a conjecture of Bohman and Keevash for
the K_4-free process and improves on previous bounds obtained by Bollob\'as and
Riordan and by Osthus and Taraz. Combined with results of Bohman and Keevash
this shows that with high probability G has $\Theta(n^{8/5}\sqrt[5]{\log n})$
edges and is `nearly regular', i.e., every vertex has degree
$\Theta(n^{3/5}\sqrt[5]{\log n})$. This answers a question of Erd\H{o}s, Suen
and Winkler for the K_4-free process. We furthermore deduce an additional
structural property: we show that whp the independence number of G is at least
$\Omega(n^{2/5}(\log n)^{4/5}/\log \log n)$, which matches an upper bound
obtained by Bohman up to a factor of $\Theta(\log \log n)$. Our analysis of the
K_4-free process also yields a new result in Ramsey theory: for a special case
of a well-studied function introduced by Erd\H{o}s and Rogers we slightly
improve the best known upper bound.Comment: 39 pages, 3 figures. Minor edits. To appear in Random Structures and
  Algorithm