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Simon Griffiths

Publications and source records attributed to Simon Griffiths.

33 records · Page 2Linked to original sources
arXiv2013

On explosions in heavy-tailed branching random walks

Consider a branching random walk on $\mathbb{R}$, with offspring distribution Z and nonnegative displacement distribution W. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution Z is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs (Z, W) for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur. We also investigate, under additional smoothness assumptions, the behavior of $M_n$, the position of the particle in generation n closest to the origin, when explosion does not occur (and hence $\lim_{n\rightarrow\infty}M_n=\infty$).

math.PR↗
arXiv2013

The Ramsey number of the clique and the hypercube

The Ramsey number r(K_s,Q_n) is the smallest positive integer N such that every red-blue colouring of the edges of the complete graph K_N on N vertices contains either a red n-dimensional hypercube, or a blue clique on s vertices. Answering a question of Burr and Erdős from 1983, and improving on recent results of Conlon, Fox, Lee and Sudakov, and of the current authors, we show that r(K_s,Q_n) = (s-1) (2^n - 1) + 1 for every s \in \N and every sufficiently large n \in \N.

math.CO↗
arXiv2013

Noise Sensitivity in Continuum Percolation

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c \ne 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p_c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.

math.PR↗
arXiv2013

On the Ramsey number of the triangle and the cube

The Ramsey number r(K_3,Q_n) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K_N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erdős conjectured that r(K_3,Q_n) = 2^{n+1} - 1 for every n \in \N, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K_3,Q_n) \le 7000 \cdot 2^n. Here we show that r(K_3,Q_n) = (1 + o(1)) 2^{n+1} as n \to \infty.

math.CO↗
arXiv2012

Invasion percolation on the Poisson-weighted infinite tree

We study invasion percolation on Aldous' Poisson-weighted infinite tree, and derive two distinct Markovian representations of the resulting process. One of these is the $σ\to\infty$ limit of a representation discovered by Angel et al. [Ann. Appl. Probab. 36 (2008) 420-466]. We also introduce an exploration process of a randomly weighted Poisson incipient infinite cluster. The dynamics of the new process are much more straightforward to describe than those of invasion percolation, but it turns out that the two processes have extremely similar behavior. Finally, we introduce two new "stationary" representations of the Poisson incipient infinite cluster as random graphs on $\mathbb {Z}$ which are, in particular, factors of a homogeneous Poisson point process on the upper half-plane $\mathbb {R}\times[0,\infty)$.

math.PR↗
arXiv2012

Tight inequalities among set hitting times in Markov chains

Given an irreducible discrete-time Markov chain on a finite state space, we consider the largest expected hitting time $T(α)$ of a set of stationary measure at least $α$ for $α\in(0,1)$. We obtain tight inequalities among the values of $T(α)$ for different choices of $α$. One consequence is that $T(α) \le T(1/2)/α$ for all $α< 1/2$. As a corollary we have that, if the chain is lazy in a certain sense as well as reversible, then $T(1/2)$ is equivalent to the chain's mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions $T(α)$ over the domain $α\in(0,1/2]$.

math.PR↗
arXiv2012

k-Sums in abelian groups

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.

math.CO↗
arXiv2012

(k+1)-sums versus k-sums

A $k$-sum of a set $A\subseteq \mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $k\wedge A$ for the set of $k$-sums of $A$ we prove that \[ \frac{|(k+1)\wedge A|}{|k\wedge A|}\, \le \, \frac{|A|-k}{k+1} \] whenever $|A|\ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.

math.NT↗
arXiv2011

Subset sums in abelian groups

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is best possible, and we obtain the stronger (exact best possible) bound in almost all cases. We prove similar results in the case |G| is even. Our proof requires us to extend a theorem of Olson on the number of subset sums of anti-symmetric subsets S from the case of Z_p to the case of a general finite abelian group. To do so, we adapt Olson's method using a generalisation of Vosper's Theorem proved by Hamidoune and Plagne.

math.CO↗
arXiv2011

The spectrum of random lifts

For a fixed d-regular graph H, a random n-lift is obtained by replacing each vertex v of H by a "fibre" containing n vertices, then placing a uniformly random matching between fibres corresponding to adjacent vertices of H. We show that with extremely high probability, all eigenvalues of the lift that are not eigenvalues of H, have order O(sqrt(d)). In particular, if H is Ramanujan then its n-lift is with high probability nearly Ramanujan. We also show that any exceptionally large eigenvalues of the n-lift that are not eigenvalues of H, are overwhelmingly likely to have been caused by a dense subgraph of size O(|E(H)|).

math.CO↗
arXiv2011

Quasi-random oriented graphs

We show that a number of conditions on oriented graphs, all of which are satisfied with high probability by randomly oriented graphs, are equivalent. These equivalences are similar to those given by Chung, Graham and Wilson in the case of unoriented graphs, and by Chung and Graham in the case of tournaments. Indeed, our main theorem extends to the case of a general underlying graph G the main result of Chung and Graham which corresponds to the case that G is complete. One interesting aspect of these results is that exactly two of the four orientations of a four-cycle can be used for a quasi-randomness condition, i.e., if the number of appearances they make in D is close to the expected number in a random orientation of the same underlying graph, then the same is true for every small oriented graph H

math.CO↗
arXiv2011

The chromatic thresholds of graphs

The chromatic threshold delta_chi(H) of a graph H is the infimum of d>0 such that there exists C=C(H,d) for which every H-free graph G with minimum degree at least d|G| satisfies chi(G) 2. We moreover characterise the graphs H with a given chromatic threshold, and thus determine delta_chi(H) for every graph H. This answers a question of Erdős and Simonovits [Discrete Math. 5 (1973), 323-334], and confirms a conjecture of Łuczak and Thomassé [preprint (2010), 18pp].

math.CO↗
arXiv2010

Subgraphs of weakly quasi-random oriented graphs

It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an active area of research, and is relatively well-understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs, and provide some results on the global behavior of the orientation of $D$. For the case that $H$ is an oriented four-cycle we prove: in every $H$-free oriented graph $D$, there is a pair $A,B\ssq V(D)$ such that $e(A,B)\ge e(D)^{2}/32|D|^{2}$ and $e(B,A)\le e(A,B)/2$. We give a random construction which shows that this bound on $e(A,B)$ is best possible (up to the constant). In addition, we prove a similar result for the case $H$ is an oriented six-cycle, and a more precise result in the case $D$ is dense and $H$ is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph $D$, and provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures.

math.CO↗
arXiv2008

Asymptotically tight bounds on subset sums

For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further improve the bound to the asymptotically best possible result |A|<= (2+o(1))sqrt{n}.

math.NT↗