Polynomial sequences for bond percolation critical thresholds

In this paper, I compute the inhomogeneous (multi-probability) bond critical surfaces for the (4,6,12) and (3^4,6) lattices using the linearity approximation described in (Scullard and Ziff, J. Stat. Mech. P03021), implemented as a branching process of lattices. I find the estimates for the bond percolation thresholds, p_c(4,6,12)=0.69377849... and p_c(3^4,6)=0.43437077..., compared with Parviainen's numerical results of p_c \approx 0.69373383 and p_c \approx 0.43430621 . These deviations are of the order 10^{-5}, as is standard for this method, although they are outside Parviainen's typical standard error of 10^{-7}. Deriving thresholds in this way for a given lattice leads to a polynomial with integer coefficients, the root in [0,1] of which gives the estimate for the bond threshold. I show how the method can be refined, leading to a sequence of higher order polynomials making predictions that likely converge to the exact answer. Finally, I discuss how this fact hints that for certain graphs, such as the kagome lattice, the exact bond threshold may not be the root of any polynomial with integer coefficients.Comment: submitted to Journal of Statistical Mechanic

The structure of typical clusters in large sparse random configurations

The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution of concentrations of particles in a medium where particles coalesce pairwise as time passes and each particle can only perform a given number of aggregations. Under appropriate assumptions, the concentrations of particles converge as time tends to infinity to some measure which bears a striking resemblance with the distribution of the total population of a Galton-Watson process started from two ancestors. Roughly speaking, the configuration model is a stochastic construction which aims at producing a typical graph on a set of vertices with pre-described degrees. Specifically, one attaches to each vertex a certain number of stubs, and then join pairwise the stubs uniformly at random to create edges between vertices. In this work, we use the configuration model as the stochastic counterpart of Smoluchowski's coagulation equations with limited aggregations. We establish a hydrodynamical type limit theorem for the empirical measure of the shapes of clusters in the configuration model when the number of vertices tends to $\infty$. The limit is given in terms of the distribution of a Galton-Watson process started with two ancestors

Powers of Hamilton cycles in pseudorandom graphs

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph $G$ is $(\varepsilon,p,k,\ell)$-pseudorandom if for all disjoint $X$ and $Y\subset V(G)$ with $|X|\ge\varepsilon p^kn$ and $|Y|\ge\varepsilon p^\ell n$ we have $e(X,Y)=(1\pm\varepsilon)p|X||Y|$. We prove that for all $\beta>0$ there is an $\varepsilon>0$ such that an $(\varepsilon,p,1,2)$-pseudorandom graph on $n$ vertices with minimum degree at least $\beta pn$ contains the square of a Hamilton cycle. In particular, this implies that $(n,d,\lambda)$-graphs with $\lambda\ll d^{5/2 }n^{-3/2}$ contain the square of a Hamilton cycle, and thus a triangle factor if $n$ is a multiple of $3$. This improves on a result of Krivelevich, Sudakov and Szab\'o [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403--426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.Comment: 30 pages, 1 figur

Asymptotic normality of the size of the giant component in a random hypergraph

Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-Löf, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph G(n,p) above the phase transition. Here we show that the same method applies to the analogous model of random k -uniform hypergraphs, establishing asymptotic normality throughout the (sparse) supercritical regime. Previously, asymptotic normality was known only towards the two ends of this regime. © 2012 Wiley Periodicals, Inc

The consequences of Zipf's law for syntax and symbolic reference.

Although many species possess rudimentary communication systems, humans seem to be unique with regard to making use of syntax and symbolic reference. Recent approaches to the evolution of language formalize why syntax is selectively advantageous compared with isolated signal communication systems, but do not explain how signals naturally combine. Even more recent work has shown that if a communication system maximizes communicative efficiency while minimizing the cost of communication, or if a communication system constrains ambiguity in a non-trivial way while a certain entropy is maximized, signal frequencies will be distributed according to Zipf's law. Here we show that such communication principles give rise not only to signals that have many traits in common with the linking words in real human languages, but also to a rudimentary sort of syntax and symbolic reference

On the maximum running time in graph bootstrap percolation

Graph bootstrap percolation is a simple cellular automaton introduced by Bollob´as in 1968. Given a graph H and a set G ⊆ E(Kn) we initially “infect” all edges in G and then, in consecutive steps, we infect every e ∈ Kn that completes a new infected copy of H in Kn. We say that G percolates if eventually every edge in Kn is infected. The extremal question about the size of the smallest percolating sets when H = Kr was answered independently by Alon, Kalai and Frankl. Here we consider a different question raised more recently by Bollob´as: what is the maximum time the process can run before it stabilizes? It is an easy observation that for r = 3 this maximum is ⌈log2 (n − 1)⌉. However, a new phenomenon occurs for r = 4 when, as we show, the maximum time of the process is n − 3. For r > 5 the behaviour of the dynamics is even more complex, which we demonstrate by showing that the Kr-bootstrap process can run for at least n 2−εr time steps for some εr that tends to 0 as r → ∞

A note on Linnik’s theorem on quadratic non-residues

We present a short and purely combinatorial proof of Linnik’s theorem: for any ε>0 there exists a constant Cε such that for any N, there are at most Cε primes p≤N such that the least positive quadratic non-residue modulo p exceeds Nε

Degree distribution of the FKP network model.

Power laws, in particular power-law degree distributions, have been observed in real-world networks in a very wide range of contexts, including social networks, biological networks, and artificial networks such as the physical internet or abstract world wide web. Recently, these observations have triggered much work attempting to explain the power laws in terms of new 'scale-free' random graph models. So far, perhaps the most effective mechanism for explaining power laws is the combination of growth and preferential attachment. In [A. Fabrikant, E. Koutsoupias, C.H. Papadimitriou, Heuristically optimized trade-offs: A new paradigm for power laws in the internet ICALP 2002, in: LNCS, vol. 2380, pp. 110-122], Fabrikant, Koutsoupias and Papadimitriou propose a new 'paradigm' for explaining power laws, based on trade-offs between competing objectives. They also introduce a new, simple and elegant parametrized model for the internet, and prove some kind of power-law bound on the degree sequence for a wide range of scalings of the trade-off parameter. \ Here we shall show that this model does not have the usual kind of power-law degree distribution observed in the real world: for the most interesting range of the parameter, neither the bulk of the nodes, nor the few highest degree nodes have degrees following a power law. We shall show that almost all nodes have degree 1, and that there is a strong bunching of degrees near the maximum. (c) 2007 Elsevier B.V All rights reserved