26 research outputs found
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 . 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 is
-pseudorandom if for all disjoint and with and we have
. We prove that for all there is an
such that an -pseudorandom graph on
vertices with minimum degree at least contains the square of a
Hamilton cycle. In particular, this implies that -graphs with
contain the square of a Hamilton cycle, and thus
a triangle factor if is a multiple of . 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 BollobÂŽas in 1968. Given a graph H and a set G â E(Kn) we initially âinfectâ all edges in G and then, in consecutive steps, we infect every e â 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 BollobÂŽas: what is the maximum time the process can run before it stabilizes? It is an easy observation that for r = 3 this maximum is âlog2 (n â 1)â. However, a new phenomenon occurs for r = 4 when, as we show, the maximum time of the process is n â 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 2âΔr time steps for some Δr that tends to 0 as r â â
A note on Linnikâs theorem on quadratic non-residues
We present a short and purely combinatorial proof of Linnikâs theorem: for any Δ>0 there exists a constant CΔ such that for any N, there are at most CΔ primes pâ€N such that the least positive quadratic non-residue modulo p exceeds NΔ
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