361 research outputs found

### Diameter of the stochastic mean-field model of distance

We consider the complete graph \cK_n on $n$ vertices with exponential mean
$n$ edge lengths. Writing $C_{ij}$ for the weight of the smallest-weight path
between vertex $i,j\in [n]$, Janson showed that $\max_{i,j\in [n]}
C_{ij}/\log{n}$ converges in probability to 3. We extend this result by showing
that $\max_{i,j\in [n]} C_{ij} - 3\log{n}$ converges in distribution to a
limiting random variable that can be identified via a maximization procedure on
a limiting infinite random structure. Interestingly, this limiting random
variable has also appeared as the weak limit of the re-centered graph diameter
of the barely supercritical Erd\H{o}s-R\'enyi random graph in work by Riordan
and Wormald.Comment: 27 page

### Infinite canonical super-Brownian motion and scaling limits

We construct a measure valued Markov process which we call infinite canonical
super-Brownian motion, and which corresponds to the canonical measure of
super-Brownian motion conditioned on non-extinction. Infinite canonical
super-Brownian motion is a natural candidate for the scaling limit of various
random branching objects on $\Z^d$ when these objects are (a) critical; (b)
mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the
spread-out oriented percolation incipient infinite cluster above 4 dimensions
and of incipient infinite branching random walk in any dimension. We conjecture
that it also arises as the scaling limit in various other models above the
upper-critical dimension, such as the incipient infinite lattice tree above 8
dimensions, the incipient infinite cluster for unoriented percolation, uniform
spanning trees above 4 dimensions, and invasion percolation above 6 dimensions.
This paper also serves as a survey of recent results linking super-Brownian to
scaling limits in statistical mechanics.Comment: 34 page

### Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. The edge
probabilities are moderated by vertex weights, and are such that the degree of
vertex i is close in distribution to a Poisson random variable with parameter
w_i, where w_i denotes the weight of vertex i. We choose the weights such that
the weight of a uniformly chosen vertex converges in distribution to a limiting
random variable W, in which case the proportion of vertices with degree k is
close to the probability that a Poisson random variable with random parameter W
takes the value k. We pay special attention to the power-law case, in which
P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3,
a property which is then inherited by the asymptotic degree distribution.
We show that the critical behavior depends sensitively on the properties of
the asymptotic degree distribution moderated by the asymptotic weight
distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and
some \tau>4 and c>0, the largest critical connected component in a graph of
size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When,
instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4)
and c>0, the largest critical connected component is of the much smaller order
n^{(\tau-2)/(\tau-1)}.Comment: 26 page

### The winner takes it all

We study competing first passage percolation on graphs generated by the
configuration model. At time 0, vertex 1 and vertex 2 are infected with the
type 1 and the type 2 infection, respectively, and an uninfected vertex then
becomes type 1 (2) infected at rate $\lambda_1$ ($\lambda_2$) times the number
of edges connecting it to a type 1 (2) infected neighbor. Our main result is
that, if the degree distribution is a power-law with exponent $\tau\in(2,3)$,
then, as the number of vertices tends to infinity and with high probability,
one of the infection types will occupy all but a finite number of vertices.
Furthermore, which one of the infections wins is random and both infections
have a positive probability of winning regardless of the values of $\lambda_1$
and $\lambda_2$. The picture is similar with multiple starting points for the
infections

### Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions

We consider the critical spread-out contact process in Z^d with d\ge1, whose
infection range is denoted by L\ge1. In this paper, we investigate the r-point
function \tau_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that,
for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time
t_i by the individual at the origin o\in Z^d at time 0. Together with the
results of the 2-point function in [van der Hofstad and Sakai, Electron. J.
Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially
rely, we prove that the r-point functions converge to the moment measures of
the canonical measure of super-Brownian motion above the upper-critical
dimension 4. We also prove partial results for d\le4 in a local mean-field
setting.Comment: 75 pages, 12 figure

### From trees to graphs: collapsing continuous-time branching processes

Continuous-time branching processes (CTBPs) are powerful tools in random
graph theory, but are not appropriate to describe real-world networks, since
they produce trees rather than (multi)graphs. In this paper we analyze
collapsed branching processes (CBPs), obtained by a collapsing procedure on
CTBPs, in order to define multigraphs where vertices have fixed out-degree
$m\geq 2$. A key example consists of preferential attachment models (PAMs), as
well as generalized PAMs where vertices are chosen according to their degree
and age. We identify the degree distribution of CBPs, showing that it is
closely related to the limiting distribution of the CTBP before collapsing. In
particular, this is the first time that CTBPs are used to investigate the
degree distribution of PAMs beyond the tree setting.Comment: 18 pages, 3 figure

### Gaussian scaling for the critical spread-out contact process above the upper critical dimension

We consider the critical spread-out contact process in \Zd with $d\geq 1$,
whose infection range is denoted by $L\geq1$. The two-point function
$\tau_t(x)$ is the probability that x\in\Zd is infected at time $t$ by the
infected individual located at the origin o\in\Zd at time 0. We prove
Gaussian behavior for the two-point function with $L\geq L_0$ for some finite
$L_0=L_0(d)$ for $d>4$. When $d\leq 4$, we also perform a local mean-field
limit to obtain Gaussian behaviour for $\tau_{tT}$ with $t>0$ fixed and $T\to
\infty$ when the infection range depends on $T$ such that $L_T=LT^b$ for any
$b>(4-d)/2d$.
The proof is based on the lace expansion and an adaptation of the inductive
approach applied to the discretized contact process. We prove the existence of
several critical exponents and show that they take on mean-field values. The
results in this paper provide crucial ingredients to prove convergence of the
finite-dimensional distributions for the contact process towards the canonical
measure of super-Brownian motion, which we defer to a sequel of this paper.Comment: 50 pages, 5 figure

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