A moment-generating formula for Erdős-Rényi component sizes

We derive a simple formula characterizing the distribution of the size of the connected component of a fixed vertex in the Erdős-Rényi random graph which allows us to give elementary proofs of some results of Federico, van der Hofstad, den Hollander and Hulshof as well as Janson and Luczak about the susceptibility in the subcritical graph and the central limit theorem of Barraez, Boucheron and De La Vega for the size of the giant component in the supercritical graph

Multigraph limit of the dense configuration model and the preferential attachment graph

The configuration model is the most natural model to generate a random multigraph with a given degree sequence. We use the notion of dense graph limits to characterize the special form of limit objects of convergent sequences of configuration models. We apply these results to calculate the limit object corresponding to the dense preferential attachment graph and the edge reconnecting model. Our main tools in doing so are (1) the relation between the theory of graph limits and that of partially exchangeable random arrays (2) an explicit construction of our random graphs that uses urn model

Percolation of worms

We introduce a new correlated percolation model on the $d$-dimensional lattice $\mathbb{Z}^d$ called the random length worms model. Assume given a probability distribution on the set of positive integers (the length distribution) and $v \in (0,\infty)$ (the intensity parameter). From each site of $\mathbb{Z}^d$ we start $\mathrm{POI}(v)$ independent simple random walks with this length distribution. We investigate the connectivity properties of the set $\mathcal{S}^v$ of sites visited by this cloud of random walks. It is easy to show that if the second moment of the length distribution is finite then $\mathcal{S}^v$ undergoes a percolation phase transition as $v$ varies. Our main contribution is a sufficient condition on the length distribution which guarantees that $\mathcal{S}^v$ percolates for all $v>0$ if $d \geq 5$. E.g., if the probability mass function of the length distribution is $m(\ell)= c \cdot \ln(\ln(\ell))^{\varepsilon}/ (\ell^3 \ln(\ell)) 1[\ell \geq \ell_0]$ for some $\ell_0>e^e$ and $\varepsilon>0$ then $\mathcal{S}^v$ percolates for all $v>0$. Note that the second moment of this length distribution is only "barely" infinite. In order to put our result in the context of earlier results about similar models (e.g., finitary random interlacements, loop percolation, Poisson Boolean model, ellipses percolation, etc.), we define a natural family of percolation models called the Poisson zoo and argue that the percolative behaviour of the random length worms model is quite close to being "extremal" in this family of models.Comment: 50 page

On the threshold of spread-out voter model percolation

In the $R$-spread out, $d$-dimensional voter model, each site $x$ of $\mathbb{Z}^d$ has state (or `opinion') 0 or 1 and, with rate 1, updates its opinion by copying that of some site $y$ chosen uniformly at random among all sites within distance $R$ from $x$. If $d \geq 3$, the set of (extremal) stationary measures of this model is given by a family $\mu_{\alpha, R}$, where $\alpha \in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0's and 1's in which the density of 1's is $\alpha$ and the correlation weakens as $R$ becomes larger. We study these configurations from the point of view of nearest neighbor site percolation on $\mathbb{Z}^d$, focusing on asymptotics as $R \to \infty$. In [Ráth, Valesin, AoP, 2017] we have shown that, if $R$ is large, there is a critical value $\alpha_c(R)$ such that there is percolation if $\alpha > \alpha_c(R)$ and no percolation if $\alpha < \alpha_c(R)$. Here we prove that, as $R \to \infty$, $\alpha_c(R)$ converges to the critical probability for Bernoulli site percolation on $\mathbb{Z}^d$. Our proof relies on a new upper bound on the joint occurrence of events under $\mu_{\alpha,R}$ which is of independent interest

Feller property of the multiplicative coalescent with linear deletion

We modify the definition of Aldous' multiplicative coalescent process and introduce the multiplicative coalescent with linear deletion (MCLD). A state of this process is a square-summable decreasing sequence of cluster sizes. Pairs of clusters merge with a rate equal to the product of their sizes and clusters are deleted with a rate linearly proportional to their size. We prove that the MCLD is a Feller process. This result is a key ingredient in the description of scaling limits of the evolution of component sizes of the mean field frozen percolation model and the so-called rigid representation of such scaling limits

Feller property of the multiplicative coalescent with linear deletion

We modify the definition of Aldous' multiplicative coalescent process and introduce the multiplicative coalescent with linear deletion (MCLD). A state of this process is a square-summable decreasing sequence of cluster sizes. Pairs of clusters merge with a rate equal to the product of their sizes and clusters are deleted with a rate linearly proportional to their size. We prove that the MCLD is a Feller process. This result is a key ingredient in the description of scaling limits of the evolution of component sizes of the mean field frozen percolation model and the so-called rigid representation of such scaling limits.Comment: 23 pages, 1 figur