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Balazs Rath

Publications and source records attributed to Balazs Rath.

18 recordsLinked to original sources
arXiv2021

On the threshold of spread-out contact process percolation

We study the stationary distribution of the (spread-out) $d$-dimensional contact process from the point of view of site percolation. In this process, vertices of $\mathbb{Z}^d$ can be healthy (state 0) or infected (state 1). With rate one infected sites recover, and with rate $λ$ they transmit the infection to some other vertex chosen uniformly within a ball of radius $R$. The classical phase transition result for this process states that there is a critical value $λ_c(R)$ such that the process has a non-trivial stationary distribution if and only if $λ> λ_c(R)$. In configurations sampled from this stationary distribution, we study nearest-neighbor site percolation of the set of infected sites; the associated percolation threshold is denoted $λ_p(R)$. We prove that $λ_p(R)$ converges to $1/(1-p_c)$ as $R$ tends to infinity, where $p_c$ is the threshold for Bernoulli site percolation on $\mathbb{Z}^d$. As a consequence, we prove that $λ_p(R) > λ_c(R)$ for large enough $R$, answering an open question of Liggett and Steif in the spread-out case.

math.PR↗
arXiv2020

Age evolution in the mean field forest fire model via multitype branching processes

We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős-Rényi edge-addition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a deterministic solution of an autonomous measure-valued differential equation. The main technique is to observe that, conditioned on the vertex ages, the graph is an inhomogeneous random graph in the sense of Bollobás, Janson and Riordan. We then study the evolution of the ages via the multitype Galton--Watson trees that arise as the limit in law of the component of an identified vertex at any fixed time. These trees are critical from the gelation time onwards.

math.PR↗
arXiv2020

One-dimensional Multi-particle DLA -- a PDE approach

In the present note we analyze the one-dimensional multi-particle diffusion limited aggregation (MDLA) model: the initial number of particles at each positive integer site has Poisson distribution with mean $μ$, independently of all other sites. Particles perform independent continuous-time simple symmetric random walks until they come to the site neighbouring the sticky aggregate, which initially consists only of the origin. If a particle tries to jump on the aggregate, the size of the aggregate increases by one, i.e., its rightmost point moves to the right by one unit. All particles which are present at the site neighbouring the aggregate at the moment when the aggregate advances, are immediately deleted. The $d-$dimensional MDLA model, $d \geq 1$, was introduced in 1980 by Rosenstock and Marquardt, and studied numerically by Voss (1984). The one dimensional model exhibits a phase transition for the rate of growth of the aggregate: it was proven by Kesten and Sidoravicius (2008) that if $μ<1$ then the size $R(t)$ of the aggregate grows like $\sqrt{t}$ and Sly (2016+) proved that if $μ>1$ then $R(t)$ grows linearly. In this note we give heuristic predictions about the constant $c(μ)$ for which $R(t)\approx c(μ)\sqrt{t}$ in the subcritical case $μ<1$, $R(t)\approx c(1+\varepsilon)t$ in the barely supercritical case $μ=1+\varepsilon$ and $R(t) \approx c(1) t^{2/3}$ in the critical case $μ=1$. We compare our predictions with new computer simulation results of the 1-dimensional multi-particle DLA model.

math-ph↗
arXiv2018

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.

math.PR↗
arXiv2018

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 Pittel for the size of the giant component in the supercritical graph.

math.PR↗
arXiv2016

Rigid representations of the multiplicative coalescent with linear deletion

We introduce the multiplicative coalescent with linear deletion, a continuous-time Markov process describing the evolution of a collection of blocks. Any two blocks of sizes $x$ and $y$ merge at rate $xy$, and any block of size $x$ is deleted with rate $λx$ (where $λ\geq 0$ is a fixed parameter). This process arises for example in connection with a variety of random-graph models which exhibit self-organised criticality. We focus on results describing states of the process in terms of collections of excursion lengths of random functions. For the case $λ=0$ (the coalescent without deletion) we revisit and generalise previous works by authors including Aldous, Limic, Armendariz, Uribe Bravo, and Broutin and Marckert, in which the coalescence is related to a "tilt" of a random function, which increases with time; for $λ>0$ we find a novel representation in which this tilt is complemented by a "shift" mechanism which produces the deletion of blocks. We describe and illustrate other representations which, like the tilt-and-shift representation, are "rigid", in the sense that the coalescent process is constructed as a projection of some process which has all of its randomness in its initial state. We explain some applications of these constructions to models including mean-field forest-fire and frozen-percolation processes.

math.PR↗
arXiv2016

Percolation on the stationary distributions of the voter model

The voter model on $\mathbb{Z}^d$ is a particle system that serves as a rough model for changes of opinions among social agents or, alternatively, competition between biological species occupying space. When $d \geq 3$, the set of (extremal) stationary distributions is a family of measures $μ_α$, for $α$ between 0 and 1. A configuration sampled from $μ_α$ is a strongly correlated field of 0's and 1's on $\mathbb{Z}^d$ in which the density of 1's is $α$. We consider such a configuration as a site percolation model on $\mathbb{Z}^d$. We prove that if $d \geq 5$, the probability of existence of an infinite percolation cluster of 1's exhibits a phase transition in $α$. If the voter model is allowed to have sufficiently spread-out interactions, we prove the same result for $d \geq 3$.

math.PR↗
arXiv2015

A short proof of the phase transition for the vacant set of random interlacements

The vacant set of random interlacements at level $u>0$, introduced in arXiv:0704.2560, is a percolation model on $\mathbb{Z}^d$, $d \geq 3$ which arises as the set of sites avoided by a Poissonian cloud of doubly infinite trajectories, where $u$ is a parameter controlling the density of the cloud. It was proved in arXiv:0704.2560 and arXiv:0808.3344 that for any $d \geq 3$ there exists a positive and finite threshold $u_*$ such that if $u u_*$ then the vacant set does not percolate. We give an elementary proof of these facts. Our method also gives simple upper and lower bounds on the value of $u_*$ for any $d \geq 3$.

math.PR↗
arXiv2014

On chemical distances and shape theorems in percolation models with long-range correlations

In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. We also obtain alternative proofs to the main results in arXiv:1111.3979. Finally, as a corollary, we obtain new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.

math.PR↗
arXiv2013

Local percolative properties of the vacant set of random interlacements with small intensity

Random interlacements at level u is a one parameter family of connected random subsets of Z^d, d>=3 introduced in arXiv:0704.2560. Its complement, the vacant set at level u, exhibits a non-trivial percolation phase transition in u, as shown in arXiv:0704.2560 and arXiv:0808.3344, and the infinite connected component, when it exists, is almost surely unique, see arXiv:0805.4106. In this paper we study local percolative properties of the vacant set of random interlacements at level u for all dimensions d>=3 and small intensity parameter u>0. We give a stretched exponential bound on the probability that a large (hyper)cube contains two distinct macroscopic components of the vacant set at level u. Our results imply that finite connected components of the vacant set at level u are unlikely to be large. These results were proved in arXiv:1002.4995 for d>=5. Our approach is different from that of arXiv:1002.4995 and works for all d>=3. One of the main ingredients in the proof is a certain conditional independence property of the random interlacements, which is interesting in its own right.

math.PR↗
arXiv2013

The effect of small quenched noise on connectivity properties of random interlacements

The random interlacements (at level u) is a one parameter family of random subsets of Z^d introduced by Sznitman in arXiv:0704.2560. The vacant set at level u is the complement of the random interlacement at level u. In this paper, we study the effect of small quenched noise on connectivity properties of the random interlacement and the vacant set. While the random interlacement induces a connected subgraph of Z^d for all levels u, the vacant set has a non-trivial phase transition in u, as shown in arXiv:0704.2560 and arXiv:0808.3344. For a positive epsilon, we allow each vertex of the random interlacement (referred to as occupied) to become vacant, and each vertex of the vacant set to become occupied with probability epsilon, independently of the randomness of the interlacement, and independently for different vertices. We prove that for any d>=3 and u>0, almost surely, the perturbed random interlacement percolates for small enough noise parameter epsilon. In fact, we prove the stronger statement that Bernoulli percolation on the random interlacement graph has a non-trivial phase transition in wide enough slabs. As a byproduct, we show that any electric network with i.i.d. positive resistances on the interlacement graph is transient, which strengthens our result in arXiv:1102.4758. As for the vacant set, we show that for any d>=3, there is still a non-trivial phase transition in u when the noise parameter epsilon is small enough, and we give explicit upper and lower bounds on the value of the critical threshold, when epsilon tends to 0.

math.PR↗
arXiv2012

Limits of permutation sequences

A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.

math.CO↗
arXiv2012

Time evolution of dense multigraph limits under edge-conservative preferential attachment dynamics

We define the edge reconnecting model, a random multigraph evolving in time. At each time step we change one endpoint of a uniformly chosen edge: the new endpoint is chosen by linear preferential attachment. We consider a sequence of edge reconnecting models where the sequence of initial multigraphs is convergent in a sense which is a natural generalization of the notion of convergence of dense graph sequences, defined by Lovasz and Szegedy in arXiv:math/0408173. We investigate how the limit object evolves under the edge reconnecting dynamics if we rescale time properly: we give the complete characterization of the time evolution of the limit object from its initial state up to the stationary state, which is described in the companion paper arXiv:1106.2058. In our proofs we use the theory of exchangeable arrays, queuing and diffusion processes. The number of parallel edges and the degrees evolve on different timescales and because of this the model exhibits subaging.

math.PR↗
arXiv2012

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 models.

math.PR↗
arXiv2010

Multigraph limits and exchangeability

The theory of limits of dense graph sequences was initiated by Lovasz and Szegedy. We give a possible generalization of this theory to multigraphs. Our proofs are based on the correspondence between dense graph limits and countable, exchangeable arrays of random variables observed by Diaconis and Janson. The main ingredient in the construction of the limit object is Aldous' representation theorem for exchangeable arrays.

math.PR↗
arXiv2009

Erdos-Renyi random graphs + forest fires = self-organized criticality

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system sticks to a permanent critical state.

math.PR↗
arXiv2009

Mean field frozen percolation

We define a modification of the Erdos-Renyi random graph process which can be regarded as the mean field frozen percolation process. We describe the behavior of the process using differential equations and investigate their solutions in order to show the self-organized critical and extremum properties of the critical frozen percolation model. We prove two limit theorems about the distribution of the size of the component of a typical frozen vertex.

math.PR↗