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Balázs Ráth

Publications and source records attributed to Balázs Ráth.

17 recordsLinked to original sources
arXiv2026

Law of large numbers for activated random walk on villages

We consider activated random walk (ARW), an interacting particle system and prototypical model of self-organized criticality in a setting which combines mean-field behavior with the geometry of an arbitrary graph, which we call the village model of ARW, or VARW for short. VARW is obtained from a fixed graph by replacing each vertex with a 'village' that consists of n replicas of that vertex. We focus on VARW where particles walk according to a strictly sub-stochastic transition kernel on a finite underlying graph, so mass is sometimes lost (which guarantees that the system eventually stabilizes almost surely). Under a subcriticality assumption on the initial state we prove a law of large numbers as n goes to infinity for the resulting stable configuration of particles and the odometer of the process, to a limit which is uniquely characterized by a system of non-linear equations.

math.PR↗
arXiv2026

Critical time of the almost 2-regular random degree constrained process

We study the phase transition of the random degree constrained process (RDCP), a time-evolving random graph model introduced by Ruciński and Wormald that generalizes the random $d$-process to the non-regular setting: each vertex of the complete graph $K_n$ has its pre-assigned degree constraint (i.e., a number from the set $\{2,\dots,Δ\}$), we attempt to add the edges one-by-one in a uniform random order, but a new edge is added only if it does not violate the degree constraints at its end-vertices. Warnke and Wormald identified the critical time of the RDCP when the giant component emerges as $n \to \infty$. Ráth, Szőke and Warnke identified the local weak limit of the RDCP and gave an alternative characterization of the critical time in terms of the principal eigenvalue of the branching operator of the multi-type branching process that arises as the local limit object. In the current paper we use this spectral characterization to study the critical time of the RDCP in the almost 2-regular case, i.e., when the degree constraint of most of the vertices is equal to 2. In this case the giant component emerges quite late, and our main result provides the precise asymptotics of the critical time as the model approaches 2-regularity. Interestingly, our formula asymptotically matches the well-known Molloy-Reed formula, despite the fact that Molloy, Surya and Warnke proved that the final graph of the RDCP is not contiguous to the configuration model with the same degree sequence.

math.PR↗
arXiv2025

Local limit of the random degree constrained process

In this paper we show that the random degree constrained process (a time-evolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover, identify the limit object as a multi-type branching process, by combining coupling arguments with the analysis of a certain recursive tree process. Using a spectral characterization, we also give an asymptotic expansion of the critical time when the giant component emerges in the so-called random $d$-process, resolving a problem of Warnke and Wormald for large $d$.

math.PR↗
arXiv2025

Asymptotics for the percolation threshold of finitary random interlacements in four and higher dimensions

We establish sharp asymptotic bounds for the critical intensity of the Finitary Random Interlacements (FRI) model in four and higher dimensions with general trajectory length distributions. Our proof reveals that the construction of near-critical FRI clusters in four and higher dimensions is essentially analogous to a Galton-Watson process, whose expected number of offspring corresponds to the capacity of a random walk killed at the given length.

math.PR↗
arXiv2025

Permuton limit of a generalization of the Mallows and $k$-card-minimum models

We introduce and study a new random permutation model that generalizes the $k$-card minimum model defined by Travers and the Mallows model. We calculate the permuton limit of such a sequence of random permutations. As a corollary, we deduce the law of large numbers for pattern densities. Moreover, we prove a universality result about the band structure of the limiting permuton, confirming a conjecture of Travers about the $k$-card minimum model. More specifically, we show that if a certain model parameter goes to infinity then the appropriately scaled restriction of the permuton measure to a line that intersects the diagonal perpendicularly converges weakly to the logistic distribution.

math.PR↗
arXiv2024

Concentration and mean field approximation results for Markov processes on large networks

We study Markov processes on weighted directed hypergraphs where the state of at most one vertex can change at a time. Our setting is general enough to include simplicial epidemic processes, processes on multilayered networks or even the dynamics of the edges of a graph. Our results are twofold. Firstly, we prove concentration bounds for the number of vertices in a certain state under mild assumptions. Our results imply that even the empirical averages of subpopulations of diverging but possibly sublinear size are well concentrated around their mean. In the case of undirected weighted graphs, we completely characterize when said averages concentrate around their expected value. Secondly, we prove (under assumptions which are tight in some significant cases) upper bounds for the error of the N-Intertwined Mean Field Approximation (NIMFA). In particular, for symmetric unweighted graphs, the error has the same order of magnitude as the reciprocal of the average degree, improving the previously known state of the art bound of the inverse square root of the average degree.

math.PR↗
arXiv2024

Path Percolation in Quantum Communication Networks

In a quantum communication network, links represent entanglement between qubits located at different nodes. Even if two nodes are not directly linked by shared entanglement, communication channels can be established between them via quantum routing protocols. However, in contrast to classical communication networks, each communication event removes all participating links along the communication path, disrupting the quantum network. Here, we propose a simple model, where randomly selected pairs of nodes communicate through shortest paths. Each time such a path is used, all participating links are eliminated, leading to a correlated percolation process that we call ``path percolation.'' We study path percolation both numerically and analytically and present the phase diagram of the steady states as a function of the rate at which new links are being added to the quantum communication network. As a key result, the steady state is found to be independent from the initial network topology when new link are added randomly between disconnected components. We close by discussing extensions of path percolation and their potential applications.

quant-ph↗
arXiv2022

Color-avoiding percolation in edge-colored Erdős-Rényi graphs

We study a variant of the color-avoiding percolation model introduced by Krause et al., namely we investigate the color-avoiding bond percolation setup on (not necessarily properly) edge-colored Erdős-Rényi random graphs. We say that two vertices are color-avoiding connected in an edge-colored graph if after the removal of the edges of any color, they are in the same component in the remaining graph. The color-avoiding connected components of an edge-colored graph are maximal sets of vertices such that any two of them are color-avoiding connected. We consider the fraction of vertices contained in color-avoiding connected components of a given size as well as the fraction of vertices contained in the giant color-avoiding connected component. Under some mild assumptions on the color-densities, we prove that these quantities converge and the limits can be expressed in terms of probabilities associated to edge-colored branching process trees. We provide explicit formulas for the limit of the normalized size of the giant color-avoiding component, and in the two-colored case we also provide explicit formulas for the limit of the fraction of vertices contained in color-avoiding connected components of a given size.

math.PR↗
arXiv2022

A phase transition between endogeny and nonendogeny

The Marked Binary Branching Tree (MBBT) is the family tree of a rate one binary branching process, on which points have been generated according to a rate one Poisson point process, with i.i.d. uniformly distributed activation times assigned to the points. In frozen percolation on the MBBT, initially, all points are closed, but as time progresses points can become either frozen or open. Points become open at their activation times provided they have not become frozen before. Open points connect the parts of the tree below and above it and one says that a point percolates if the tree above it is infinite. We consider a version of frozen percolation on the MBBT in which at times of the form $θ^n$, all points that percolate are frozen. The limiting model for $θ\to 1$, in which points freeze as soon as they percolate, has been studied before by Ráth, Swart, and Terpai. We extend their results by showing that there exists a $0<θ^\ast<1$ such that the model is endogenous for $θ\leqθ^\ast$ but not for $θ>θ^\ast$. This means that for $θ\leq θ^\ast$, frozen percolation is a.s. determined by the MBBT but for $θ>θ^\ast$ one needs additional randomness to describe it.

math.PR↗
arXiv2022

Random interlacement is a factor of i.i.d

The random interlacement point process (introduced by Sznitman, generalized by Teixeira) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph $G$. We show that the random interlacement point process on any transient transitive graph $G$ is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced by Popov and Teixeira) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if $G$ is non-unimodular.

math.PR↗
arXiv2022

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.

math.PR↗
arXiv2020

Frozen percolation on the binary tree is nonendogenous

In frozen percolation, i.i.d. uniformly distributed activation times are assigned to the edges of a graph. At its assigned time, an edge opens provided neither of its endvertices is part of an infinite open cluster; in the opposite case, it freezes. Aldous (2000) showed that such a process can be constructed on the infinite 3-regular tree and asked whether the event that a given edge freezes is a measurable function of the activation times assigned to all edges. We give a negative answer to this question, or, using an equivalent formulation and terminology introduced by Aldous and Bandyopadhyay (2005), we show that the recursive tree process associated with frozen percolation on the oriented binary tree is nonendogenous. An essential role in our proofs is played by a frozen percolation process on a continuous-time binary Galton Watson tree that has nice scale invariant properties.

math.PR↗
arXiv2017

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 $μ_{α, R}$, where $α\in [0,1]$. Configurations sampled from this measure are polynomially correlated fields of 0's and 1's in which the density of 1's is $α$ 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 \cite{RV15}, we have shown that, if $R$ is large, there is a critical value $α_c(R)$ such that there is percolation if $α> α_c(R)$ and no percolation if $α< α_c(R)$. Here we prove that, as $R \to \infty$, $α_c(R)$ 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 $μ_{α,R}$ which is of independent interest.

math.PR↗
arXiv2012

Connectivity properties of random interlacement and intersection of random walks

We consider the interlacement Poisson point process on the space of doubly-infinite Z^d-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The set of vertices and edges visited by at least one of these trajectories is the random interlacement at level u of Sznitman arXiv:0704.2560 . We prove that for any u>0, almost surely, (1) any two vertices in the random interlacement at level u are connected via at most ceiling(d/2) trajectories of the point process, and (2) there are vertices in the random interlacement at level u which can only be connected via at least ceiling(d/2) trajectories of the point process. In particular, this implies the already known result of Sznitman arXiv:0704.2560 that the random interlacement at level u is connected.

math.PR↗
arXiv2012

An equation-free approach to coarse-graining the dynamics of networks

We propose and illustrate an approach to coarse-graining the dynamics of evolving networks (networks whose connectivity changes dynamically). The approach is based on the equation-free framework: short bursts of detailed network evolution simulations are coupled with lifting and restriction operators that translate between actual network realizations and their (appropriately chosen) coarse observables. This framework is used here to accelerate temporal simulations (through coarse projective integration), and to implement coarsegrained fixed point algorithms (through matrix-free Newton-Krylov GMRES). The approach is illustrated through a simple network evolution example, for which analytical approximations to the coarse-grained dynamics can be independently obtained, so as to validate the computational results. The scope and applicability of the approach, as well as the issue of selection of good coarse observables are discussed.

cs.SI↗
arXiv2011

On the transience of random interlacements

We consider the interlacement Poisson point process on the space of doubly-infinite Z^d-valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The set of vertices and edges visited by at least one of these trajectories is the graph induced by the random interlacements at level u of Sznitman arXiv:0704.2560. We prove that for any u>0, almost surely, the random interlacement graph is transient.

math.PR↗