arXiv subjects

Jan M. Swart

Publications and source records attributed to Jan M. Swart.

At least 19 recordsLinked to original sources

arXiv2026

Universality of the Brownian net

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the universal scaling limit of one-dimensional branching-coalescing random walks with weak binary branching and arbitrary increment distributions that have finite $(3+\varepsilon)$-th moment. This gives the first example in the domain of attraction of the Brownian net where paths can cross without coalescing, which poses fundamental technical challenges not present in the non-crossing case.

math.PR↗

arXiv2026

Tightness criteria for random compact sets of cadlag paths

We give tightness criteria for random variables taking values in the space of all compact sets of cadlag real-valued paths, in terms of both the Skorohod J1 and M1 topologies. This extends earlier work motivated by the study of the Brownian web that was concerned only with continuous paths. In the M1 case, we give a natural extension of our tightness criteria which ensures that non-crossing systems of paths have weak limit points that are also non-crossing. This last result is exemplified through a rescaling of heavy tailed Poisson trees and a more general application to weaves.

math.PR↗

arXiv2026

Percolation representations of additive particle systems

It is well-known that additive interacting particle systems with a local state space of cardinality two have a percolation representation in terms of open paths in a graphical representation. In this paper, it is shown how such a percolation representation can be constructed more generally when the local state space is a finite distributive lattice. The theory is demonstrated on Krone's two-stage contact process.

math.PR↗

arXiv2026

Peierls bounds from Toom contours

For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. The proof of sufficiency is based on an intricate Peierls argument. We present a simplified version of this Peierls argument. Our main motivation is the open problem of determining stability of monotone cellular automata with intrinsic randomness, in which for the unperturbed evolution the local update rules at different space-time points are chosen in an i.i.d. fashion according to some fixed law. We apply Toom's Peierls argument to prove stability of a class of cellular automata with intrinsic randomness and also derive lower bounds on the critical parameter for some deterministic cellular automata.

math.PR↗

arXiv2025

A Course in Interacting Particle Systems

These lecture notes give an introduction to the theory of interacting particle systems. The main subjects are the construction using generators and graphical representations, the mean field limit, stochastic order, duality, and the relation to oriented percolation. An attempt is made to give a large number of examples beyond the classical voter, contact and Ising processes and to illustrate these based on numerical simulations.

math.PR↗

arXiv2025

Monotone duality of interacting particle systems

The duality theory for monotone interacting particle systems was initiated by Gray (1986) and further developed by Sturm and Swart (2018). It contains the better known additive duality as a special case but differs in the sense that the dual process contains not only single particles but also pairs, triples, and general $n$-tuples of particles, which correspond to the fact that in the forward process sometimes several particles are needed to create one particle at a later time. In earlier work, the dual process was constructed for finite initial states only, but, assuming that the empty state is a trap for the forward process, we show that the dual process can be started in infinite initial states and has an upper invariant law. It can therefore be viewed as some sort of interacting particle system in its own right. For the monotone dual of a cooperative contact process, we show that the upper invariant law is the long-time limit started from any nontrivial homogeneous invariant law. We use this to prove continuity of the survival probability of the forward process as a function of its parameters.

math.PR↗

arXiv2025

Skorokhod's topologies on path space

Skorokhod's J1 and M1 topologies are standard tools in proving limit theorems for stochastic processes. Motivated by applications, we extend these topologies so that they are capable of describing the convergence of a sequence of functions that are not all defined on the same domain. Traditionally, the J1 and M1 topologies are defined using time changes. Instead, we base our definitions on the point of view that the graph of a cadlag function can naturally be viewed as a compact set that is equipped with a total order. The distance between two graphs is then measured by matching points on one graph with points on the other graph in a way that respects the total order. We treat the J1 and M1 topologies in a unified framework and simplify the existing theory. We introduce a space of paths, elements of which are cadlag functions defined on an arbitrary closed subset of the real line. We show that this space is Polish and derive compactness criteria. Specialised to functions that are all defined on the same domain, this yields new proofs of known results.

math.GN↗

arXiv2025

A note on extendable sets of colorings and rooted minors

DeVos and Seymour (2003) proved that for every set $C$ of 3-colorings of a set $X$ of vertices, there exists a plane graph $G$ with vertices of $X$ incident with the outer face such that a 3-coloring of $X$ extends to a 3-coloring of $G$ if and only if it belongs to $C$. We prove a generalization of this claim for $k$-colorings of $X$-rooted-$K_{k+1}$-minor-free $K_{k+2}$-minor-free graphs.

math.CO↗

arXiv2025

Peierls bounds from random Toom contours

For deterministic monotone cellular automata on the $d$-dimensional integer lattice, Toom (1980) has given necessary and sufficient conditions for the all-one fixed point to be stable against small random perturbations. We are interested in the open problem of extending Toom's result to monotone cellular automata with intrinsic randomness, where the unperturbed evolution is random with i.i.d. update rules attached to the space-time points. For some applications it is also desirable to consider a more general graph structure, so we assume that the underlying lattice is an arbitrary countable group. Toom's proof of stability is based on a Peierls argument. In previous work, we demonstrated that this Peierls argument can also be used to prove stability for cellular automata with intrinsic randomness, but in this case estimating the Peierls sum becomes much harder than in the deterministic case. In the present paper, we develop a method based on random contours to estimate the Peierls sum and apply it to prove new stability results for monotone cellular automata with intrisic randomness. We also demonstrate the limitations of the method by constructing an example where the Peierls sum is infinite for arbitrary small perturbations even though stability is believed to hold.

math.PR↗

arXiv2024

A min-max random game on a graph that is not a tree

We study a random game in which two players in turn play a fixed number of moves. For each move, there are two possible choices. To each possible outcome of the game we assign a winner in an i.i.d. fashion with a fixed parameter p. In the case where all different game histories lead to different outcomes, a classical result due to Pearl (1980) says that in the limit when the number of moves is large, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. We are interested in a modification of this game where the outcome is determined by the exact sequence of moves played by the first player and by the number of times the second player has played each of the two possible moves. We show that also in this case, there is a sharp threshold in the parameter p that separates the regimes in which either player has with high probability a winning strategy. Since in the modified game, different game histories can lead to the same outcome, the graph associated with the game is no longer a tree which means independence is lost. As a result, the analysis becomes more complicated and open problems remain.

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

Applying monoid duality to a double contact process

In this paper we use duality techniques to study a combination of the well-known contact process (CP) and the somewhat less-known annihilating branching process. As the latter can be seen as a cancellative version of the contact process, we rebrand it as the cancellative contact process (cCP). Our process of interest will consist of two entries, the first being a CP and the second being a cCP. We call this process the double contact process (2CP) and prove that it has (depending on the model parameters) at most one invariant law under which ones are present in both processes. In particular, we can choose the model parameter in such a way that CP and cCP are monotonely coupled. In this case also the above mentioned invariant law will have the property that, under it, ones in the cCP can only be present at sites where there are also ones in the CP. Along the way we extend the dualities for Markov processes discovered in our paper "Commutative monoid duality" to processes on infinite state spaces so that they, in particular, can be used for interacting particle systems.

math.PR↗

arXiv2022

Commutative monoid duality

We introduce two partially overlapping classes of pathwise dualities between interacting particle systems that are based on commutative monoids (semigroups with a neutral element) and semirings, respectively. For interacting particle systems whose local state space has two elements, this approach yields a unified treatment of the well-known additive and cancellative dualities. For local state spaces with three or more elements, we discover several new dualities.

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↗

arXiv2020

An invariance principle for biased voter model interfaces

We consider one-dimensional biased voter models, where 1's replace 0's at a faster rate than the other way round, started in a Heaviside initial state describing the interface between two infinite populations of 0's and 1's. In the limit of weak bias, for a diffusively rescaled process, we consider a measure-valued process describing the local fraction of type 1 sites as a function of time. Under a finite second moment condition on the rates, we show that in the diffusive scaling limit there is a drifted Brownian path with the property that all but a vanishingly small fraction of the sites on the left (resp. right) of this path are of type 0 (resp. 1). This extends known results for unbiased voter models. Our proofs depend crucially on recent results about interface tightness for biased voter models.

math.PR↗

arXiv2020

Recursive tree processes and the mean-field limit of stochastic flows

Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. We consider interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. This turns out to be closely related to recursive tree processes as studied by Aldous and Bandyopadyay in discrete time. We here develop an analogue theory for recursive tree processes in continuous time. We illustrate the abstract theory on an example of a particle system with cooperative branching. This yields an interesting new example of a recursive tree process that is not endogenous.

math.PR↗

arXiv2019

Necessary and sufficient conditions for a nonnegative matrix to be strongly R-positive

Using the Perron-Frobenius eigenfunction and eigenvalue, each finite irreducible nonnegative matrix $A$ can be transformed into a probability kernel $P$. This was generalized by David Vere-Jones who gave necessary and sufficient conditions for a countably infinite irreducible nonnegative matrix $A$ to be transformable into a recurrent probability kernel $P$, and showed uniqueness of $P$. Such $A$ are called R-recurrent. Let us say that $A$ is strongly R-positive if the return times of the Markov chain with kernel $P$ have exponential moments of some positive order. Then it is known that strong R-positivity is equivalent to the property that lowering the value of finitely many entries of $A$ lowers the spectral radius. This paper gives a short and largely self-contained proof of this fact.

math.PR↗

arXiv2018

Equilibrium interfaces of biased voter models

A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest.

math.PR↗