We consider a random walk on top of the contact process on $\mathbb{Z}^d$ with $d\geq 1$. In particular, we focus on the "contact process as seen from the random walk". Under the assumption that the infection rate of the contact process is large or the jump rate of the random walk is small, we show that this process has at most two extremal measures. Moreover, the convergence to these extremal measures is characterised by whether the contact process survives or dies out, similar to the complete convergence theorem known for the ordinary contact process. Using this, we furthermore provide a law of large numbers for the random walk which holds under general assumptions on the jump probabilities of the random walk.
We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos and den Hollander, dos Santos, Sidoravicius. We show the CLT for such a random walk, valid for all supercritical infection rates for the contact process environment.
We consider branching random walks and contact processes on infinite, connected, locally finite graphs whose reproduction and infectivity rates across edges are inversely proportional to vertex degree. We show that when the ambient graph is a Galton-Watson tree then, in certain circumstances, the branching random walks and contact processes will have weak survival phases. We also provide bounds on critical values.
We consider the subcritical contact branching random walk on Zd in continuous time with the arbitrary number of offspring and with immigration. We prove the existence of the steady state (statistical equilibrium).
A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these cones. For the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk eventually gets trapped inside a single cone. This in turn leads …
In this paper we prove that, under the assumption of quasi-transitivity, if a branching random walk on ${\mathbb{Z}^d}$ survives locally (at arbitrarily large times there are individuals alive at the origin), then so does the same process when restricted to the infinite percolation cluster ${\mathcal{C}_{\infty}}$ of a supercritical Bernoulli percolation. When no more than $k$ individuals per site are allowed, we obtain the $k$-type contact process, which can be derived from the branching random walk by killing all particles that are born at a site where already $k$ individuals are present. We…
The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
This paper studies contact processes on general countable groups. It is shown that any such contact process has a well-defined exponential growth rate, and this quantity is used to study the process. In particular, it is proved that on any nonamenable group, the critical contact process dies out.
Random walks on $\mathbb{Z}$ with metastable Gaussian distribution caused by linear drift with application to the contact process on the complete graph
Considering a critical branching random walk on the real line. In a recent paper, Aidekon [3] developed a powerful method to obtain the convergence in law of its minimum after a log-factor normalization. By an adaptation of this method, we show that the point process formed by the branching random walk and its minimum converge in law to a Poisson point process colored by a certain point process. This result, confirming a conjecture of Brunet and Derrida [10], can be viewed as a discrete analog of the corresponding results for the branching brownian motion, previously established by Arguin et a…
Consider a walker performing a random walk in an i.i.d. random environment, and assume that the walker tells us at each time the environment it sees at its present location. Given this history of the transition probabilities seen from the walker - but not its trajectory - can we tell if the RWRE is recurrent or transient? Can we reconstruct the law of the environment? We show that in a one-dimensional environment, the law of the environment can be reconstructed, and we know in particular if the RWRE is recurrent or transient.
What is the long-time behavior of the law of a contact process started with a single infected site, distributed according to counting measure on the lattice? This question is related to the configuration as seen from a typical infected site and gives rise to the definition of so-called eigenmeasures, which are possibly infinite measures on the set of nonempty configurations that are preserved under the dynamics up to a multiplicative constant. In this paper, we study eigenmeasures of contact processes on general countable groups in the subcritical regime. We prove that in this regime, the proc…
We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on $\mathbb{Z}^4$, for sufficiently small attraction. We prove that the susceptibility and correlation length of order $p$ (for any $p>0$) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of $|x|^{-2}$. This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric r…
We consider a one-dimensional continuous time random walk with transition rates depending on an underlying autonomous simple symmetric exclusion process starting out of equilibrium. This model represents an example of a random walk in a slowly non-uniform mixing dynamic random environment. Under a proper space-time rescaling in which the exclusion is speeded up compared to the random walk, we prove a hydrodynamic limit theorem for the exclusion as seen by this walk and we derive an ODE describing the macroscopic evolution of the walk. The main difficulty is the proof of a replacement lemma for…
The loop-erased random walk (LERW) in $ \Z^d, d \geq 2$, is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the "middle" of the path.
We give an expression of the speed of the biased random walk on a Galton--Watson tree. In the particular case of the simple random walk, we recover the result of Lyons, Pemantle and Peres \cite{LyPePe95}. The proof uses a description of the invariant distribution of the environment seen from the particle.
We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. We show that the distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branchi…
In this paper we study threshold-one contact processes on lattices and regular trees. The asymptotic behavior of the critical infection rates as the degrees of the graphs growing to infinity are obtained. Defining λ_c as the supremum of infection rates which causes extinction of the process at equilibrium, we prove that nλ_c^{T^n}\rightarrow1 and 2dλ_c^{Z^d}\rightarrow1 as n,d\rightarrow+\infty. Our result is a development of the conclusion that λ_c^{Z^d}\leq\frac{2.18}{d} shown in \cite{Dur1991}. To prove our main result, a crucial lemma about the probability of a simple random walk on a latt…
We consider self-avoiding walk and percolation in $\Zd$, oriented percolation in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point $\pc$. We investigate the value of $\pc$ when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that $\pc=1+C(D)+O(L^{-2d})$…
An intrinsic multitype branching structure within the transient (1;R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the environments seen from the particle and reprove the LLN with an drift explicitly in terms of the environment, comparing with the results in Brémont (2002).
We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration inequalities for the environment as seen from the position of the walker, i.e the environment process. We also obtain ergodicity of the uniquely ergodic measure of the environment process as well as continuity as a function of the jump rates of the walker. As a consequence we obtain several limit theorems, such as law of large numbers, Einstein relation, central limit theorem and concentration propert…
This paper is a collection of recent results on discrete-time and continuous-time branching random walks. Some results are new and others are known. Many aspects of this theory are considered: local, global and strong local survival, the existence of a pure global survival phase and the approximation of branching random walks by means of multitype contact processes or spatially confined branching random walks. Most results are obtained using a generating function approach: the probabilities of extinction are seen as fixed points of an infinite dimensional power series. Throughout this paper we…
We study the random walk $X$ on the range of a simple random walk on $\mathbb{Z}^d$ in dimensions $d\geq 4$. When $d\geq 5$ we establish quenched and annealed scaling limits for the process $X$, which show that the intersections of the original simple random walk path are essentially unimportant. For $d=4$ our results are less precise, but we are able to show that any scaling limit for $X$ will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when $d=4$ similar logarithmic corrections are necessary in describing the a…
In this paper, we consider a random walk and a random color scenery on Z. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times. We focus on the case where the random walk has increments 0, +1 or -1 with probability epsilon, (1-epsilon)p and (1-epsilon)(1-…
We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the ``ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e.\ for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.
We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical parameter. We prove that, at the strong critical value, the process dies out locally almost surely; while, at the weak critical value, global survival and global extinction are both possible.
We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the "environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.
We consider the random walk in an independent and identically distributed (i.i.d.) random environment on a Cayley graph of a finite free product of copies of $\mathbb{Z}$ and $\mathbb{Z}_2$. Such a Cayley graph is readily seen to be a regular tree. Under a uniform elipticity assumption on the i.i.d. environment we show that the walk has positive speed and establish the annealed central limit theorem for the graph distance of the walker from the starting point.
We prove a law of large numbers for certain random walks on certain attractive dynamic random environments when initialised from all sites equal to the same state. This result applies to random walks on $\mathbb{Z}^d$ with $d\geq1$. We further provide sufficient mixing conditions under which the assumption on the initial state can be relaxed, and obtain estimates on the large deviation behaviour of the random walk. As prime example we study the random walk on the contact process, for which we obtain a law of large numbers in arbitrary dimension. For this model, further properties about the spe…
We prove that the dimension drop phenomenon holds for the harmonic measure associated to a transient random walk in a random environment (as defined by R. Lyons and R. Pemantle in 1992 and generalized by G. Faraud in 2011) on an infinite Galton-Watson tree without leaves. We use regeneration times and ergodic theory techniques from the work of R. Lyons, R. Pemantle and Y. Peres in 1996 to give an explicit construction of the invariant measure for the forward environment seen by the particule at exit times which is absolutely continuous with respect to the joint law of the tree and the path of …
This paper considers contact processes with additional voter model dynamics. For such models, results of Lloyd and Sudbury can be applied to find a self-duality, as well as dualities and thinning relations with systems of random walks with annihilation, branching, coalescence, and deaths. We show that similar relations, which are known from the literature for certain interacting SDE's, can be derived as local mean field limits of the relations of Lloyd and Sudbury.
We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. The case of Galton-Watson processes is particularly included so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering. As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be interesting on its own and which correct, generalize and simplify some earlier works.
We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly f…
In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Kni…
In this paper we study random walks on dynamical random environments in $1 + 1$ dimensions. Assuming that the environment is invariant under space-time shifts and fulfills a mild mixing hypothesis, we establish a law of large numbers and a concentration inequality around the asymptotic speed. The mixing hypothesis imposes a polynomial decay rate of covariances on the environment with sufficiently high exponent but does not impose uniform mixing. Examples of environments for which our methods apply include the contact process and Markovian environments with a positive spectral gap, such as the …
We consider simple random walks on random graphs embedded in $\mathbb{R}^d$ and generated by point processes such as Delaunay triangulations, Gabriel graphs and the creek-crossing graphs. Under suitable assumptions on the point process, we show an annealed invariance principle for these random walks. These results hold for a large variety of point processes including Poisson point processes, Matérn cluster and Matérn hardcore processes which have respectively clustering and repulsiveness properties. The proof relies on the use the process of the environment seen from the particle. It allows to…
We introduce via perturbation a class of random walks in reversible dynamic environments having a spectral gap. In this setting one can apply the mathematical results derived in http://arxiv.org/abs/1602.06322. As first results, we show that the asymptotic velocity is antisymmetric in the perturbative parameter and, for a subclass of random walks, we characterize the velocity and a stationary distribution of the environment seen from the walker as suitable series in the perturbative parameter. We then consider as a special case a random walk on the East model that tends to follow dynamical int…
In this paper we consider a d-dimensional scenery seen along a simple symmetric branching random walk, where at each time each particle gives the color record it is seeing. We show that we can a.s. reconstruct the scenery up to equivalence from the color record of all the particles. For this we assume that the scenery has at least 2d + 1 colors which are i.i.d. with uniform probability. This is an improvement in comparison to [22] where the particles needed to see at each time a window around their current position. In [11] the reconstruction is done for d = 2 with only one particle instead of…
We consider a branching-selection particle system on the real line. In this model the total size of the population at time $n$ is limited by $\exp\left(a n^{1/3}\right)$. At each step $n$, every individual dies while reproducing independently, making children around their current position according to i.i.d. point processes. Only the $\exp\left(a(n+1)^{1/3}\right)$ rightmost children survive to form the $(n+1)^\mathrm{th}$ generation. This process can be seen as a generalisation of the branching random walk with selection of the $N$ rightmost individuals, introduced by Brunet and Derrida. We o…
We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in [BCDG13] where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution to the Brownia…
We consider random walks in dynamic random environments given by Markovian dynamics on $\mathbb{Z}^d$. We assume that the environment has a stationary distribution $μ$ and satisfies the Poincaré inequality w.r.t. $μ$. The random walk is a perturbation of another random walk (called "unperturbed"). We assume that also the environment viewed from the unperturbed random walk has stationary distribution $μ$. Both perturbed and unperturbed random walks can depend heavily on the environment and are not assumed to be finite-range. We derive a law of large numbers, an averaged invariance principle for…
We consider two interacting random walks on $\mathbb{Z}$ such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter $β\geq 0$. When $β= 0$ both processes are independent symmetric random walks on $\mathbb{Z}$, and hence recurrent. We show that both random walks are further recurrent if $β\in (0,1]$. We also show that these processes …
Let $\mathcal{T}$ be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean $μ>1$, conditioned to survive. Let $φ_{\mathcal{T}}$ be a random embedding of $\mathcal{T}$ into $\mathbb{Z}^d$ according to a simple random walk step distribution. Let $\mathcal{T}_p$ be percolation on $\mathcal{T}$ with parameter $p$, and let $p_c = μ^{-1}$ be the critical percolation parameter. We consider a random walk $(X_n)_{n \ge 1}$ on $\mathcal{T}_p$ and investigate the behavior of the embedded process $φ_{\mathcal{T}_p}(X_n)$ as $n\to \infty$ and simultaneously, $\mathcal{T}_p$…
We define two families of Poissonian soups of bidirectional trajectories on $\mathbb{Z}^2$, which can be seen to adequately describe the local picture of the trace left by a random walk on the two-dimensional torus $(\mathbb{Z}/N \mathbb{Z})^2$, started from the uniform distribution, run up to a time of order $(N\log N)^2$ and forced to avoid a fixed point. The local limit of the latter was recently established in arXiv:1502.03470. Our construction proceeds by considering, somewhat in the spirit of statistical mechanics, a sequence of finite-volume approximations, consisting of random walks av…
We study a random walk on $\mathbb{F}_p$ defined by $X_{n+1}=1/X_n+\varepsilon_{n+1}$ if $X_n\neq 0$, and $X_{n+1}=\varepsilon_{n+1}$ if $X_n=0$, where $\varepsilon_{n+1}$ are independent and identically distributed. This can be seen as a non-linear analogue of the Chung--Diaconis--Graham process. We show that the mixing time is of order $\log p$, answering a question of Chatterjee and Diaconis.
Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on $Z^d$ where $d \ge 2$. The first new result is a type of uniqueness theorem: for every pair $x$ and $y$ of vertices which lie in infinite open paths, there exists almost surely a third vertex $z$ which is joined to infinity and which is attainable from $x$ and $y$ along directed open paths. Secondly, it is proved that a random walk on an infinite directed clust…
We consider a model of open quantum random walk and together with a quantum trajectory approach we are able to examine a notion of hitting time. We see that many constructions, such as minimal solutions to hitting time problems, are variations of well-known classical probability results, but the density matrix degree of freedom on each site gives rise to systems which are seen to be nonclassical. As a more specific application we study the collection of walks induced by normal commuting contractions, for which the corresponding probability expressions are obtained. We examine quantum versions …
On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $Δ_ω$, as the divergence of the horizontal gradient, once a volume $ω$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of random walks, where \emph{all} sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement $\mathbf{c}$ to the sub-Riemannian distribution, and is denoted $L^c$. We address the problem of equivalence of the two operators. This problem is int…
We study a model of competition between two types evolving as branching random walks on $\mathbb{Z}^d$. The two types are represented by red and blue balls respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of $\mathbb{Z}^d$ contain one ball each, which are independently coloured red with probability $p$ and blue otherwise. We address the question of \emph{fixation}, referring to the sites eventually settling for a given colour, or not. Under a mild moment condition on the branching rule, we prove that the p…
The spherical functions of the noncompact Grassmann manifolds over the real or complex numbers or the quaternions with rank q and dimension parameter p can be seen as Heckman-Opdam hypergeometric functions of type BC, when the double coset space is identified with some Weyl chamber of type B. The associated double coset hypergroups may be embedded into a continuous family of commutative hypergroups with these hypergeometric functions as multiplicative functions with p in some continuous parameter range by a result of Rösler. Several limit theorems for random walks associated with these hypergr…
