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Time scale. The horizontal axis represents time and the vertical axis represents the scale. Note that γ (1)
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Let $(G,\mu)$ be a uniformly elliptic random conductance graph on $\mathbb{Z}^d$ with a Poisson point process of particles at time $t=0$ that perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^d$ and time into intervals indexed by $\tau$. Given a local event $E(i,\tau)$ that depends only on the parti...
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Citations
... In order to handle the additional complexities of high-dimensional geometry, we found it necessary to develop a new method that is somewhat simpler than the one used in [3], and which we call 'pinching a hyperplane'. We remark that various techniques for constructing random Lipschitz surfaces have previously been developed in the literature (see, e.g., [7,[9][10][11][12]); interestingly, our method appears to be different from these earlier approaches. ...
... Let A be a p-random subset of Z and set k ∶= k (A). Now, for each k ≥ 1 and Q ∈ k , let T k (Q) be the set given by Proposition 3.3, and let ⊂ R be defined as in (9). In particular, by the proposition, the set Z ⊂ Z is -closed. ...
We study monotone cellular automata (also known as 𝒰‐bootstrap percolation) in ℤd$$ {\mathbb{Z}}^d $$ with random initial configurations. Confirming a conjecture of Balister, Bollobás, Przykucki and Smith, who proved the corresponding result in two dimensions, we show that the critical probability is non‐zero for all subcritical models.
... Multi-scale analysis is a powerful technique that has been used to analyse a wide range of difficult processes, especially those with slow decay of correlations; recent examples include random interlacements, dependent percolation processes, and interacting particle systems [9,16,[22][23][24]26,28]. Despite its power, multi-scale analyses usually rely strongly on certain crucial properties, such as stationarity (that is, the process is in equilibrium) and monotonicity. ...
... If one is able to do this, then this robust framework is able to handle systems with strong correlations that otherwise are not tractable to analysis. Examples of such an approach range from random interlacements and dependent percolation [9,26,28] to interacting particle systems [16,[22][23][24]. ...
... In this case, they exhibit an example where correlations decay as d − 1 2 T (see Equation (2.11) from [8]). Remark 1. 6. We remark that the statement of the theorem above can be extended to allow for the case when the function f 1 depends not only on the configuration inside the box B 1 , but actually on the whole past of the process up to the upper time limit given by the ball B 1 . ...
... Regarding other works, Gracar and Stauffer [7,6] analyzed a more general situation where the random walks move on top of the random conductance model. They prove the existence of a percolation structure (which they call Lipschitz surface) and use this to conclude that the infection spreads with positive speed for d ≥ 2. A less structured percolating argument was obtained by Stauffer [19] in continuous space, where particles move as independent Brownian motions. ...
... Overview of the proofs. The proofs of Theorems 1.2 and 1.4 are based on a multi-scale framework developed by Gracar and Stauffer in [4], and provide a novel application of this technique. Given a local monotone event, the results in [4] provide the existence of a Lipschitz surface (see Definition 2.2) formed by boxes where translations of this event occur. ...
... The proofs of Theorems 1.2 and 1.4 are based on a multi-scale framework developed by Gracar and Stauffer in [4], and provide a novel application of this technique. Given a local monotone event, the results in [4] provide the existence of a Lipschitz surface (see Definition 2.2) formed by boxes where translations of this event occur. To show that the infection survives locally and establish Theorem 1.2, we consider a local event which guarantees that if the infection reaches a space-time box where this event holds, then the infection not only survives for a long time but also spreads to nearby boxes. ...
... Gracar and Stauffer develop the Lipschitz surface approach in [4,5]. The existence of a Lipschitz surface (see Theorem 2.3 in the current paper) was proved in [4], while [5] used this structure to study infection spread on top of the random conductance model. ...
We treat infection models with recovery and establish two open problems from Kesten and Sidoravicius [8]. Particles are initially placed on the $d$-dimensional integer lattice with a given density and evolve as independent continuous-time simple random walks. Particles initially placed at the origin are declared as infected, and healthy particles immediately become infected when sharing a site with an already infected particle. Besides, infected particles become healthy with a positive rate. We prove that, provided the recovery rate is small enough, the infection process not only survives, but also visits the origin infinitely many times on the event of survival. Second, we establish the existence of density parameters for which the infection survives for all possible choices of the recovery rate.
... Regarding other works, Gracar and Stauffer [7,6] analyzed a more general situation where the random walks move on top of the random conductance model. They prove the existence of a percolation structure (which they call Lipschitz surface) and use this to conclude that the infection spreads with positive speed for d ≥ 2. A less structured percolating argument was obtained by Stauffer [19] in continuous space, where particles move as independent Brownian motions. ...
We study infection spread among biased random walks on $\mathbb{Z}^{d}$. The random walks move independently and an infected particle is placed at the origin at time zero. Infection spreads instantaneously when particles share the same site and there is no recovery. If the initial density of particles is small enough, the infected cloud travels in the direction of the bias of the random walks, implying that the infection does not survive locally. When the density is large, the infection spreads to the whole $\mathbb{Z}^{d}$. The proofs rely on two different techniques. For the small density case, we use a description of the infected cloud through genealogical paths, while the large density case relies on a renormalization scheme.
... Multi-scale analysis is a powerful technique that has been used to analyse a wide range of difficult processes, especially those with slow decay of correlations; recent examples include random interlacements, dependent percolation processes, and interacting particle systems [20,21,22,26,24,9,14]. Despite its power, multi-scale analyses usually rely strongly on certain crucial properties, such as stationarity (that is, the process is in equilibrium) and monotonicity. ...
... If one is able to do this, then this robust framework is able to handle systems with strong correlations that otherwise are not tractable to analysis. Examples of such an approach range from random interlacements and dependent percolation [26,24,9] to interacting particle systems [20,21,22,14]. ...
The main contribution of this paper is the development of a novel approach to multi-scale analysis that we believe can be used to analyse processes with non-equilibrium dynamics. Our approach will be referred to as \emph{multi-scale analysis with non-equilibrium feedback} and will be used to analyse a natural random growth process with competition on $\mathbb{Z}^d$ called \emph{first passage percolation in a hostile environment} that consists of two first passage percolation processes $FPP_1$ and $FPP_{\lambda}$ that compete for the occupancy of sites. Initially, $FPP_1$ occupies the origin and spreads through the edges of $\mathbb{Z}^d$ at rate 1, while $FPP_{\lambda}$ is initialised at sites called \emph{seeds} that are distributed according to a product of Bernoulli measures of parameter $p\in(0,1)$, where a seed remains dormant until $FPP_1$ or $FPP_{\lambda}$ attempts to occupy it before then spreading through the edges of $\mathbb{Z}^d$ at rate $\lambda>0$. Particularly challenging aspects of FPPHE are its non-equilibrium dynamics and its lack of monotonicity (for instance, adding seeds could be benefitial to $FPP_1$ instead of $FPP_\lambda$); such characteristics, for example, prevent the application of a more standard multi-scale analysis. As a consequence of our main result for FPPHE, we establish a coexistence phase for the model for $d\geq3$, answering an open question in \cite{sidoravicius2019multi}. This exhibits a rare situation where a natural random competition model on $\mathbb{Z}^d$ observes coexistence for processes with \emph{different} speeds. Moreover, we are able to establish the stronger result that $FPP_1$ and $FPP_{\lambda}$ can both occupy a \emph{positive density} of sites with positive probability, which is in stark contrast with other competition processes.
... Then, in Section 3, we use this framework to analyze the spread of information. Due to space limitations, we will not be able to give full proofs of the above framework, for which we refer to the full version [6]. This extended abstract has yet one additional result with respect to [6], which is the construction and proof of the Lipschitz net, which is adapted to analyzing processes on finite graphs. ...
... Due to space limitations, we will not be able to give full proofs of the above framework, for which we refer to the full version [6]. This extended abstract has yet one additional result with respect to [6], which is the construction and proof of the Lipschitz net, which is adapted to analyzing processes on finite graphs. ...
... The following theorem establishes the existence of the Lipschitz surface. Due to space limitations, the proof is given in [6]. ...
We consider the problem of spread of information among mobile agents on the torus. The agents are initially distributed as a Poisson point process on the torus, and move as independent simple random walks. Two agents can share information whenever they are at the same vertex of the torus. We study the so-called flooding time: the amount of time it takes for information to be known by all agents. We establish a tight upper bound on the flooding time, and introduce a technique which we believe can be applicable to analyze other processes involving mobile agents. 2012 ACM Subject Classification Mathematics of computing → Probability and statistics We consider the problem of spread of information between mobile agents on a d-dimensional torus of side-length n. We will denote by N = n d the number of vertices on the torus, and will refer to the agents as particles. At time 0, the particles are distributed on the vertices of the torus as a Poisson point process of intensity λ. Then, particles move by performing independent continuous-time simple random walks on the torus; that is, at rate 1 a particle chooses a neighboring vertex uniformly at random and jumps there. It is not difficult to check that this system of particles is in stationarity. Thus, at any given time t, the location of the particles is a Poisson point process of intensity λ on the torus. However, the configuration of particles at time t is not independent of the configuration of particles at time 0, and as we will explain below, it is this dependence that makes this model challenging to analyze. Assume that at time 0 there is a particle at the origin with a piece of information that has to be distributed to all other particles. Then, any uninformed particle (a particle that does not know the information) receives the information whenever it is at the same vertex as an informed particle (a particle that knows the information). We study the time it takes the information to reach all the particles, which is commonly referred to as the flooding time.
... Then, in Section 3, we use this framework to analyze the spread of information. Due to space limitations, we will not be able to give full proofs of the above framework, for which we refer to the full version [6]. This extended abstract has yet one additional result with respect to [6], which is the construction and proof of the Lipschitz net, which is adapted to analyzing processes on finite graphs. ...
... Due to space limitations, we will not be able to give full proofs of the above framework, for which we refer to the full version [6]. This extended abstract has yet one additional result with respect to [6], which is the construction and proof of the Lipschitz net, which is adapted to analyzing processes on finite graphs. ...
... The following theorem establishes the existence of the Lipschitz surface. Due to space limitations, the proof is given in [6]. ...
We consider the problem of spread of information among mobile agents on the torus. The agents are initially distributed as a Poisson point process on the torus, and move as independent simple random walks. Two agents can share information whenever they are at the same vertex of the torus. We study the so-called flooding time: the amount of time it takes for information to be known by all agents. We establish a tight upper bound on the flooding time, and introduce a technique which we believe can be applicable to analyze other processes involving mobile agents.
... The above result has been established on the square lattice (i.e., µ x,y = 1 for all (x, y) ∈ E) by Kesten and Sidoravicius [8] via an intricate multi-scale analysis; see also [9] for a shape theorem. In a companion paper [6], we develop a framework, based on a multi-scale analysis, which can be used to analyze processes on this setting without the need of carrying out a multi-scale analysis from scratch. We prove our 1.1 via this framework, showing the applicability of our technique from [6]. ...
... In a companion paper [6], we develop a framework, based on a multi-scale analysis, which can be used to analyze processes on this setting without the need of carrying out a multi-scale analysis from scratch. We prove our 1.1 via this framework, showing the applicability of our technique from [6]. We also apply this technique to analyze the spread of an infection with recovery. ...
... In Section 3, we state a more precise version of 1.3 and prove it. In Section 4 we prove an extension of the local mixing result to random walks whose displacement is conditioned to be bounded, which is particularly useful in applications [12,6]. In Section 5, we use the local mixing result and results from our companion paper [6] to prove Theorems 1.1 and 1.2 for graphs satisfying (1.1). ...
Let $(G; \mu)$ be a uniformly elliptic random conductance graph on $\mathbb{Z}^d$ with a Poisson point process of particles at time $t = 0$ that perform independent simple random walks. We show that inside a cube $Q_K$ of side length $K$ , if all subcubes of side length $\ell < K$ inside $Q_K$ have sufficiently many particles, the particles return to stationarity after $c\ell^2$ time with a probability close to 1. We also show this result for percolation clusters on locally finite graphs. Using this mixing result and the results of [6], we show that in this setup, an infection spreads with positive speed in any direction. We prove the robustness of the framework by extending the result to infection with recovery, where we show positive speed and that the infection survives indefinitely with positive probability
