1,721,156 research outputs found
Ritratto di Walter Stauffer a cinquant'anni
No full text<p>Ritratto di Walter Stauffer a cinquant'anni realizzato dal pittore Max Rimboeck nel 1938</p>
Space-time percolation and detection by mobile nodes
Full text linkConsider the model where nodes are initially distributed as a Poisson point process with intensity λ over Rd and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance r of their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of Rd, is detected by at least one node. We consider the case where the target particle can move according to any continuous function and can adapt its motion based on the location of the nodes. We show that there exists a sufficiently large value of λ so that the target will eventually be detected almost surely. This means that the target cannot evade detection even if it has full information about the past, present and future locations of the nodes. Also, this establishes a phase transition for λ since, for small enough λ, with positive probability the target can avoid detection forever. A key ingredient of our proof is to use fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time.</p
Phase transition for finite-speed detection among moving particles
Full text linkConsider the model where particles are initially distributed on Zd,d≥2, according to a Poisson point process of intensity λ>0, and are moving in continuous time as independent simple symmetric random walks. We study the escape versus detection problem, in which the target, initially placed at the origin of Zd,d≥2, and changing its location on the lattice in time according to some rule, is said to be detected if at some finite time its position coincides with the position of a particle. For any given S>0S>0, we consider the case where the target can move with speed at most SS, according to any continuous function and can adapt its motion based on the location of the particles. We show that, for any S>0, there exists a sufficiently small λ∗>0, so that if the initial density of particles λ<λ∗, then the target can avoid detection forever
Perturbing the hexagonal circle packing: a percolation perspective
Full text linkWe consider the hexagonal circle packing with radius 1/2 and perturb it by letting the circles move as independent Brownian motions for time t. It is shown that, for large enough t, if Πt is the point process given by the center of the circles at time t, then, as t → ∞, the critical radius for circles centered at Πt to contain an infinite component converges to that of continuum percolation (which was shown - based on a Monte Carlo estimate - by Balister, Bollobás and Walters to be strictly bigger than 1/2). On the other hand, for small enough t, we show (using a Monte Carlo estimate for a fixed but high dimensional integral) that the union of the circles contains an infinite connected component. We discuss some extensions and open problems
Characterizing optimal sampling of binary contingency tables via the configuration model
Full text linkA binary contingency table is an m × n array of binary entries with row sums r = (r1,..., rm) and column sums c = (c1,..., cn). The configuration model generates a contingency table by considering ri tokens of type 1 for each row i and cj tokens of type 2 for each column j, and then taking a uniformly random pairing between type-1 and type-2 tokens. We give a necessary and sufficient condition so that the probability that the configuration model outputs a binary contingency table remains bounded away from 0 as goes to ∞. Our finding shows surprising differences from recent results for binary symmetric contingency tables
Multi-particle diffusion limited aggregation
Full text linkWe consider a stochastic aggregation model on Zd. Start with particles distributed according to the product Bernoulli measure with parameter μ. In addition, start with an aggregate at the origin. Non-aggregated particles move as continuous-time simple random walks obeying the exclusion rule, whereas aggregated particles do not move. The aggregate grows by attaching particles to its surface whenever a particle attempts to jump onto it. This evolution is called multi-particle diffusion limited aggregation. Our main result states that if on d> 1 the initial density of particles is large enough, then with positive probability the aggregate has linearly growing arms; that is, there exists a constant c> 0 so that at time t the aggregate contains a point of distance at least ct from the origin, for all t. The key conceptual element of our analysis is the introduction and study of a new growth process. Consider a first passage percolation process, called type 1, starting from the origin. Whenever type 1 is about to occupy a new vertex, with positive probability, instead of doing it, it gives rise to another first passage percolation process, called type 2, which starts to spread from that vertex. Each vertex gets occupied only by the process that arrives to it first. This process may have three phases: extinction (type 1 gets eventually surrounded by type 2), coexistence (infinite clusters of both types emerge), and strong survival (type 1 produces an infinite cluster which entraps all type 2 clusters). Understanding the various phases of this process is of mathematical interest on its own right. We establish the existence of a strong survival phase, and use this to show our main result
Polynomial mixing time of edge flips on quadrangulations
Full text linkWe establish the first polynomial upper bound for the mixing time of random edge flips on rooted quadrangulations: we show that the spectral gap of the edge flip Markov chain on quadrangulations with n faces admits, up to constants, an upper bound of n- 5 / 4 and a lower bound of n- 11 / 2. In order to obtain the lower bound, we also consider a very natural Markov chain on plane trees—or, equivalently, on Dyck paths—and improve the previous lower bound for its spectral gap by Shor and Movassagh
Random walks in random conductances: Decoupling and spread of infection
Full text linkLet (G,μ) be a uniformly elliptic random conductance graph on 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 ℓ<K inside Q K have sufficiently many particles, the particles return to stationarity after cℓ 2 time with a probability close to 1. We show that in this setup, an infection spreads with positive speed in any direction. Our framework is robust enough to allow us to also extend the result to infection with recovery. </p
Critical density of activated random walks on transitive graphs
Full text linkWe consider the activated random walk model on general vertextransitive graphs. A central question in this model is whether the critical density μ c for sustained activity is strictly between 0 and 1. It was known that μ c > 0 on Z d, d = 1, and that μ c < 1 on Z for small enough sleeping rate. We show that μ c → 0 as λ → 0 in all vertex-transitive transient graphs, implying that μ c < 1 for small enough sleeping rate. We also show that μ c < 1 for any sleeping rate in any vertex-transitive graph in which simple random walk has positive speed. Furthermore, we prove that μc > 0 in any vertex-transitive amenable graph, and that μ c ∞ (0, 1) for any sleeping rate on regular trees. </p
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