First Passage Percolation on $$\mathbb {Z}^2$$ Z 2 : A Simulation Study

Alm, Sven Erick; Deijfen, Maria

doi:10.1007/s10955-015-1356-0

Cited by 14 publications

(14 citation statements)

References 37 publications

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“…A major contribution was done recently in the beautiful and extensive work of Alm and Deifjen [9] for FPP in two dimensions. Running 19 years of CPU time, in a cluster of 28 Linux machines, they investigate the value of the time constant and the limit shape for several continuous distributions.…”

Section: Simulationsmentioning

confidence: 99%

50 Years of First-Passage Percolation

Auffinger

Damron

Hanson

2017

University Lecture Series

230

405

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We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2 + log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.

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Section: Simulationsmentioning

confidence: 99%

50 Years of First-Passage Percolation

Auffinger

Damron

Hanson

2017

University Lecture Series

230

405

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“…Example -Forward-pass. Let ỹ = f θ (x 1 , x 2 , x 3 , x 4 , x 5 ) be a single-layer neural network function with parameters θ = {(W (1) , b (1) ), (W (2) b (2) )}, as illustrated in Figure 2. Assume that f θ has ReLu activation, i.e g(z) = max{0, z} and that the output is continuous, e.g.…”

Section: Neural Network and Deepmentioning

confidence: 99%

“…This type of results are called shape theorems. Let B(t) = {x + [− 1 2 , 1 2 ] d : x ∈ B(t)} be the continuum version of B(t). Theorem 1 (Cox and Durrett [3]).…”

Section: Introductionmentioning

confidence: 99%

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Predicting First Passage Percolation Shapes Using Neural Networks

Rosengren¹

2020

Preprint

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Many random growth models have the property that the set of discovered sites, scaled properly, converges to some deterministic set as time grows. Such results are known as shape theorems. Typically, not much is known about the shapes. For first passage percolation on Z d we only know that the shape is convex, compact, and inherits all the symmetries of Z d . Using simulated data we construct and fit a neural network able to adequately predict the shape of the set of discovered sites from the mean, standard deviation, and percentiles of the distribution of the passage times. The purpose of the note is two-fold. The main purpose is to give researchers a new tool for quickly getting an impression of the shape from the distribution of the passage times -instead of having to wait some time for the simulations to run, as is the only available way today. The second purpose of the note is simply to introduce modern machine learning methods into this area of discrete probability, and a hope that it stimulates further research.

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“…Many scholars have studied path formation time using first passage percolation. For instance, taking information distribution time or path formation time as first passage percolation is simulated in [22,23]. Path formation time is also studied by using the subadditive ergodic theorems in [24,25].…”

Section: Introductionmentioning

confidence: 99%

Path Formation Time in the Noise-Limited Fractionated Spacecraft Network with FDMA

Yan

2018

International Journal of Aerospace Engineering

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Connectivity and path formation time are very important for the design and optimization in fractionated spacecraft network. Taking frequency division multiple access (FDMA) with subcarrier binary phase-shift keying (BPSK) modulation as an example, this paper focuses on the issues of constraint to orbital elements and path formation time for the noise-limited fractionated spacecraft network percolating. First, based on the proposed evolution of the dynamic topology graph in the fractionated spacecraft network, we prove the constraint condition of orbital elements for noise-limited fractionated spacecraft network percolating, and the definition of path formation time is provided and the mobility model is established. Next, we study the relationship between first docking time and spatial initial distribution and the relationship between first separating time and spatial initial distribution. These relationships provide an important basis for the orbit design in the fractionated spacecraft network. Finally, the numerical results show that the network topology for fractionated spacecraft is time-varying and dynamic. The path formation time and hop length scale linearly with path length within each orbital hyperperiod and change periodically. Besides, the time constant gradually tends to a stable value with path formation time increasing, that is, path length. These results powerfully support percolation theory further under the environment of the noise-limited fractionated spacecraft network.

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First Passage Percolation on $$\mathbb {Z}^2$$ Z 2 : A Simulation Study

Cited by 14 publications

References 37 publications

50 Years of First-Passage Percolation

50 Years of First-Passage Percolation

Predicting First Passage Percolation Shapes Using Neural Networks

Path Formation Time in the Noise-Limited Fractionated Spacecraft Network with FDMA

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