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Register a new userThe Constrained-degree percolation model
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Bernardo Nunes Borges de Lima
Publication date
2020
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English
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In the Constrained-degree percolation model on a graph $(mathbb{V},mathbb{E})$ there are a sequence, $(U_e)_{einmathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open at time $U_e$, it succeeds if both its end-vertices would have degrees at most $k-1$. We prove a phase transition theorem for this model on the square lattice $mathbb{L}^2$, as well as on the d-ary regular tree. We also prove that on the square lattice the infinite cluster is unique in the supercritical phase.
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We consider the Constrained-degree percolation model with random constraints on the square lattice and prove a non-trivial phase transition. In this model, each vertex has an independently distributed random constraint $jin {0,1,2,3}$ with probability $rho_j$. Each edge $e$ tries to open at a random uniform time $U_e$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when $rho_3$ is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, together with a coarse-graining argument.
We consider the Bernoulli Boolean discrete percolation model on the d-dimensional integer lattice. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the intensity of the underlying point process is small enough. We also study a Harris graphical procedure to construct, forward in time, particle systems with interactions of infinite range under the assumption that the corresponding generator admits a Kalikow-type decomposition. We do so by using the subcriticality of the boolean model of discrete percolation.
We consider first passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order.
We consider an anisotropic bond percolation model on $mathbb{Z}^2$, with $textbf{p}=(p_h,p_v)in [0,1]^2$, $p_v>p_h$, and declare each horizontal (respectively vertical) edge of $mathbb{Z}^2$ to be open with probability $p_h$(respectively $p_v$), and otherwise closed, independently of all other edges. Let $x=(x_1,x_2) in mathbb{Z}^2$ with $0<x_1<x_2$, and $x=(x_2,x_1)in mathbb{Z}^2$. It is natural to ask how the two point connectivity function $prob({0leftrightarrow x})$ behaves, and whether anisotropy in percolation probabilities implies the strict inequality $prob({0leftrightarrow x})>prob({0leftrightarrow x})$. In this note we give an affirmative answer in the highly supercritical regime.
We consider the bond percolation problem on a transient weighted graph induced by the excursion sets of the Gaussian free field on the corresponding cable system. Owing to the continuity of this setup and the strong Markov property of the field on the one hand, and the links with potential theory for the associated diffusion on the other, we rigorously determine the behavior of various key quantities related to the (near-)critical regime for this model. In particular, our results apply in case the base graph is the three-dimensional cubic lattice. They unveil the values of the associated critical exponents, which are explicit but not mean-field and consistent with predictions from scaling theory below the upper-critical dimension.
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