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2
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0answers
61 views

Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$. We know that $C_n$ has a giant component whose size is of ...
9
votes
0answers
266 views

First passage percolation on a random geometric graph in the large connectivity limit

Let $V_\rho\subset\mathbb{R}^2$ be a point set in the plane obtained from a Poisson process of density $\rho$. The random geometric graph $G_\rho$ is obtained from $V_\rho$ by connecting points that ...
44
votes
5answers
3k views

Escape the zombie apocalypse

Consider zombies placed uniformly at random over $\mathbb{R}^2$ with asymptotic density $\mu$ zombies/area. You are placed at a random point and can move with speed $1$. Zombies move with speed $v\leq ...
4
votes
0answers
148 views

Does the concept of connective constant make sense for any tiling of the plane?

First let me define what is the "connective constant" of a two dimensional lattice. Let $c_{n}$ denote the number of $n$ step self-avoiding walks starting from a fixed origin point in the lattice. ...
5
votes
0answers
127 views

Semi-directed first-passage percolation on $\mathbb{Z}^2$ with deterministic vertical weights

Consider the following first-passage percolation problem on the plane grid $\mathbb{Z}^2$: all the horizontal edges are directed (pointing east) and carry an independent random weight, say standard ...
5
votes
1answer
165 views

percolation probability in a hexagonal region

Suppose one takes a large hexagonal region in the tiling of the plane by unit hexagons, with $n+1$ hexagons on each side, as seen in the figure below (taken from the COMAP website) for the case $n=5$. ...
1
vote
1answer
69 views

Independent bond percolation on upper density zero subgraphs of the square lattice can have a non-trivial critical point ?

Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where edge set $\mathbb{E}^2$ is given by the pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$. Let ...
2
votes
1answer
103 views

Continuum limit of first-passage percolation paths

A few years ago, when I was working on first-passage percolation problems, I thought about the following problem. Recently it came back to my mind. Consider, for some $\delta=n^{-1}>0$, the grid ...
5
votes
1answer
241 views

Probability of two vertices to be connected in G(n,p)

A question I asked at math.SE without elliciting an answer. Let $G(n,p)$ be an Erdős–Rényi graph on $n$ vertices. Is there an explicit expression for the probability $P_{n,p}(u,v)$ that two fixed ...
5
votes
0answers
147 views

Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise. Tsirelson showed the existence of black noise through the scaling limit ...
3
votes
0answers
87 views

Critical value of semi-oriented percolation

Has it been proved that the two dimensional semi-oriented percolation process exhibits a phase transition at $p_c < 1/2$ (STRICTLY less than 1/2!!)? Semi-oriented Percolation: 2 dimensional ...
2
votes
0answers
52 views

More positive pivotal edges than negative ones at critical bond percolation on Z^2?

Consider critical bond percolation on $\mathbb{Z}^2$ inside a fixed rectangle $(0,0) - (an,n), a \geq 1$ and write $A$ for the event that there is an open crossing in the long (left to right) ...
4
votes
0answers
229 views

Expected number of components with multiple cycles in a subgraph of a square lattice

Short version Is there an understanding of the emergence and subsequent disappearance of components with zero, one, or more cycles in a random subgraph of a square or cubic lattice, as the ...
3
votes
2answers
210 views

Ergodicity for a Probabilistic Cellular Automaton on a finite space

Let's consider a Probabilistic Cellular Automaton on a one dimensional lattice $S$. Each site of the lattice can have two states, $0$ and $1$. The transition probability acting on each site is: ...
2
votes
1answer
117 views

Percolation in $Z^2$, problem in the proof of the existence of a critical probability

I have a $Z^2$ lattice. Every element $(z_1, z_2) \in Z^2$ is connected to one of its 4 neighbours $(z_1, z_2) + (0,1)$, $(z_1, z_2) + (0,-1)$, $(z_1, z_2) + (1,0)$, $(z_1, z_2) + (-1,0)$ by an edge. ...
1
vote
0answers
147 views

Percolation on infinite percolation clusters

Let's consider an infinite percolation cluster $\mathcal{C}_p$ that takes shape in the supercritical phase ($p>p_c$) of a bond percolation in $\mathbb{Z}^d$. What could be said about a bond ...
7
votes
1answer
523 views

Van Den Berg-Kesten-Reimer inequality

Statement of Van Den Berg-Kesten-Reimer inequality: Let $n$ be a positive integer. For $i\in[n]$, let $\mu_i$ be a probability measure on a finite set $\Omega_i$. Let ...
2
votes
0answers
147 views

On randomly colored random chords

Let $Q=[-1/2,1/2]^2$ be a unit square and let $(\ell_n,\varepsilon_n)_{n\geq1}$ be an iid sequence of isotropic lines intersecting $Q$ (more precisely, distributed according to a Haar measure on the ...
0
votes
0answers
165 views

Branching process question

(Cross-posted to math stackexchange question 130154) I am trying to analyze the following branching process. We start with a root (level 0) node. Each surviving node has two children, each of which ...