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### 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. ...

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### 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 ...

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141 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 $n=5$. ...

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63 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 ...

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**1**answer

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### 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 ...

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190 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 ...

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139 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 ...

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### 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 ...

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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) ...

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### 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 ...

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206 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: ...

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### 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. ...

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### 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 ...

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452 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 ...

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### 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 ...

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### 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 ...