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**11**

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385 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 ...

**5**

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149 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**

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179 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 ...

**4**

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158 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. ...

**4**

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374 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 edge-...

**3**

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62 views

### Is there any known construction of IIC as a limit from supercritical phase?

Consider a nearest-neighbor percolation model on $\Bbb{Z}^d$, where each bond is occupied with probability $p$ independent of each other. Let $\Bbb{P}_p$ denote the corresponding law on the ...

**3**

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96 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**

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93 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 ...

**2**

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65 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) ...

**2**

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152 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 ...

**1**

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163 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 ...

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39 views

### Critical probability, bond percolation on triangular lattice

Let G a graph and $p_c(G)$ the critical probability of bond percolation. Let G be a triangular lattice then $p_c(G) = 2\sin(\frac{\pi}{18})$ (Grimmett, Percolation p.65). Ramanujan find this formula :
...

**0**

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94 views

### Random Cluster Model only for bond percolation?

Can someone please tell me which of the following statements I make are true of the current state of the art:
The Random Cluster Model is a generalization of bond percolation (with possibly ...

**0**

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172 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 ...