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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 ...
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 ...
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 ...
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. ...
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-...
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 ...
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 ...
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 ...
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) ...
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 ...
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 ...
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 : ...