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Ben Golub
Ben Golub
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In percolation on a lattice, whathow is "infected" status correlated for points in a region around the local correlation structure of giant component membershiporigin?

Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster. (This can be interpreted as the probability of being infected if some site in the giant component is infected.) Let $N_d$ be the set of all points within distance $d$ of the origin in $\mathbb{Z}^2$. What is known about the joint distribution of $(Y_x)_{x\in N_d}$, rigorously or heuristically?

In percolation on a lattice, what is the local correlation structure of giant component membership?

Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster. Let $N_d$ be the set of all points within distance $d$ of the origin in $\mathbb{Z}^2$. What is known about the joint distribution of $(Y_x)_{x\in N_d}$, rigorously or heuristically?

In percolation on a lattice, how is "infected" status correlated for points in a region around the origin?

Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster. (This can be interpreted as the probability of being infected if some site in the giant component is infected.) Let $N_d$ be the set of all points within distance $d$ of the origin in $\mathbb{Z}^2$. What is known about the joint distribution of $(Y_x)_{x\in N_d}$, rigorously or heuristically?

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Ben Golub
Ben Golub
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connectivity function within In percolation on a neighborhood in lattice percolation, what is the local correlation structure of giant component membership?

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Ben Golub
Ben Golub
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connectivity function within a neighborhood in lattice percolation

Consider independent bond percolation on $\mathbb{Z}^2$, with $p>p_c$ so that the process is supercritical. For any site $x$ let $Y_x$ be the indicator of $x$ belonging to the infinite open cluster. Let $N_d$ be the set of all points within distance $d$ of the origin in $\mathbb{Z}^2$. What is known about the joint distribution of $(Y_x)_{x\in N_d}$, rigorously or heuristically?