## Poisson percolation on the square lattice

Publication
, Journal Article

Cristali, I; Junge, M; Durrett, R

Published in: Alea

January 1, 2019

Suppose that on the square lattice the edge with midpoint x becomes open at rate ∥x∥σ-1 . Let ρ(x, t) be the probability that the corresponding edge is open at time t and let n(p; t) be the distance at which edges are open with probability p at time t. We show that with probability tending to 1 as t → σ: (i) the open cluster containing the origin ℂ0(t) is contained in the square of radius n(pc-∈, t), and (ii) the cluster fills the square of radius n(pc+∈, t) with the density of points near x being close to θ(ρ(x, t)) where θ(p) is the percolation probability when bonds are open with probability p on ℤ2. Results of Nolin suggest that if N = n(pc, t) then the boundary uctuations of ℂ0(t) are of size N4/7.

### Duke Scholars

## Published In

Alea

## DOI

## ISSN

1980-0436

## Publication Date

January 1, 2019

## Volume

16

## Issue

1

## Start / End Page

429 / 437

## Related Subject Headings

- 4905 Statistics
- 4901 Applied mathematics
- 0104 Statistics
- 0102 Applied Mathematics

### Citation

APA

Chicago

ICMJE

MLA

NLM

Cristali, I., Junge, M., & Durrett, R. (2019). Poisson percolation on the square lattice.

*Alea*,*16*(1), 429–437. https://doi.org/10.30757/ALEA.V16-16Cristali, I., M. Junge, and R. Durrett. “Poisson percolation on the square lattice.”

*Alea*16, no. 1 (January 1, 2019): 429–37. https://doi.org/10.30757/ALEA.V16-16.Cristali I, Junge M, Durrett R. Poisson percolation on the square lattice. Alea. 2019 Jan 1;16(1):429–37.

Cristali, I., et al. “Poisson percolation on the square lattice.”

*Alea*, vol. 16, no. 1, Jan. 2019, pp. 429–37.*Scopus*, doi:10.30757/ALEA.V16-16.Cristali I, Junge M, Durrett R. Poisson percolation on the square lattice. Alea. 2019 Jan 1;16(1):429–437.

## Published In

Alea

## DOI

## ISSN

1980-0436

## Publication Date

January 1, 2019

## Volume

16

## Issue

1

## Start / End Page

429 / 437

## Related Subject Headings

- 4905 Statistics
- 4901 Applied mathematics
- 0104 Statistics
- 0102 Applied Mathematics