percolations examples

Question 1

rooted regular tree

Answer 1

one vertex of degree k and all others of degree k+1

Question 2

p_c of rooted tree

Answer 2

1/k

Question 3

neighbours in d-dimensional lattice

Answer 3

||u-v||=1 where ||x||=sqrt(x1^2+…+xd^2)

Question 4

dual lattice to L2

Answer 4

obtained by shifting the lattice by (1/2, 1/2)

Question 5

open in G*

Answer 5

if dual edge is closed

Question 6

e*

Answer 6

shares a common midpoint with e

Question 7

subgraph induced by W

Answer 7

vertex set W, edge set {edges that both endpoints are in W}

Question 8

C connected

Answer 8

if subgraph induced by C is connected