percolations on general graphs

Question 1

L

Answer 1

all vertices finite degree
assume infinite vertices

Question 2

neighbour

Answer 2

if there is an edge

Question 3

clusters

Answer 3

components of percolation graph

Question 4

percolation probability θ_x(p)

Answer 4

x lies in an infinite component

Question 5

right continuous

Answer 5

for all ε there exists δ such that |f(x)-f(a)|<ε for a<x<a+δ

Question 6

Ν_inf

Answer 6

number of infinite clusters

Question 7

bond percolation model

Answer 7

random spanning subgraph of L for each e, it exists with probability p
edges exists mutually independently

Question 8

open edge

Answer 8

in G

Question 9

open path

Answer 9

all edges are open

Question 10

critical probability

Answer 10

sup{p:θ_x(p)=0}

Question 11

zero-one law for percolation

Answer 11

for connected L, P[N_inf>=1] is 1 or 0

Question 12

phase transition

Answer 12

p<p_c percolation does not occur with prob 1, p>p_c percolation occurs with prob 1