Triangulations aléatoires vérifiant une propriété de Markov spatiale

Hα
Hα
Tκ
Tκ
Tκ
n n
n1/4

n+∞n
t
t
r r4
1
2
(Hα)
α∈[0,1[
(Tκ)κ∈]0,2
27 ]
κ=2
27 κ < 2 27
(Tκ)
c
G S2
G
G

3
2 5
|c|c
c
m m
3
t c
c
t S(t)t
t S(t) = S2t
S2\S(t)t m
S2\S(t)m
t S(t)
t1t2
S(t1)S(t2)t1t2
m
c v e f
c v −e+f= 2
c f = 2v−4e= 3v−6
ρ

7 10 15
II 2
II T
T
t∈Tr∈N∗r t Br(t)
r−1ρ
t t0Tr Br(t) = Br(t0)d(t, t0) =
1
1+r
dT
d
(tn)r Br(tn)
ArArBr(Ar0) = Ar
r0≥r A r Ar
r A
(T, d) (tn)
tnn d(tm, tn)=1 m6=n(tn)
T
A⊂T A
(nr)t∈Ar∈N∗|Br(t)| ≤ nr
d

Tnn
µn
p≥0ϕn,p p n
p= 2 n= 0 ϕ0,2= 0
p
2
ϕ0,2= 1
|Tn|=φn−3,3n
n−3
ϕn,p+2 =2n+1(2p+ 1)!(2p+ 3n)!
p!2n!(2p+ 2n+ 2)!
ϕn,p+2 ∼Cp+227
2nn−5/2
Cp+2 =√3(2p+ 1)!
4√πp!29
4p∼C9p√p
p≥2t≤2
27 κ p
νpνpn
pκn
Z(κ)
p
Z(κ)
p=Pn≥0ϕn,pκn
Z(κ)
pκ≤2
27
κ=θ(1 −2θ)2
Z(κ)
p+2 =(2p)!(1 −2θ)−(2p+2)((1 −6θ)p+ 2 −6θ)
p!(p+ 2)!
κ=2
27
Zp+2 =Z(2/27)
p+2 =(2p)!
p!(p+ 2)!9 4p+1