MÉMOIRE DE MAÎTRISE Zéros de fonctions analytiques aléatoires

U

Zf
Zf
Zf
Zf

UCU
U
(Ω, P )
N: Ω × B(U) →N∪ {∞},
N(ω, ·) U
BUN(·, B)
Z
B Z ∩B
N(B)N(B) = Pa∈Z1a∈BN
Z N
0 1 N(B) = card(Z∩B)
B
p N
B
E(N(B)) = ZB p(z)dz .
n n pn
B1, . . . , Bn
E(N(B1)···N(Bn)) = ZB1×···×Bn pn(z)dz .
n
z1, . . . , znU
p({z1, . . . , zn}) = lim
ε→0
1
(πε2)nPN(B(zi, ε)) ≥1, i = 1, . . . , n
Zf
fU
Zf
f f

Zf
Z(Xn)n≥0
UN
1 (Xn)n≥0{X1, . . . , XN}
1
z1, . . . , znUε > 0B(zi, ε)
{N(B(zi, ε)), i = 1, . . . , n}
N(B(zi, ε)) πε2
P(N(B(zi, ε)≥1, i = 1, . . . , n)) =
n
Y
i=11−P(N(B(zi, ε)) = 0)=
n
Y
i=1
(1 −e−πε2).
p(z1, . . . , zn) = 1
ZUk≥1p
k Z Z∧kk Z
BUk
E(card(B∩Z∧k)) = ZB p(z1, . . . , zk)dz1···dzk.
UkUkB=B1× ··· × Bk
PUMk(P)
k(z1, . . . , zk)B z1, . . . , zkP
N Z
Mk(P) = X
Q1,...,Qk
card(Q1× ··· × Qk∩B)
=X
Q1,...,Qk
card(Q1× ··· × Qk∩B∩Z∧k)
=X
Q1,...,Qk
k
Y
i=1
card(Qi∩Bi)
=X
Q1,...,Qk
k
Y
i=1
N(Qi∩Bi),
k(Q1, . . . , Qk)P