Gaussian fields and Kac-Rice formulae

147 Pages·2011·1.92 MB·English

Random Fields and Random Geometry
.
I: Gaussian fields and Kac-Rice formulae
Robert Adler
Electrical Engineering
Technion – Israel Institute of Technology
.
and
many, many others
October 25, 2011
I do not intend to cover all these slides in 75 minutes!
(Some of the material is for your later reference, and some for the
afternoon tutorial.)
Our heroes
Marc Kac Stephen O. Rice
1914–1984 1907–1986
P{ξ = (ξ ,...,ξ ) ∈ A} = (cid:90) e−(cid:107)x(cid:107)2/2 dx (cid:0)ξ ∼ N(0,I )(cid:1)
0 n−1 n×n
(2π)n/2
A
Theorem: N = the number of real zeroes of f
n
4 (cid:90) 1 [1−n2[x2(1−x2)/(1−x2n]2]1/2
E{N } = dx
n π 1−x2
0
Approx’n: For large n
2logn
E{N } ∼
n
π
Bound: For large n
2logn 14
E{N } ≤ + .
n
π π
Real roots of algebraic equations (Kac, 1943)
f(t) = ξ +ξ t +ξ t2+···+ξ tn−1
0 1 2 n−1
Theorem: N = the number of real zeroes of f
n
4 (cid:90) 1 [1−n2[x2(1−x2)/(1−x2n]2]1/2
E{N } = dx
n π 1−x2
0
Approx’n: For large n
2logn
E{N } ∼
n
π
Bound: For large n
2logn 14
E{N } ≤ + .
n
π π
Real roots of algebraic equations (Kac, 1943)
f(t) = ξ +ξ t +ξ t2+···+ξ tn−1
0 1 2 n−1
P{ξ = (ξ ,...,ξ ) ∈ A} = (cid:90) e−(cid:107)x(cid:107)2/2 dx (cid:0)ξ ∼ N(0,I )(cid:1)
0 n−1 n×n
(2π)n/2
A
Approx’n: For large n
2logn
E{N } ∼
n
π
Bound: For large n
2logn 14
E{N } ≤ + .
n
π π
Real roots of algebraic equations (Kac, 1943)
f(t) = ξ +ξ t +ξ t2+···+ξ tn−1
0 1 2 n−1
P{ξ = (ξ ,...,ξ ) ∈ A} = (cid:90) e−(cid:107)x(cid:107)2/2 dx (cid:0)ξ ∼ N(0,I )(cid:1)
0 n−1 n×n
(2π)n/2
A
Theorem: N = the number of real zeroes of f
n
4 (cid:90) 1 [1−n2[x2(1−x2)/(1−x2n]2]1/2
E{N } = dx
n π 1−x2
0
Bound: For large n
2logn 14
E{N } ≤ + .
n
π π
Real roots of algebraic equations (Kac, 1943)
f(t) = ξ +ξ t +ξ t2+···+ξ tn−1
0 1 2 n−1
P{ξ = (ξ ,...,ξ ) ∈ A} = (cid:90) e−(cid:107)x(cid:107)2/2 dx (cid:0)ξ ∼ N(0,I )(cid:1)
0 n−1 n×n
(2π)n/2
A
Theorem: N = the number of real zeroes of f
n
4 (cid:90) 1 [1−n2[x2(1−x2)/(1−x2n]2]1/2
E{N } = dx
n π 1−x2
0
Approx’n: For large n
2logn
E{N } ∼
n
π
Real roots of algebraic equations (Kac, 1943)
f(t) = ξ +ξ t +ξ t2+···+ξ tn−1
0 1 2 n−1
P{ξ = (ξ ,...,ξ ) ∈ A} = (cid:90) e−(cid:107)x(cid:107)2/2 dx (cid:0)ξ ∼ N(0,I )(cid:1)
0 n−1 n×n
(2π)n/2
A
Theorem: N = the number of real zeroes of f
n
4 (cid:90) 1 [1−n2[x2(1−x2)/(1−x2n]2]1/2
E{N } = dx
n π 1−x2
0
Approx’n: For large n
2logn
E{N } ∼
n
π
Bound: For large n
2logn 14
E{N } ≤ + .
n
π π
Shiffman
Zeroes of complex polynomials
f(z) = ξ +a ξ z+a ξ z2+···+a ξ zn−1, z ∈ C.
0 1 1 2 2 n−1 n−1