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Gaussian fields and Kac-Rice formulae PDF

147 Pages·2011·1.92 MB·English
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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

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I: Gaussian fields and Kac-Rice formulae. Robert Adler. Electrical Engineering. Technion – Israel Institute of Technology . and many, many others.
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