Random Fields and Random Geometry. I: Gaussian fields and Kac-Rice formulae

Transcription

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2 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

3 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.)

4 Our heroes Marc Kac Stephen O. Rice

5 Real roots of algebraic equations (Kac, 1943) f (t) = ξ 0 + ξ 1 t + ξ 2 t ξ n 1 t n 1

6 Real roots of algebraic equations (Kac, 1943) f (t) = ξ 0 + ξ 1 t + ξ 2 t ξ n 1 t n 1 P{ξ = (ξ 0,..., ξ n 1 ) A} = A e x 2 /2 (2π) n/2 dx ( ξ N(0, In n ) )

7 Real roots of algebraic equations (Kac, 1943) f (t) = ξ 0 + ξ 1 t + ξ 2 t ξ n 1 t n 1 P{ξ = (ξ 0,..., ξ n 1 ) A} = Theorem: N n = the number of real zeroes of f A e x 2 /2 (2π) n/2 dx ( ξ N(0, In n ) ) E{N n } = 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx

8 Real roots of algebraic equations (Kac, 1943) f (t) = ξ 0 + ξ 1 t + ξ 2 t ξ n 1 t n 1 P{ξ = (ξ 0,..., ξ n 1 ) A} = Theorem: N n = the number of real zeroes of f A e x 2 /2 (2π) n/2 dx ( ξ N(0, In n ) ) E{N n } = Approx n: For large n 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx E{N n } 2 log n π

9 Real roots of algebraic equations (Kac, 1943) f (t) = ξ 0 + ξ 1 t + ξ 2 t ξ n 1 t n 1 P{ξ = (ξ 0,..., ξ n 1 ) A} = Theorem: N n = the number of real zeroes of f A e x 2 /2 (2π) n/2 dx ( ξ N(0, In n ) ) E{N n } = Approx n: For large n Bound: For large n 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx E{N n } 2 log n π E{N n } 2 log n π + 14 π.

10 Zeroes of complex polynomials f (z) = ξ 0 +a 1 ξ 1 z+a 2 ξ 2 z 2 + +a n 1 ξ n 1 z n 1, z C.

11 Zeroes of complex polynomials f (z) = ξ 0 +a 1 ξ 1 z+a 2 ξ 2 z 2 + +a n 1 ξ n 1 z n 1, z C.

12 Zeroes of complex polynomials f (z) = ξ 0 +a 1 ξ 1 z+a 2 ξ 2 z 2 + +a n 1 ξ n 1 z n 1, z C.

13 Zeroes of complex polynomials f (z) = ξ 0 +a 1 ξ 1 z+a 2 ξ 2 z 2 + +a n 1 ξ n 1 z n 1, z C. Shiffman

14 Thinking more generally: 1: f : R N R N, random { } E #{t R N : f (t) = u} =?

15 Thinking more generally: 1: f : R N R N, random { } E #{t R N : f (t) = u} =? 2: f : M N, random, dim(m) = dim(n) E {#{p M : f (p) = q}} =?

16 Thinking more generally: 1: f : R N R N, random { } E #{t R N : f (t) = u} =? 2: f : M N, random, dim(m) = dim(n) E {#{p M : f (p) = q}} =? 3: In another notation: E { #{f 1 (q)} } =?

17 Thinking more generally: 1: f : R N R N, random { } E #{t R N : f (t) = u} =? 2: f : M N, random, dim(m) = dim(n) E {#{p M : f (p) = q}} =? 3: In another notation: E { #{f 1 (q)} } =? 4: More generally: f : M N, dim(m) dim(n), D N In this case, typically, dim ( f 1 (D) ) = dim(m) dim(n) + dim(d), and it is not clear what the corresponding question is.

18 The original (non-specific) Rice formula U u U u (f, T ) = #{t T : f (t) = u, ḟ (t) > 0}

19 The original (non-specific) Rice formula U u U u (f, T ) = #{t T : f (t) = u, ḟ (t) > 0} Basic conditions on f : Continuity and differentiability

20 The original (non-specific) Rice formula U u U u (f, T ) = #{t T : f (t) = u, ḟ (t) > 0} Basic conditions on f : Continuity and differentiability Plus (boring) side conditions to be met later

21 The original (non-specific) Rice formula U u U u (f, T ) = #{t T : f (t) = u, ḟ (t) > 0} Basic conditions on f : Continuity and differentiability Plus (boring) side conditions to be met later Kac-Rice formula: E {U u } = y p t (u, y) dydt T 0 = p t (u) y p t (y u) dydt T 0 { = p t (u) E ḟ (t) 1 (0, ) (ḟ (t)) } f (t) = u dt T where p t (x, y) is the joint density of (f (t), ḟ (t)), p t (u) is the probability density of f (t), etc.

22 The original (non-specific) Rice formula U u U u (f, T ) = #{t T : f (t) = u, ḟ (t) > 0} Basic conditions on f : Continuity and differentiability Plus (boring) side conditions to be met later Kac-Rice formula: E {U u } = y p t (u, y) dydt T 0 = p t (u) y p t (y u) dydt T 0 { = p t (u) E ḟ (t) 1 (0, ) (ḟ (t)) } f (t) = u dt T where p t (x, y) is the joint density of (f (t), ḟ (t)), p t (u) is the probability density of f (t), etc. T = [0, T ] is an interval. M is a general set (e.g. manifold)

23 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1.

24 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1.

25 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1. 1 = R δ ε (x u) dx = t u 1 t l 1 ḟ (t) δ ε (f (t) u) 1 (0, ) (f (t)) dt = lim ε 0 t u 1 t l 1 ḟ (t) δ ε(f (t) u) 1 (0, ) (f (t)) dt

26 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1. 1 = R δ ε (x u) dx = t u 1 t l 1 ḟ (t) δ ε (f (t) u) 1 (0, ) (f (t)) dt = lim ε 0 t u 1 t l 1 ḟ (t) δ ε(f (t) u) 1 (0, ) (f (t)) dt U u (f, T ) = lim ḟ (t) δ ε 0 ε(f (t) u) 1 (0, ) (f (t)) dt T

27 The original (non-specific) Rice formula: The proof So far, everything is deterministic U u (f, T ) = lim ḟ (t) δ ε (f (t) u) 1 (0, ) (f (t)) dt ε 0 T

28 The original (non-specific) Rice formula: The proof So far, everything is deterministic U u (f, T ) = lim ḟ (t) δ ε (f (t) u) 1 (0, ) (f (t)) dt ε 0 T Take expectations, with some sleight of hand { E{U u (f, T )} = E lim ε 0 T = = = T { lim E ε 0 lim T ε 0 x= T 0 } ḟ (t) δ ε(f (t) u) 1 (0, ) (f (t)) dt } ḟ (t) δ ε (f (t) u) 1 (0, ) (f (t)) dt y=0 y p t (u, y) dydt y δ ε (x u) p t (x, y) dxdy dt

29 Constructing Gaussian Processes A (separable) parameter space M.

30 Constructing Gaussian Processes A (separable) parameter space M. A finite or infinite set of functions ϕ 1, ϕ 2,..., ϕ j : M R satisfying ϕ 2 j (t) <, for all t M. j

31 Constructing Gaussian Processes A (separable) parameter space M. A finite or infinite set of functions ϕ 1, ϕ 2,..., ϕ j : M R satisfying ϕ 2 j (t) <, for all t M. j A sequence ξ 1, ξ 2,..., of independent, N(0, 1) random variables,

32 Constructing Gaussian Processes A (separable) parameter space M. A finite or infinite set of functions ϕ 1, ϕ 2,..., ϕ j : M R satisfying ϕ 2 j (t) <, for all t M. j A sequence ξ 1, ξ 2,..., of independent, N(0, 1) random variables, Define the random field f : M R by f (t) = j ξ j ϕ j (t)

33 Constructing Gaussian Processes A (separable) parameter space M. A finite or infinite set of functions ϕ 1, ϕ 2,..., ϕ j : M R satisfying ϕ 2 j (t) <, for all t M. j A sequence ξ 1, ξ 2,..., of independent, N(0, 1) random variables, Define the random field f : M R by f (t) = j ξ j ϕ j (t) Mean, covariance, and variance functions µ(t) = E{f (t)} = 0 C(s, t) = {[f (s) µ(s)] [f (t) µ(t)]} = ϕ j (s)ϕ j (t) σ 2 (t) = C(t, t) = ϕ 2 j (t)

34 Existence of Gaussian processes Theorem Let M be a topological space. Let C : M M be positive semi-definite. Then there exists a Gaussian process on f : M R with mean zero and covariance function C. Furthermore, f has a representation of the form f (t) = j ξ jϕ j (t), and if f is a.s. continuous then the sum converges uniformly, a.s., on compacts.

35 Existence of Gaussian processes Theorem Let M be a topological space. Let C : M M be positive semi-definite. Then there exists a Gaussian process on f : M R with mean zero and covariance function C. Furthermore, f has a representation of the form f (t) = j ξ jϕ j (t), and if f is a.s. continuous then the sum converges uniformly, a.s., on compacts. Corollary If there is justice in the world (smoothness and summability) ḟ (t) = t f (t) = j ξ j ϕ j (t), and so if f is Gaussian, so is ḟ. Where ḟ is any derivative on nice M.

36 Existence of Gaussian processes Theorem Let M be a topological space. Let C : M M be positive semi-definite. Then there exists a Gaussian process on f : M R with mean zero and covariance function C. Furthermore, f has a representation of the form f (t) = j ξ jϕ j (t), and if f is a.s. continuous then the sum converges uniformly, a.s., on compacts. Corollary If there is justice in the world (smoothness and summability) ḟ (t) = t f (t) = j ξ j ϕ j (t), and so if f is Gaussian, so is ḟ. Where ḟ is any derivative on nice M. Furthermore E{ḟ (s)ḟ (t)} = E{ ξ j ϕ j (s) ξ k ϕ k (t) } = ϕ j (s) ϕ j (t) E{f (s) ḟ (t)} = E { ξ j ϕ j (s) ξ k ϕ k (t) } = ϕ j (s) ϕ j (t)

37 Example: The Brownian sheet on RN

38 Example: The Brownian sheet on R N E{W (t)} = 0 E{W (s)w (t)} = (s 1 t 1 ) (s N t N ). where t = (t 1,..., t N ).

39 Example: The Brownian sheet on R N E{W (t)} = 0 E{W (s)w (t)} = (s 1 t 1 ) (s N t N ). where t = (t 1,..., t N ). Replace j by j = (j 1,..., j N ) ϕ j (t) = 2 N/2 N i=1 2 (2j i + 1)π sin ( 1 2 (2j i + 1) πt i ) W (t) = ξ j1,...,j N ϕ j1,...,j N (t) j 1 j N

40 Example: The Brownian sheet on R N E{W (t)} = 0 E{W (s)w (t)} = (s 1 t 1 ) (s N t N ). where t = (t 1,..., t N ). Replace j by j = (j 1,..., j N ) ϕ j (t) = 2 N/2 N i=1 2 (2j i + 1)π sin ( 1 2 (2j i + 1) πt i ) W (t) = ξ j1,...,j N ϕ j1,...,j N (t) j 1 N = 1 W is standard Brownian motion. The corresponding expansion is due to Lévy, and the corresponding RKHS is known as Cameron-Martin space. j N

41 Constant variance Gaussian processes We know that f (t) = ξj ϕ j (t) ḟ (t) = ξ j ϕ j (t) E{ḟ (s)ḟ (t)} = ϕ j (s) ϕ j (t) E{f (s)ḟ (t)} = ϕ j (s) ϕ j (t)

42 Constant variance Gaussian processes We know that f (t) = ξj ϕ j (t) ḟ (t) = ξ j ϕ j (t) E{ḟ (s)ḟ (t)} = ϕ j (s) ϕ j (t) E{f (s)ḟ (t)} = ϕ j (s) ϕ j (t) Thus, if σ 2 (t) = E{f 2 (t)} = ϕ 2 j (t) is a constant, say 1, so that {ϕ j (t)} l2 = 1

43 Constant variance Gaussian processes We know that f (t) = ξj ϕ j (t) ḟ (t) = ξ j ϕ j (t) E{ḟ (s)ḟ (t)} = ϕ j (s) ϕ j (t) E{f (s)ḟ (t)} = ϕ j (s) ϕ j (t) Thus, if σ 2 (t) = E{f 2 (t)} = ϕ 2 j (t) is a constant, say 1, so that {ϕ j (t)} l2 = 1 then E{f (t)ḟ (t)} = ϕ j (t) ϕ j (t) = 1 2 t ϕ2 j (t) = 1 2 ϕ 2 t j (t) = 0

44 Constant variance Gaussian processes We know that f (t) = ξj ϕ j (t) ḟ (t) = ξ j ϕ j (t) E{ḟ (s)ḟ (t)} = ϕ j (s) ϕ j (t) E{f (s)ḟ (t)} = ϕ j (s) ϕ j (t) Thus, if σ 2 (t) = E{f 2 (t)} = ϕ 2 j (t) is a constant, say 1, so that {ϕ j (t)} l2 = 1 then E{f (t)ḟ (t)} = ϕ j (t) ϕ j (t) = 1 2 t ϕ2 j (t) = 1 2 ϕ 2 t j (t) = 0 f (t) and its derivative ḟ (t) are INDEPENDENT. (uncorrelated)

45 Constant variance Gaussian processes and Kac-Rice Generic Kac-Rice formula { E {U u } = p t (u) E ḟ (t) 1 (0, ) (ḟ (t)) } f (t) = u dt T

46 Constant variance Gaussian processes and Kac-Rice Generic Kac-Rice formula { E {U u } = p t (u) E ḟ (t) 1 (0, ) (ḟ (t)) } f (t) = u dt T But f (t) and ḟ (t) are independent!

47 Constant variance Gaussian processes and Kac-Rice Generic Kac-Rice formula { E {U u } = p t (u) E ḟ (t) 1 (0, ) (ḟ (t)) } f (t) = u dt T But f (t) and ḟ (t) are independent! Notation 1 = σ 2 (t), λ(t) = E{[ḟ (t)] 2 }

48 Constant variance Gaussian processes and Kac-Rice Generic Kac-Rice formula { E {U u } = p t (u) E ḟ (t) 1 (0, ) (ḟ (t)) } f (t) = u dt T But f (t) and ḟ (t) are independent! Notation 1 = σ 2 (t), λ(t) = E{[ḟ (t)] 2 } Rice formula (+ε) E {U u } = e u2 /2 2π T [λ(t)] 1/2 dt.

49 Constant variance Gaussian processes and Kac-Rice Generic Kac-Rice formula { E {U u } = p t (u) E ḟ (t) 1 (0, ) (ḟ (t)) } f (t) = u dt T But f (t) and ḟ (t) are independent! Notation Rice formula (+ε) 1 = σ 2 (t), λ(t) = E{[ḟ (t)] 2 } E {U u } = e u2 /2 2π If λ(t) λ, we have the Rice formula T E {U u } = T λ1/2 2π [λ(t)] 1/2 dt. e u2 /2

50 Gaussian Kac-Rice with no simplification E {U u (T )} = T λ 1/2 (t)σ 1 (t)[1 µ 2 (t)] 1/2 ϕ ( ) m(t) σ(t) [ 2ϕ(η(t)) + 2η(t)[2Φ(η(t)) 1] ] dt

51 Gaussian Kac-Rice with no simplification E {U u (T )} = T λ 1/2 (t)σ 1 (t)[1 µ 2 (t)] 1/2 ϕ ( ) m(t) σ(t) [ 2ϕ(η(t)) + 2η(t)[2Φ(η(t)) 1] ] dt m(t) = E{f (t)} σ 2 (t) = E{[f (t)] 2 } λ(t) = E{[ḟ (t)] 2 } µ(t) = E{[f (t) m(t)] [ḟ (t)]} λ 1/2 (t)σ(t) η(t) = ṁ(t) λ1/2 (t)µ(t)m(t)/σ(t) λ 1/2 (t)[1 µ 2 (t)] 1/2

52 Gaussian Kac-Rice with no simplification E {U u (T )} = T λ 1/2 (t)σ 1 (t)[1 µ 2 (t)] 1/2 ϕ ( ) m(t) σ(t) [ 2ϕ(η(t)) + 2η(t)[2Φ(η(t)) 1] ] dt m(t) = E{f (t)} σ 2 (t) = E{[f (t)] 2 } λ(t) = E{[ḟ (t)] 2 } µ(t) = E{[f (t) m(t)] [ḟ (t)]} λ 1/2 (t)σ(t) η(t) = ṁ(t) λ1/2 (t)µ(t)m(t)/σ(t) λ 1/2 (t)[1 µ 2 (t)] 1/2 Very important fact: Long term covariances do not appear in any of these formulae.

53 Some pertinent thoughts Real roots of Gaussian polynomials The original Kac result now makes sense: f (t) = ξ 0 + ξ 1 t + + ξ n 1 t n 1 E{N n } = 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx

54 Some pertinent thoughts Real roots of Gaussian polynomials The original Kac result now makes sense: f (t) = ξ 0 + ξ 1 t + + ξ n 1 t n 1 E{N n } = 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx Downcrossings, crossings, critical points Critical points of different kinds are just zeroes of ḟ with different side conditions. But now second derivatives appear, calculations will be harder. (f (t), f (t) are not independent.)

55 Some pertinent thoughts Real roots of Gaussian polynomials The original Kac result now makes sense: f (t) = ξ 0 + ξ 1 t + + ξ n 1 t n 1 E{N n } = 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx Downcrossings, crossings, critical points Critical points of different kinds are just zeroes of ḟ with different side conditions. But now second derivatives appear, calculations will be harder. (f (t), f (t) are not independent.) Weakening the conditions In the Gaussian case, only absolute continuity and the finiteness of λ is needed. (Itô, Ylvisaker)

56 Some pertinent thoughts Real roots of Gaussian polynomials The original Kac result now makes sense: f (t) = ξ 0 + ξ 1 t + + ξ n 1 t n 1 E{N n } = 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx Downcrossings, crossings, critical points Critical points of different kinds are just zeroes of ḟ with different side conditions. But now second derivatives appear, calculations will be harder. (f (t), f (t) are not independent.) Weakening the conditions In the Gaussian case, only absolute continuity and the finiteness of λ is needed. (Itô, Ylvisaker) Higher moments?

57 Some pertinent thoughts Real roots of Gaussian polynomials The original Kac result now makes sense: f (t) = ξ 0 + ξ 1 t + + ξ n 1 t n 1 E{N n } = 4 π 1 0 [1 n 2 [x 2 (1 x 2 )/(1 x 2n ] 2 ] 1/2 1 x 2 dx Downcrossings, crossings, critical points Critical points of different kinds are just zeroes of ḟ with different side conditions. But now second derivatives appear, calculations will be harder. (f (t), f (t) are not independent.) Weakening the conditions In the Gaussian case, only absolute continuity and the finiteness of λ is needed. (Itô, Ylvisaker) Higher moments? Fixed points of vector valued processes?

58 The Kac-Rice Metatheorem

59 The Kac-Rice Metatheorem

60 The Kac-Rice Metatheorem The setup f = (f 1,..., f N ) : M R N R N g = (g 1,..., g K ) : M R N R K

61 The Kac-Rice Metatheorem The setup Number of points: f = (f 1,..., f N ) : M R N R N g = (g 1,..., g K ) : M R N R K N u N u (M) N u (f, g : M, B) = E { # {t M : f (t) = u, g(t) B} }.

62 The Kac-Rice Metatheorem The setup Number of points: f = (f 1,..., f N ) : M R N R N g = (g 1,..., g K ) : M R N R K N u N u (M) N u (f, g : M, B) = E { # {t M : f (t) = u, g(t) B} }. The metatheorem, or generalised Kac-Rice E{N u } = det y 1 B (v) p t (u, y, v) d( y) dv dt M R { D } = E det f (t) 1 B (g(t)) f (t) = u p t (u) dt, M p t(x, y, v) is the joint density of (f t, f t, g t) ( f )(t) f (t) ( f i j (t) ) i,j=1,...,n ( f i (t) t j )i,j=1,...,n.

63 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1.

64 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1.

65 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1. 1 = R δ ε (x u) dx = t u 1 t l 1 ḟ (t) δ ε (f (t) u) 1 (0, ) (f (t)) dt = lim ε 0 t u 1 t l 1 ḟ (t) δ ε(f (t) u) 1 (0, ) (f (t)) dt

66 The original (non-specific) Rice formula: The proof Take a (positive) approximate delta function, δ ε, supported on [ ε, +ε], and R δ ε(x) dx = 1. 1 = R δ ε (x u) dx = t u 1 t l 1 ḟ (t) δ ε (f (t) u) 1 (0, ) (f (t)) dt = lim ε 0 t u 1 t l 1 ḟ (t) δ ε(f (t) u) 1 (0, ) (f (t)) dt U u (f, T ) = lim ḟ (t) δ ε 0 ε(f (t) u) 1 (0, ) (f (t)) dt T

67 The Kac-Rice Conditions (the fine print).

68 The Kac-Rice Conditions (the fine print). Let f, g, M and B be as above, with the additional assumption that the boundaries of M and B have finite N 1 and K 1 dimensional measures, respectively. Furthermore, assume that the following conditions are satisfied for some u R N : (a) All components of f, f, and g are a.s. continuous and have finite variances (over M). (b) For all t M, the marginal densities p t (x) of f (t) (implicitly assumed to exist) are continuous at x = u. (c) The conditional densities p t (x f (t), g(t)) of f (t) given g(t) and f (t) (implicitly assumed to exist) are bounded above and continuous at x = u, uniformly in t M. (d) The conditional densities p t (z f (t) = x) of det f (t) given f (t) = x, are continuous for z and x in neighbourhoods of 0 and u, respectively, uniformly in t M. (e) The conditional densities p t (z f (t) = x) of g(t) given f (t) = x, are continuous for all z and for x in a neighbourhood u, uniformly in t M. (f) The following moment condition holds: { f sup max E i j (t) N } <. t M 1 i,j N (g) The moduli of continuity of each of the components of f, f, and g satisfy ( P { ω(η) > ε } = o η N ), as η 0, for any ε > 0.

69 Higher (factorial) moments Factorial notation (x) k = x(x 1)... (x k + 1).

70 Higher (factorial) moments Factorial notation (x) k = x(x 1)... (x k + 1). Kac-Rice (again?) E{(N u ) k } = = M k E { k det f (t j ) 1 B (g(t j )) f } ( t) = ũ p t (ũ)d t j=1 M k R kd j=1 k detd j 1 B (v j ) p t (ũ, D, ṽ) d D dṽ d t,

71 Higher (factorial) moments Factorial notation (x) k = x(x 1)... (x k + 1). Kac-Rice (again?) E{(N u ) k } = = M k E M k { k det f (t j ) 1 B (g(t j )) f } ( t) = ũ p t (ũ)d t j=1 R kd j=1 k detd j 1 B (v j ) p t (ũ, D, ṽ) d D dṽ d t, M k = { t = (t 1,..., t k ) : t j M, 1 j k} f ( t) = (f (t 1),..., f (t k )) : M k R Nk g( t) = (g(t 1),..., g(t k )) : M k R Kk D = N(N + 1)/2 + K

72 The Gaussian case: What can/can t be explicitly computed General mean and covariance functions

73 The Gaussian case: What can/can t be explicitly computed General mean and covariance functions Isotropic fields (N = 2, 3)

74 The Gaussian case: What can/can t be explicitly computed General mean and covariance functions Isotropic fields (N = 2, 3) Zero mean and constant variance (via the induced metric )

75 The Gaussian case: What can/can t be explicitly computed General mean and covariance functions Isotropic fields (N = 2, 3) Zero mean and constant variance (via the induced metric ) Gaussian related processes

76 The Gaussian case: What can/can t be explicitly computed General mean and covariance functions Isotropic fields (N = 2, 3) Zero mean and constant variance (via the induced metric ) Gaussian related processes Perturbed Gaussian processes ( Approximately )

77 The Gaussian case: What can/can t be explicitly computed General mean and covariance functions Isotropic fields (N = 2, 3) Zero mean and constant variance (via the induced metric ) Gaussian related processes Perturbed Gaussian processes ( Approximately ) E{No. of critical points of index k above the level u}

78 The Gaussian case: What can/can t be explicitly computed General mean and covariance functions Isotropic fields (N = 2, 3) Zero mean and constant variance (via the induced metric ) Gaussian related processes Perturbed Gaussian processes ( Approximately ) E{No. of critical points of index k above the level u} N } E{ k=1 ( 1)k (No. of critical points of index k above u)) Useful for Morse theory

79 The Gaussian-related case

80 The Gaussian-related case f (t) = (f 1 (t),..., f k (t)) : T R k F : R k R d

81 The Gaussian-related case f (t) = (f 1 (t),..., f k (t)) : T R k F : R k R d g(t) = F (g(t)) = F (g 1 (t),..., g k (t)),

82 The Gaussian-related case f (t) = (f 1 (t),..., f k (t)) : T R k F : R k R d g(t) = F (g(t)) = F (g 1 (t),..., g k (t)), F (x) = k xi 2, 1 x 1 k 1 ( k 2 x, m n 1 x i 2 i 2)1/2 n n+m n+1 x 2 i i.e. χ 2 fields with k degrees of freedom, T field with k 1 degrees of freedom, F field with n and m degrees of freedom..

83 The Gaussian Kinematic Formula (GKF)

84 The Gaussian Kinematic Formula (GKF) Jonathan s lecture

85 The perturbed-gaussian case A physics approach [ ] ϕ(x) = ϕ G (x) 1 + Tr [E G {h n (X )} h n (x)] ϕ G is iid Gaussian n=3 h n (x) = ( 1) n 1 n ϕ G (x) ϕ G (x) x n are Hermite tensors of rank n with coefficients constructed from the moments E G {h n (X )}

86 The perturbed-gaussian case A physics approach [ ] ϕ(x) = ϕ G (x) 1 + Tr [E G {h n (X )} h n (x)] ϕ G is iid Gaussian n=3 h n (x) = ( 1) n 1 n ϕ G (x) ϕ G (x) x n are Hermite tensors of rank n with coefficients constructed from the moments E G {h n (X )} A statistical (Gaussian related) approach f (t) = f G (t) + J j=1 p j ε j fj GR (t)

87 Applications I: Exceedence probabilities via level crossings P { sup f (t) u } 0 t T

88 Applications I: Exceedence probabilities via level crossings P { sup 0 t T f (t) u }

89 Applications I: Exceedence probabilities via level crossings P { sup 0 t T f (t) u } P { sup f (t) u } = P {f (0) u} + P {f (0) < u, N u 1} 0 t T = P {f (0) u) + P {f (0) < u, N u 1} P {f (0) u) + E{N u } = E{# of connected components in A u (T )}

90 Applications I: Exceedence probabilities via level crossings P { sup 0 t T f (t) u } P { sup f (t) u } = P {f (0) u} + P {f (0) < u, N u 1} 0 t T = P {f (0) u) + P {f (0) < u, N u 1} Note: Nothing is Gaussian here! P {f (0) u) + E{N u } = E{# of connected components in A u (T )}

91 Applications I: Exceedence probabilities via level crossings P { sup 0 t T f (t) u } P { sup f (t) u } = P {f (0) u} + P {f (0) < u, N u 1} 0 t T = P {f (0) u) + P {f (0) < u, N u 1} Note: Nothing is Gaussian here! P {f (0) u) + E{N u } = E{# of connected components in A u (T )} Inequality is usually an approximation, for large u.

92 Applications II: Local maxima on the line Number of local maxima above the level u { } M u (T ) = # t [0, T ] : ḟ (t) = 0, f (t) < 0, f (t) u

93 Applications II: Local maxima on the line Number of local maxima above the level u { } M u (T ) = # t [0, T ] : ḟ (t) = 0, f (t) < 0, f (t) u f stationary mean 0, variance 1, λ 4 = E{[f (t)] 4 } E {M (T )} = T λ1/2 4 2πλ 1/2 2

94 Applications II: Local maxima on the line Number of local maxima above the level u { } M u (T ) = # t [0, T ] : ḟ (t) = 0, f (t) < 0, f (t) u f stationary mean 0, variance 1, λ 4 = E{[f (t)] 4 } E {M (T )} = T λ1/2 4 2πλ 1/2 2 Similarly, with = λ 4 λ 2 2 and λ 2 λ. ( ) E {M u (T )} = T λ1/2 4 λ 1/2 4 u Ψ 1/2 2πλ 1/2 2 ( ) T λ1/2 2 λ2 u ϕ(u)φ 2π 1/2,

95 Applications II: Local maxima on the line Number of local maxima above the level u { } M u (T ) = # t [0, T ] : ḟ (t) = 0, f (t) < 0, f (t) u f stationary mean 0, variance 1, λ 4 = E{[f (t)] 4 } E {M (T )} = T λ1/2 4 2πλ 1/2 2 Similarly, with = λ 4 λ 2 2 and λ 2 λ. ( ) E {M u (T )} = T λ1/2 4 λ 1/2 4 u Ψ 1/2 2πλ 1/2 2 An easy computation E {M u (T )} lim u E {N u (T )} = 1. ( ) T λ1/2 2 λ2 u ϕ(u)φ 2π 1/2,

96 Applications II: Local maxima on the line Number of local maxima above the level u { } M u (T ) = # t [0, T ] : ḟ (t) = 0, f (t) < 0, f (t) u f stationary mean 0, variance 1, λ 4 = E{[f (t)] 4 } E {M (T )} = T λ1/2 4 2πλ 1/2 2 Similarly, with = λ 4 λ 2 2 and λ 2 λ. ( ) E {M u (T )} = T λ1/2 4 λ 1/2 4 u Ψ 1/2 2πλ 1/2 2 An easy computation E {M u (T )} lim u E {N u (T )} = 1. which holds in very wide generality. ( ) T λ1/2 2 λ2 u ϕ(u)φ 2π 1/2,

97 Applications III: Local maxima on M R N M u (M) Number of local maxima on M above the level u

98 Applications III: Local maxima on M R N M u (M) Number of local maxima on M above the level u N = 2 f isotropic on unit square E {M } = 1 6π 3 λ 4 λ 2

99 Applications III: Local maxima on M R N M u (M) Number of local maxima on M above the level u N = 2 f isotropic on unit square E {M } = N = 2, f stationary on unit square 1 6π 3 λ 4 λ 2 E {M } = 1 2π 2 d 1 Λ 1/2 G( d 1/d 2 ) Λ the covariance matrix of first order derivatives of f d 1, d 2 eigenvalues of cov matrix of second order derivatives G involves first and second order elliptic integrals

100 Applications III: Local maxima on M R N M u (M) Number of local maxima on M above the level u N = 2 f isotropic on unit square E {M } = N = 2, f stationary on unit square 1 6π 3 λ 4 λ 2 E {M } = 1 2π 2 d 1 Λ 1/2 G( d 1/d 2 ) Λ the covariance matrix of first order derivatives of f d 1, d 2 eigenvalues of cov matrix of second order derivatives G involves first and second order elliptic integrals N = 2, f isotropic on unit square E {M u } =???

101 Applications III: Local maxima on M R N M u (M) Number of local maxima on M above the level u N = 2 f isotropic on unit square E {M } = N = 2, f stationary on unit square 1 6π 3 λ 4 λ 2 E {M } = 1 2π 2 d 1 Λ 1/2 G( d 1/d 2 ) Λ the covariance matrix of first order derivatives of f d 1, d 2 eigenvalues of cov matrix of second order derivatives G involves first and second order elliptic integrals N = 2, f isotropic on unit square Is there a replacement for E {M u } =??? E {M u (T )} lim u E {N u (T )} = 1.

102 Applications IV: Longuet-Higgins and oceanography

103 Applications IV: Longuet-Higgins and oceanography

104 Applications IV: Longuet-Higgins and oceanography There are precise computations for the expected numbers of specular points, mainly by M.S. Longuet-Higgins,

105 Applications IV: Longuet-Higgins and oceanography There are precise computations for the expected numbers of specular points, mainly by M.S. Longuet-Higgins, M.S. Longuet-Higgins, 1925

106 Applications IV: Longuet-Higgins and oceanography There are precise computations for the expected numbers of specular points, mainly by M.S. Longuet-Higgins, Mark Dennis M.S. Longuet-Higgins, 1925

107 Applications V: Higher moments and complex polynomials f (z) = ξ 0 +a 1 ξ 1 z+a 2 ξ 2 z 2 + +a n 1 ξ n 1 z n 1, z C.

108 Applications V: Higher moments and complex polynomials f (z) = ξ 0 +a 1 ξ 1 z+a 2 ξ 2 z 2 + +a n 1 ξ n 1 z n 1, z C. Means tell us where we expect the roots to be, but variances are needed to give concentration information. Balint Virag

109 Applications VI: Poisson limits and Slepian models

110 Applications VI: Poisson limits and Slepian models Theorem 1: Sequences of increasingly rare events such as the existence of high level local maxima in N dimensions or level crossings in 1 dimension, looked at over long time periods or large regions so that a few of them still occur have an asymptotic Poisson distribution as long as dependence in time or space is not too strong. 2: The normalisations and the parameters of the limiting Poisson depend only on the expected number of events in a given region or time interval.

111 Applications VI: Poisson limits and Slepian models Theorem 1: Sequences of increasingly rare events such as the existence of high level local maxima in N dimensions or level crossings in 1 dimension, looked at over long time periods or large regions so that a few of them still occur have an asymptotic Poisson distribution as long as dependence in time or space is not too strong. 2: The normalisations and the parameters of the limiting Poisson depend only on the expected number of events in a given region or time interval. Theorem If f is stationary and ergodic although less will do P { f (τ) A correct τ is a local maximum of f }

112 Applications VI: Poisson limits and Slepian models Theorem 1: Sequences of increasingly rare events such as the existence of high level local maxima in N dimensions or level crossings in 1 dimension, looked at over long time periods or large regions so that a few of them still occur have an asymptotic Poisson distribution as long as dependence in time or space is not too strong. 2: The normalisations and the parameters of the limiting Poisson depend only on the expected number of events in a given region or time interval. Theorem If f is stationary and ergodic although less will do P { f (τ) A correct τ is a local maximum of f } = E{#{t B : t is a local maximum of f and f (t) A}} E{#{t B : t is a local maximum of f }} B is any ball

113 Applications VII: Eigenvalues of random matrices A a n n matrix

114 Applications VII: Eigenvalues of random matrices A a n n matrix Define f A (t) = At, t, t M If A is random, then f A is a random field. If A is Gaussian (i.e. has Gaussian components) then f A is Gaussian.

115 Applications VII: Eigenvalues of random matrices A a n n matrix Define f A (t) = At, t, t M If A is random, then f A is a random field. If A is Gaussian (i.e. has Gaussian components) then f A is Gaussian. Algebraic fact If A has no repeated eigenvalues, there are exactly 2n critical points of f A, which occur at ± the eigenvectors of A. The values of f A at critical points are the eigenvalues of M

116 Applications VII: Eigenvalues of random matrices A a n n matrix Define f A (t) = At, t, t M If A is random, then f A is a random field. If A is Gaussian (i.e. has Gaussian components) then f A is Gaussian. Algebraic fact If A has no repeated eigenvalues, there are exactly 2n critical points of f A, which occur at ± the eigenvectors of A. The values of f A at critical points are the eigenvalues of M Finding the maximum eigenvalue λ max (A) = sup f A (t) t S n 1

117 Applications VII: Eigenvalues of random matrices A a n n matrix Define f A (t) = At, t, t M If A is random, then f A is a random field. If A is Gaussian (i.e. has Gaussian components) then f A is Gaussian. Algebraic fact If A has no repeated eigenvalues, there are exactly 2n critical points of f A, which occur at ± the eigenvectors of A. The values of f A at critical points are the eigenvalues of M Finding the maximum eigenvalue λ max (A) = sup f A (t) t S n 1 (Some) random matrix problems are equivalent to random field problems, and vice versa

118 Appendix I: The canonical Gaussian process Consider f (t) = l j=1 ξ jϕ j (t)

119 Appendix I: The canonical Gaussian process Consider f (t) = l j=1 ξ jϕ j (t) σ 2 (t) 1 ϕ(t) = (ϕ 1 (t),..., π l (t)) S l 1.

120 Appendix I: The canonical Gaussian process Consider f (t) = l j=1 ξ jϕ j (t) σ 2 (t) 1 ϕ(t) = (ϕ 1 (t),..., π l (t)) S l 1. A crucial mapping

121 Appendix I: The canonical Gaussian process Consider f (t) = l j=1 ξ jϕ j (t) σ 2 (t) 1 ϕ(t) = (ϕ 1 (t),..., π l (t)) S l 1. A crucial mapping Define a new Gaussian process f on ϕ(m) f (x) = f ( ϕ 1 (x) ),

122 Appendix I: The canonical Gaussian process Consider f (t) = l j=1 ξ jϕ j (t) σ 2 (t) 1 ϕ(t) = (ϕ 1 (t),..., π l (t)) S l 1. A crucial mapping Define a new Gaussian process f on ϕ(m) ( f (x) = f ϕ 1 (x) ), } E { f (x) f (y) = E { f ( ϕ 1 (x) ) f ( ϕ 1 (y) )} = ( ϕ j ϕ 1 (x) ) ( ϕ j ϕ 1 (y) ) = x j y j = x, y

123 The canonical Gaussian process on S l 1 1: Has mean zero and covariance E {f (s)f (s)} = s, t for s, t S l 1. 2: It can be realised as f (t) = l t j ξ j. j=1 3: It is stationary and isotropic since the covariance is function of only the (geodesic) distance between s and t.

124 Exceedence probabilities for canonical process: M S l 1 P { sup f t u } = t M = = = = 0 0 u u u P { sup f t u ξ = r } P ξ (dr) t M P { sup ξ, t u ξ = r } P ξ (dr) t M P { sup ξ, t u } ξ = r P ξ (dr) t M P { sup ξ/r, t u/r ξ = r } P ξ (dr) t M P { sup U, t u/r } P ξ (dr) t M where U is uniform on S l 1.

125 We need P { sup U, t u/r } t M

126 We need P { sup U, t u/r } t M Working with tubes The tube of radius ρ around a closed set M S l 1 ) is { } Tube(M, ρ) = t S l 1 : τ(t, M) ρ { } = t S l 1 : s M such that s, t cos(ρ) { } = t S l 1 : sup s, t cos(ρ) s M.

127 We need P { sup U, t u/r } t M Working with tubes The tube of radius ρ around a closed set M S l 1 ) is { } Tube(M, ρ) = t S l 1 : τ(t, M) ρ { } = t S l 1 : s M such that s, t cos(ρ) { } = t S l 1 : sup s, t cos(ρ) s M. And so... P { sup f t u } t M = u η ( l Tube(M, cos 1 (u/r)) ) P ξ (dr) and geometry has entered the picture, in a serious fashion!

128 Appendix II: Stationary and isotropic fields Definition: M has a group structure, µ(t) = const and C(s, t) = C(s t).

129 Appendix II: Stationary and isotropic fields Definition: M has a group structure, µ(t) = const and C(s, t) = C(s t). Gaussian case: (Weak) stationarity also implies strong stationarity.

130 Appendix II: Stationary and isotropic fields Definition: M has a group structure, µ(t) = const and C(s, t) = C(s t). Gaussian case: (Weak) stationarity also implies strong stationarity. M = R N : C : R N R is non-negative definite there exists a finite measure ν such that C(t) = e i t,λ ν(dλ), R N ν is called the spectral measure and, since C is real, must be symmetric. i.e. ν(a) = ν( A) for all A B N.

131 Appendix II: Stationary and isotropic fields Definition: M has a group structure, µ(t) = const and C(s, t) = C(s t). Gaussian case: (Weak) stationarity also implies strong stationarity. M = R N : C : R N R is non-negative definite there exists a finite measure ν such that C(t) = e i t,λ ν(dλ), R N ν is called the spectral measure and, since C is real, must be symmetric. i.e. ν(a) = ν( A) for all A B N. Spectral moments λ i1...i N = R N λ i 1 1 λ i N N ν(dλ) ν is symmetric odd ordered spectral moments are zero.

132 Elementary considerations give { k f (s) k } f (t) E = s i1 s i1... s ik t i1 t i1... t ik 2k C(s, t) s i1 t i1... s ik t ik.

133 Elementary considerations give { k f (s) k } f (t) E = s i1 s i1... s ik t i1 t i1... t ik 2k C(s, t) s i1 t i1... s ik t ik. When f is stationary, and α, β, γ, δ {0, 1, 2,... }, then { α+β f (t) γ+δ } f (t) E α t i β t j γ t k δ t l = ( 1) α+β α+β+γ+δ α t i β t j γ t k δ C(t) t l t=0 = ( 1) α+β i α+β+γ+δ R N λ α i λ β j λγ k λδ l ν(dλ).

134 Elementary considerations give { k f (s) k } f (t) E = s i1 s i1... s ik t i1 t i1... t ik 2k C(s, t) s i1 t i1... s ik t ik. When f is stationary, and α, β, γ, δ {0, 1, 2,... }, then { α+β f (t) γ+δ } f (t) E α t i β t j γ t k δ t l = ( 1) α+β α+β+γ+δ α t i β t j γ t k δ C(t) t l Write f j = f / t j, t=0 = ( 1) α+β i α+β+γ+δ R N λ α i λ β j λγ k λδ l ν(dλ). f ij = 2 f / t i t j Then f (t) and f j (t) are uncorrelated, f i (t) and f jk (t) are uncorrelated

135 Elementary considerations give { k f (s) k } f (t) E = s i1 s i1... s ik t i1 t i1... t ik 2k C(s, t) s i1 t i1... s ik t ik. When f is stationary, and α, β, γ, δ {0, 1, 2,... }, then { α+β f (t) γ+δ } f (t) E α t i β t j γ t k δ t l = ( 1) α+β α+β+γ+δ α t i β t j γ t k δ C(t) t l Write f j = f / t j, t=0 = ( 1) α+β i α+β+γ+δ R N λ α i λ β j λγ k λδ l ν(dλ). f ij = 2 f / t i t j Then f (t) and f j (t) are uncorrelated, f i (t) and f jk (t) are uncorrelated Isotropy (C(t) = C( t ) ν is spherically symmetric E {f i (t)f j (t)} = E {f (t)f ij (t)} = λδ ij

136 Appendix III: Regularity of Gaussian processes The canonical metric, d d(s, t) = [ E {( f (s) f (t) ) 2}] 1 2, A ball of radius ε and centered at t M is denoted by B d (t, ε) = {s M : d(s, t) ε}.

137 Appendix III: Regularity of Gaussian processes The canonical metric, d d(s, t) = [ E {( f (s) f (t) ) 2}] 1 2, A ball of radius ε and centered at t M is denoted by Compactness assumption B d (t, ε) = {s M : d(s, t) ε}. diam(m) = sup d(s, t) <. s,t M

138 Appendix III: Regularity of Gaussian processes The canonical metric, d d(s, t) = [ E {( f (s) f (t) ) 2}] 1 2, A ball of radius ε and centered at t M is denoted by Compactness assumption B d (t, ε) = {s M : d(s, t) ε}. diam(m) = sup d(s, t) <. s,t M Entropy Fix ε > 0 and let N(M, d, ε) N(ε) denote the smallest number of d-balls of radius ε whose union covers M. Set H(M, d, ε) H(ε) = ln (N(ε)). Then N and H are called the (metric) entropy and log-entropy functions for M (or f ).

139 Dudley s theorem Let f be a centered Gaussian field on a d-compact M Then there exists a universal K such that and where E { sup t M } diam(m) f t K H 1/2 (ε) dε, 0 E {ω f,d (δ)} K ω f,d (δ) = δ 0 H 1/2 (ε) dε, sup f (t) f (s), δ > 0, d(s,t) δ Furthermore, there exists a random η (0, ) and a universal K such that for all δ < η. ω f,d (δ) K δ 0 H 1/2 (ε) dε,

140 Special cases of the entropy result If f is also stationary f is a.s. continuous on M f is a.s. bounded on M δ 0 H 1/2 (ε) dε <, δ > 0

141 Special cases of the entropy result If f is also stationary f is a.s. continuous on M f is a.s. bounded on M If M R N, and p 2 (u) δ = sup E { f s f t 2}, s t u 0 H 1/2 (ε) dε <, δ > 0 continuity & boundedness follow if, for some δ > 0, either δ ( ( ln u) 1 2 dp(u) < or p e u2) du <. 0 δ

142 Special cases of the entropy result If f is also stationary f is a.s. continuous on M f is a.s. bounded on M If M R N, and p 2 (u) δ = sup E { f s f t 2}, s t u 0 H 1/2 (ε) dε <, δ > 0 continuity & boundedness follow if, for some δ > 0, either δ ( ( ln u) 1 2 dp(u) < or p e u2) du <. 0 A sufficient condition For some 0 < K < and α, η > 0, E { f s f t 2} for all s, t with s t < η. δ K log s t 1+α,

143 Appendix IV: Borell-Tsirelson inequality Finiteness theorem: f = sup t M f t P{ f < } = 1 E{ f } <,

144 Appendix IV: Borell-Tsirelson inequality Finiteness theorem: f = sup t M f t P{ f < } = 1 E{ f } <, THE inequality: For all u > 0, σ 2 M = sup t M E{f 2 t } P{ f E{ f } > u } e u2 /2σ 2 M.

145 Appendix IV: Borell-Tsirelson inequality Finiteness theorem: f = sup t M f t P{ f < } = 1 E{ f } <, THE inequality: For all u > 0, σ 2 M = sup t M E{f 2 t } This implies P{ f E{ f } > u } e u2 /2σ 2 M. P{ f u} e µu u2 /2σ 2 M, µ u = (2uE{ f } [E{ f }] 2 )/σ 2 M

146 Appendix IV: Borell-Tsirelson inequality Finiteness theorem: f = sup t M f t P{ f < } = 1 E{ f } <, THE inequality: For all u > 0, σ 2 M = sup t M E{f 2 t } This implies P{ f E{ f } > u } e u2 /2σ 2 M. P{ f u} e µu u2 /2σ 2 M, µ u = (2uE{ f } [E{ f }] 2 )/σ 2 M Asymptotics: For high levels u, the dominant behavior of all Gaussian exceedence probabilities is determined by e u2 /2σ 2 M.

147 Places to start reading and to find other references 1. R.J. Adler and J.E. Taylor, Random Fields and Geometry, Springer, R.J. Adler, and J.E. Taylor, Topological Complexity of Random Functions, Springer Lecture Notes in Mathematics, Vol. 2019, Springer, D. Aldous, Probability Approximations via the Poisson Clumping Heuristic, Applied Mathematical Sciences, Vol. 77, Springer-Verlag, J-M. Azaïs and M. Wschebor. Level Sets and Extrema of Random Processes and Fields, Wiley, H. Cramér and M.R. Leadbetter, Stationary and Related Stochastic Processes, Wiley, M.R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, 1983.