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Random walks in the plane Armin Straub Tulane University, New - - PowerPoint PPT Presentation

Mar 02, 2024 96 likes •610 views

Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Random walks in the plane Armin Straub Tulane University, New Orleans August 2, 2010 Joint work with : Jon Borwein Dirk Nuyens James Wan U. of Newcastle, AU

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

Armin Straub

Tulane University, New Orleans

August 2, 2010 Joint work with: Jon Borwein Dirk Nuyens James Wan

  • U. of Newcastle, AU

K.U.Leuven, BE

  • U. of Newcastle, AU

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Random walks in the plane

We study random walks in the plane consisting of n steps. Each step is of length 1 and is taken in a randomly chosen direction. We are interested in the distance traveled in n steps. For instance, how large is this distance on average?

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Long walks

Asked by Karl Pearson in Nature in 1905

  • K. Pearson. “The random walk.” Nature, 72, 1905.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Long walks

Asked by Karl Pearson in Nature in 1905 For long walks, the probability density is approximately 2x n e−x2/n For instance, for n = 200:

10 20 30 40 50 0.01 0.02 0.03 0.04 0.05 0.06

  • K. Pearson. “The random walk.” Nature, 72, 1905.

Lord Rayleigh. “The problem of the random walk.” Nature, 72, 1905.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Densities

n = 2

0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8

n = 3

0.5 1.0 1.5 2.0 2.5 3.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

n = 4

1 2 3 4 0.1 0.2 0.3 0.4 0.5

n = 5

1 2 3 4 5 0.05 0.10 0.15 0.20 0.25 0.30 0.35

n = 6

1 2 3 4 5 6 0.05 0.10 0.15 0.20 0.25 0.30 0.35

n = 7

1 2 3 4 5 6 7 0.05 0.10 0.15 0.20 0.25 0.30

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Densities

n = 2

0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8

n = 3

0.5 1.0 1.5 2.0 2.5 3.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

n = 4

1 2 3 4 0.1 0.2 0.3 0.4 0.5

n = 5

1 2 3 4 5 0.05 0.10 0.15 0.20 0.25 0.30 0.35

n = 6

1 2 3 4 5 6 0.05 0.10 0.15 0.20 0.25 0.30 0.35

n = 7

1 2 3 4 5 6 7 0.05 0.10 0.15 0.20 0.25 0.30

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Moments

Fact from probability theory: the distribution of the distance is determined by its moments.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Moments

Fact from probability theory: the distribution of the distance is determined by its moments. Represent the kth step by the complex number e2πixk. The sth moment of the distance after n steps is: Wn(s) :=

  • [0,1]n
  • n
  • k=1

e2πxki

  • s

dx In particular, Wn(1) is the average distance after n steps.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Moments

Fact from probability theory: the distribution of the distance is determined by its moments. Represent the kth step by the complex number e2πixk. The sth moment of the distance after n steps is: Wn(s) :=

  • [0,1]n
  • n
  • k=1

e2πxki

  • s

dx In particular, Wn(1) is the average distance after n steps. This is hard to evaluate numerically to high precision. For instance, Monte-Carlo integration gives approximations with an asymptotic error of O(1/ √ N) where N is the number of sample points.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Moments

The sth moment of the distance after n steps: Wn(s) :=

  • [0,1]n
  • n
  • k=1

e2πxki

  • s

dx n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Moments

The sth moment of the distance after n steps: Wn(s) :=

  • [0,1]n
  • n
  • k=1

e2πxki

  • s

dx n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186. W2(1) = 4 π

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Moments

The sth moment of the distance after n steps: Wn(s) :=

  • [0,1]n
  • n
  • k=1

e2πxki

  • s

dx n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186. W2(1) = 4 π W3(1) = 1.57459723755189 . . . = ?

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Moments

The sth moment of the distance after n steps: Wn(s) :=

  • [0,1]n
  • n
  • k=1

e2πxki

  • s

dx n s = 1 s = 2 s = 3 s = 4 s = 5 s = 6 s = 7 2 1.273 2.000 3.395 6.000 10.87 20.00 37.25 3 1.575 3.000 6.452 15.00 36.71 93.00 241.5 4 1.799 4.000 10.12 28.00 82.65 256.0 822.3 5 2.008 5.000 14.29 45.00 152.3 545.0 2037. 6 2.194 6.000 18.91 66.00 248.8 996.0 4186. W2(1) = 4 π W3(1) = 1.57459723755189 . . . = ?

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Even moments

n s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 2 6 20 70 252 A000984 3 3 15 93 639 4653 A002893 4 4 28 256 2716 31504 A002895 5 5 45 545 7885 127905 6 6 66 996 18306 384156

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Even moments

n s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 2 6 20 70 252 A000984 3 3 15 93 639 4653 A002893 4 4 28 256 2716 31504 A002895 5 5 45 545 7885 127905 6 6 66 996 18306 384156 Sloane’s, etc.: W2(2k) = 2k

k

  • W3(2k) = k

j=0

k

j

22j

j

  • W4(2k) = k

j=0

k

j

22j

j

2(k−j)

k−j

  • Armin Straub

Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Even moments

n s = 2 s = 4 s = 6 s = 8 s = 10 Sloane’s 2 2 6 20 70 252 A000984 3 3 15 93 639 4653 A002893 4 4 28 256 2716 31504 A002895 5 5 45 545 7885 127905 6 6 66 996 18306 384156 Sloane’s, etc.: W2(2k) = 2k

k

  • W3(2k) = k

j=0

k

j

22j

j

  • W4(2k) = k

j=0

k

j

22j

j

2(k−j)

k−j

  • W5(2k) = k

j=0

k

j

22(k−j)

k−j

j

ℓ=0

j

ℓ

22ℓ

ℓ

  • Armin Straub

Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Combinatorics

Theorem (Borwein-Nuyens-S-Wan) Wn(2k) =

  • a1+···+an=k
  • k

a1, . . . , an 2 .

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Combinatorics

Theorem (Borwein-Nuyens-S-Wan) Wn(2k) =

  • a1+···+an=k
  • k

a1, . . . , an 2 . fn(k) := Wn(2k) counts the number of abelian squares: strings xy

  • f length 2k from an alphabet with n letters such that y is a

permutation of x.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Combinatorics

Theorem (Borwein-Nuyens-S-Wan) Wn(2k) =

  • a1+···+an=k
  • k

a1, . . . , an 2 . fn(k) := Wn(2k) counts the number of abelian squares: strings xy

  • f length 2k from an alphabet with n letters such that y is a

permutation of x. Introduced by Erd˝

  • s and studied by others.
  • L. B. Richmond and J. Shallit. “Counting abelian squares.” The

Electronic Journal of Combinatorics, 16, 2009.

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Combinatorics

Theorem (Borwein-Nuyens-S-Wan) Wn(2k) =

  • a1+···+an=k
  • k

a1, . . . , an 2 . fn(k) := Wn(2k) counts the number of abelian squares: strings xy

  • f length 2k from an alphabet with n letters such that y is a

permutation of x. Introduced by Erd˝

  • s and studied by others.

fn(k) satisfies recurrences and convolutions.

  • L. B. Richmond and J. Shallit. “Counting abelian squares.” The

Electronic Journal of Combinatorics, 16, 2009.

  • P. Barrucand. “Sur la somme des puissances des coefficients

multinomiaux et les puissances successives d’une fonction de Bessel.”

  • C. R. Acad. Sci. Paris, 258, 5318–5320, 1964.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Functional Equations for Wn(s)

For integers k 0, (k + 2)2W3(2k + 4) − (10k2 + 30k + 23)W3(2k + 2) + 9(k + 1)2W3(2k) = 0.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Functional Equations for Wn(s)

For integers k 0, (k + 2)2W3(2k + 4) − (10k2 + 30k + 23)W3(2k + 2) + 9(k + 1)2W3(2k) = 0. Theorem (Carlson) If f(z) is analytic for Re (z) 0, “nice”, and f(0) = 0, f(1) = 0, f(2) = 0, . . . , then f(z) = 0 identically.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Functional Equations for Wn(s)

For integers k 0, (k + 2)2W3(2k + 4) − (10k2 + 30k + 23)W3(2k + 2) + 9(k + 1)2W3(2k) = 0. Theorem (Carlson) If f(z) is analytic for Re (z) 0, “nice”, and f(0) = 0, f(1) = 0, f(2) = 0, . . . , then f(z) = 0 identically. |f(z)| Aeα|z|, and |f(iy)| Beβ|y| for β < π

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Functional Equations for Wn(s)

For integers k 0, (k + 2)2W3(2k + 4) − (10k2 + 30k + 23)W3(2k + 2) + 9(k + 1)2W3(2k) = 0. Theorem (Carlson) If f(z) is analytic for Re (z) 0, “nice”, and f(0) = 0, f(1) = 0, f(2) = 0, . . . , then f(z) = 0 identically. Wn(s) is nice! |f(z)| Aeα|z|, and |f(iy)| Beβ|y| for β < π

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Functional Equations for Wn(s)

So we get complex functional equations like (s+4)2W3(s+4)−2(5s2+30s+46)W3(s+2)+9(s+2)2W3(s) = 0.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Functional Equations for Wn(s)

So we get complex functional equations like (s+4)2W3(s+4)−2(5s2+30s+46)W3(s+2)+9(s+2)2W3(s) = 0. This gives analytic continuations of Wn(s) to the complex plane, with poles at certain negative integers.

6 4 2 2 3 2 1 1 2 3 4 6 4 2 2 3 2 1 1 2 3 4

W3(s) W4(s)

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

W3(1) = 1.57459723755189 . . . = ?

Easy: W2(2k) = 2k k

  • . In fact, W2(s) =

s s/2

  • .

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

W3(1) = 1.57459723755189 . . . = ?

Easy: W2(2k) = 2k k

  • . In fact, W2(s) =

s s/2

  • .

Again, from combinatorics: W3(2k) =

k

  • j=0

k j 22j j

  • = 3F2

1

2, −k, −k

1, 1

  • 4
  • =:V3(2k)

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

W3(1) = 1.57459723755189 . . . = ?

Easy: W2(2k) = 2k k

  • . In fact, W2(s) =

s s/2

  • .

Again, from combinatorics: W3(2k) =

k

  • j=0

k j 22j j

  • = 3F2

1

2, −k, −k

1, 1

  • 4
  • =:V3(2k)

We discovered numerically that V3(1) ≈ 1.574597 − .126027i.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

W3(1) = 1.57459723755189 . . . = ?

Easy: W2(2k) = 2k k

  • . In fact, W2(s) =

s s/2

  • .

Again, from combinatorics: W3(2k) =

k

  • j=0

k j 22j j

  • = 3F2

1

2, −k, −k

1, 1

  • 4
  • =:V3(2k)

We discovered numerically that V3(1) ≈ 1.574597 − .126027i. Theorem (Borwein-Nuyens-S-Wan) For integers k we have W3(k) = Re 3F2 1

2, − k 2, − k 2

1, 1

  • 4
  • .

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

W3(1) = 1.57459723755189 . . . = ?

Corollary (Borwein-Nuyens-S-Wan) W3(1) = 3 16 21/3 π4 Γ6 1 3

  • + 27

4 22/3 π4 Γ6 2 3

  • Similar formulas for W3(3), W3(5), . . .

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

A generating function

Recall: Wn(2k) =

  • a1+···+an=k
  • k

a1, . . . , an 2

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

A generating function

Recall: Wn(2k) =

  • a1+···+an=k
  • k

a1, . . . , an 2 Therefore:

∞

  • k=0

Wn(2k)(−x)k (k!)2 =

∞

  • k=0
  • a1+···+an=k

(−x)k (a1!)2 · · · (an!)2 = ∞

  • a=0

(−x)a (a!)2 n = J0(2√x)n

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Ramanujan’s Master Theorem

Theorem (Ramanujan’s Master Theorem) For “nice” analytic functions ϕ, ∞ xν−1 ∞

  • k=0

(−1)k k! ϕ(k)xk

  • dx = Γ(ν)ϕ(−ν).

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Ramanujan’s Master Theorem

Theorem (Ramanujan’s Master Theorem) For “nice” analytic functions ϕ, ∞ xν−1 ∞

  • k=0

(−1)k k! ϕ(k)xk

  • dx = Γ(ν)ϕ(−ν).

Begs to be applied to

∞

  • k=0

Wn(2k)(−x)k (k!)2 = J0(2√x)n by setting ϕ(k) = Wn(2k) k!

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Ramanujan’s Master Theorem

We find: Wn(−s) = 21−s Γ(1 − s/2) Γ(s/2) ∞ xs−1Jn

0 (x) dx

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Ramanujan’s Master Theorem

We find: Wn(−s) = 21−s Γ(1 − s/2) Γ(s/2) ∞ xs−1Jn

0 (x) dx

A 1-dimensional representation! Useful for symbolical computations as well as for high-precision integration

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

Ramanujan’s Master Theorem

We find: Wn(−s) = 21−s Γ(1 − s/2) Γ(s/2) ∞ xs−1Jn

0 (x) dx

A 1-dimensional representation! Useful for symbolical computations as well as for high-precision integration First and more inspiredly found by David Broadhurst building on work of J.C. Kluyver David Broadhurst. “Bessel moments, random walks and Calabi-Yau equations.” Preprint, Nov 2009. J.C. Kluyver. “A local probability problem.” Nederl. Acad.

  • Wetensch. Proc., 8, 341–350, 1906.

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

A convolution formula

Conjecture For even n, Wn(s) ? =

∞

  • j=0

s/2 j 2 Wn−1(s − 2j).

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

A convolution formula

Conjecture For even n, Wn(s) ? =

∞

  • j=0

s/2 j 2 Wn−1(s − 2j). Inspired by the combinatorial convolution for fn(k) = Wn(2k): fn+m(k) =

k

  • j=0

k j 2 fn(j) fm(k − j)

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

A convolution formula

Conjecture For even n, Wn(s) ? =

∞

  • j=0

s/2 j 2 Wn−1(s − 2j). Inspired by the combinatorial convolution for fn(k) = Wn(2k): fn+m(k) =

k

  • j=0

k j 2 fn(j) fm(k − j) True for even s True for n = 2 Now proven up to some technical growth conditions

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

You will have to look at the papers to find. . .

a hyper-closed form for W4(1), Meijer-G and hypergeometric expressions for W3(s) and W4(s), evaluations of derivatives including W ′

3(0) = 1

π Cl π 3

  • ,

W ′

4(0) = 7ζ(3)

2π2 , expressions for residues at the poles of Wn(s), . . .

Armin Straub Random walks in the plane

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Introduction Combinatorics Recursions W3(1) A Bessel Integral Outro

References

  • J. Borwein, D. Nuyens, A. Straub, and J. Wan. “Random Walk

Integrals.” Preprint, Oct 2009.

  • J. Borwein, A. Straub, and J. Wan. “Three-Step and Four-Step

Random Walk Integrals.” Preprint, May 2010. Both preprints as well as this talk are/will be available from: http://arminstraub.com

THANK YOU!

Special thanks to:

Tewodros Amdeberhan, David Bailey, David Broadhurst, Richard Crandall, Peter Donovan, Victor Moll, Michael Mossinghoff, Sinai Robins, Bruno Salvy, Wadim Zudilin

Armin Straub Random walks in the plane