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

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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

  1. 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 K.U.Leuven, BE U. of Newcastle, AU Armin Straub Random walks in the plane

  2. Introduction Combinatorics Recursions W 3 (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

  3. Introduction Combinatorics Recursions W 3 (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

  4. Introduction Combinatorics Recursions W 3 (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

  5. Introduction Combinatorics Recursions W 3 (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

  6. Introduction Combinatorics Recursions W 3 (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

  7. Introduction Combinatorics Recursions W 3 (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

  8. Introduction Combinatorics Recursions W 3 (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

  9. Introduction Combinatorics Recursions W 3 (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

  10. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Long walks Asked by Karl Pearson in Nature in 1905 For long walks, the probability density is approximately 2 x n e − x 2 /n For instance, for n = 200 : 0.06 0.05 0.04 0.03 0.02 0.01 10 20 30 40 50 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

  11. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Densities n = 4 n = 2 n = 3 0.5 0.7 0.8 0.6 0.4 0.6 0.5 0.3 0.4 0.4 0.2 0.3 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 n = 5 n = 6 n = 7 0.35 0.30 0.35 0.30 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.05 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 Armin Straub Random walks in the plane

  12. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Densities n = 4 n = 2 n = 3 0.5 0.7 0.8 0.6 0.4 0.6 0.5 0.3 0.4 0.4 0.2 0.3 0.2 0.2 0.1 0.1 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 n = 5 n = 6 n = 7 0.35 0.30 0.35 0.30 0.30 0.25 0.25 0.25 0.20 0.20 0.20 0.15 0.15 0.15 0.10 0.10 0.10 0.05 0.05 0.05 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 5 6 7 Armin Straub Random walks in the plane

  13. Introduction Combinatorics Recursions W 3 (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

  14. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Moments Fact from probability theory: the distribution of the distance is determined by its moments. Represent the k th step by the complex number e 2 πix k . The s th moment of the distance after n steps is: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 In particular, W n (1) is the average distance after n steps. Armin Straub Random walks in the plane

  15. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Moments Fact from probability theory: the distribution of the distance is determined by its moments. Represent the k th step by the complex number e 2 πix k . The s th moment of the distance after n steps is: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 In particular, W n (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

  16. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 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

  17. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 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 . W 2 (1) = 4 π Armin Straub Random walks in the plane

  18. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 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 . W 2 (1) = 4 W 3 (1) = 1 . 57459723755189 . . . = ? π Armin Straub Random walks in the plane

  19. Introduction Combinatorics Recursions W 3 (1) A Bessel Integral Outro Moments The s th moment of the distance after n steps: � n � � s � � � e 2 πx k i W n ( s ) := d x � � � � [0 , 1] n k =1 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 . W 2 (1) = 4 W 3 (1) = 1 . 57459723755189 . . . = ? π Armin Straub Random walks in the plane

  20. Introduction Combinatorics Recursions W 3 (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

  21. Introduction Combinatorics Recursions W 3 (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.: � 2 k � W 2 (2 k ) = k � 2 � 2 j W 3 (2 k ) = � k � k � j =0 j j � k � 2 � 2 j �� 2( k − j ) � W 4 (2 k ) = � k j =0 j j k − j Armin Straub Random walks in the plane

  22. Introduction Combinatorics Recursions W 3 (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.: � 2 k � W 2 (2 k ) = k � 2 � 2 j W 3 (2 k ) = � k � k � j =0 j j � k � 2 � 2 j �� 2( k − j ) � W 4 (2 k ) = � k j =0 j j k − j � 2 � 2( k − j ) � k �� j � j � 2 � 2 ℓ � W 5 (2 k ) = � k j =0 j ℓ =0 ℓ ℓ k − j Armin Straub Random walks in the plane

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