Short Walks in Higher Dimensions Ghislain McKay Febuary 3, 2015 - PowerPoint PPT Presentation

Short Walks in Higher Dimensions Ghislain McKay Febuary 3, 2015

What is a Random Walk? A path formed by a succession of n steps (of unit length) in random directions. Figure: A 26-step random walk in the plane

What is a Random Walk? A path formed by a succession of n steps (of unit length) in random directions. In 1905, Karl Pearson was interested in the distribution of the distance from the origin for an n -step random walk. Figure: A 26-step random walk in the plane

What is a Random Walk? A path formed by a succession of n steps (of unit length) in random directions. In 1905, Karl Pearson was interested in the distribution of the distance from the origin for an n -step random walk. We look at two functions: • p n ( x ) the probability density function • W n ( s ) the moment function

Probability Density Functions For a continuous random variable X , the probability density function (pdf) describes the relative likelyhood that X takes on a given value. The probability of X falling within a range of values is given by the integral of the pdf over that range. Figure: Probability of X taking on a value in the interval from a to b

Moment Functions Definition The s -th moment function of a real-valued continuous function p ( x ) is � ∞ x s p ( x )d x W ( s ) = −∞

Moment Functions Definition The s -th moment function of a real-valued continuous function p ( x ) is � ∞ x s p ( x )d x W ( s ) = −∞ When p ( x ) is a probability density function of a random variable X , we have W ( s ) = E[ X s ] where E[ · ] is the expected value.

Moment Functions Definition The s -th moment function of a real-valued continuous function p ( x ) is � ∞ x s p ( x )d x W ( s ) = −∞ When p ( x ) is a probability density function of a random variable X , we have W ( s ) = E[ X s ] where E[ · ] is the expected value. The moments describe the shape of the distribution independent of translation.

The 2-Dimensional Case In 2 dimensions we can represent a random walk in the following way n � e 2 πix k where x ∈ [0 , 1] n k =1

The 2-Dimensional Case In 2 dimensions we can represent a random walk in the following way n � e 2 πix k where x ∈ [0 , 1] n k =1 Definition The moments of the distance from the origin after an n -step random walk in 2-dimensions is given by s � n � � � � � e 2 πix k W n ( s ) := d x � � � � [0 , 1] n � � k =1

Even Moments in 2 Dimensions The even moments in 2 dimensions are all integral. W 2 (0; 2 k ) : 1 , 2 , 6 , 20 , 70 , 252 , 924 , 3432 , 12870 , . . . W 3 (0; 2 k ) : 1 , 3 , 15 , 93 , 639 , 4653 , 35169 , 272835 , 2157759 , . . . W 4 (0; 2 k ) : 1 , 4 , 28 , 256 , 2716 , 31504 , 387136 , 4951552 , 65218204 , . . . W 5 (0; 2 k ) : 1 , 5 , 45 , 545 , 7885 , 127905 , 2241225 , 41467725 , 798562125 , . . . W 6 (0; 2 k ) : 1 , 6 , 66 , 996 , 18306 , 384156 , 8848236 , 218040696 , 5651108226 , . . .

Even Moments in 2 Dimensions The even moments in 2 dimensions are all integral. W 2 (0; 2 k ) : 1 , 2 , 6 , 20 , 70 , 252 , 924 , 3432 , 12870 , . . . W 3 (0; 2 k ) : 1 , 3 , 15 , 93 , 639 , 4653 , 35169 , 272835 , 2157759 , . . . W 4 (0; 2 k ) : 1 , 4 , 28 , 256 , 2716 , 31504 , 387136 , 4951552 , 65218204 , . . . W 5 (0; 2 k ) : 1 , 5 , 45 , 545 , 7885 , 127905 , 2241225 , 41467725 , 798562125 , . . . W 6 (0; 2 k ) : 1 , 6 , 66 , 996 , 18306 , 384156 , 8848236 , 218040696 , 5651108226 , . . . they are given by � 2 � 2 � � k k ! � � W n (2 k ) = = k 1 , . . . , k n k 1 ! · k 2 ! · · · k n ! k 1 + ··· + k n = k k 1 + ··· + k n = k which counts abelian squares (strings of length 2 k over an n letter alphabet where the first k letters are a permutation of the last k letters.)

The Probability Density Function In 1905, Lord Rayleigh gave an asymptotic form for large n � − x 2 p n ( x ) ∼ 2 x � n exp as n → ∞ n a Rayleigh distribution with mean � nπ 4 .

The Probability Density Function Figure: p n ( x ) for n = 3 , 4 , . . . , 8

The Probability Density Function In 1905, Lord Rayleigh gave an asymptotic form for large n � − x 2 p n ( x ) ∼ 2 x � n exp as n → ∞ n a Rayleigh distribution with mean � nπ 4 . For walks of 7 steps or more this is a very good approximation.

The Probability Density Function In 1905, Lord Rayleigh gave an asymptotic form for large n � − x 2 p n ( x ) ∼ 2 x � n exp as n → ∞ n a Rayleigh distribution with mean � nπ 4 . For walks of 7 steps or more this is a very good approximation. For this reason we will restrict ourselves to n -step walks where 2 ≤ n ≤ 6 (hence the name “short” walks).

Gamma Function Definition The Gamma function is an extension of the factorial function such that for a positive integer n Γ( n ) = ( n − 1)! For complex numbers z with positive real part it can be defined by (Euler’s definition) � ∞ t z − 1 e − t d t Γ( z ) = 0

Bessel Functions of the First Kind Definition The Besel function of the first kind J ν ( x ) is a solution to the differential equation x 2 d 2 y d x 2 + x d y d x + ( x 2 − ν 2 ) y = 0 we can define them by their taylor series around x = 0 ∞ ( − 1) k � x � 2 k + ν � J ν ( x ) = k ! Γ( k + ν + 1) 2 k =0

Bessel Functions of the First Kind Figure: J ν ( x ) for ν = 0 , 1 , 2 , 3 , 4

Towards Higher Dimensions For walks in d ≥ 2 dimensions, we define ν = d 2 − 1 notice that when d = 2 we have ν = 0 .

Towards Higher Dimensions For walks in d ≥ 2 dimensions, we define ν = d 2 − 1 notice that when d = 2 we have ν = 0 . We also define � ν � 2 j ν ( x ) = ν ! J ν ( x ) x where J ν ( x ) is the Bessel function of the first kind.

In Higher Dimensions Definition The probability density function of the distance to the origin in d ≥ 2 dimensions after n ≥ 2 steps is � ∞ 1 ( tx ) ν +1 J v ( tx )j n p n ( ν ; x ) = ν ( t )d t for x > 0 2 ν ν ! 0

In Higher Dimensions Figure: p 3 ( ν, x ) for ν = 0 , 1 2 , 1 , . . . , 7 2

In Higher Dimensions Figure: p 4 ( ν, x ) for ν = 0 , 1 2 , 1 , . . . , 7 2

In Higher Dimensions Definition The probability density function of the distance to the origin in d ≥ 2 dimensions after n ≥ 2 steps is � ∞ 1 ( tx ) ν +1 J v ( tx )j n p n ( ν ; x ) = ν ( t )d t for x > 0 2 ν ν ! 0

In Higher Dimensions Definition The probability density function of the distance to the origin in d ≥ 2 dimensions after n ≥ 2 steps is � ∞ 1 ( tx ) ν +1 J v ( tx )j n p n ( ν ; x ) = ν ( t )d t for x > 0 2 ν ν ! 0 Asymptotically, for x > 0 , as ν → ∞ � ν +1 2 − ν � 2 ν + 1 � − 2 ν + 1 � x 2 ν +1 exp x 2 p n ( ν ; x ) ∼ Γ( ν + 1) n 2 n Γ( ν + 3 Γ( ν +1) → √ n as ν → ∞ . 2 ) � 2 n a Chi distribution with mean 2 ν +1

In Higher Dimensions Definition The probability density function of the distance to the origin in d ≥ 2 dimensions after n ≥ 2 steps is � ∞ 1 ( tx ) ν +1 J v ( tx )j n p n ( ν ; x ) = ν ( t )d t for x > 0 2 ν ν ! 0 Asymptotically, for x > 0 , as ν → ∞ � ν +1 2 − ν � 2 ν + 1 � − 2 ν + 1 � x 2 ν +1 exp x 2 p n ( ν ; x ) ∼ Γ( ν + 1) n 2 n Γ( ν + 3 Γ( ν +1) → √ n as ν → ∞ . 2 ) � 2 n a Chi distribution with mean 2 ν +1 The proof follows from � − t 2 � j ν ( t ) ∼ exp as ν → ∞ 4 ν + 2

The Moment function By definition the moment function is � ∞ x s p n ( ν ; x )d x W n ( ν ; s ) = 0 Theorem Let n ≥ 2 and d ≥ 2 . For any nonnegative integer k , � ∞ W n ( ν ; s ) = 2 s − k +1 Γ( s � k 2 + ν + 1) � − 1 d x 2 k − s − 1 j n ν ( x )d x Γ( ν + 1)Γ( k − s 2 ) x d x 0

Combinatorial Intepretation of the Moments Theorem The even moments of an n -step random walk in d dimensions are W n ( ν ; 2 k ) = ( k + ν )! ν ! n − 1 � k �� k + nν � � ( k + nν )! k 1 , . . . , k n k 1 + ν, . . . , k n + ν k 1 + ··· + k n = k Proof. Replace k by k + 1 and set s = 2 k , we obtain � ∞ � k W n ( ν ; 2 k ) = 2 k ( k + ν )! � − d − 1 d j n d x ν ( x )d x ν ! d x x 0 � � � k ( k + ν )! � − 2 d j n = ν ( x ) ν ! x d x x =0

Combinatorial Intepretation of the Moments Proof. Replace k by k + 1 and set s = 2 k , we obtain � � � k ( k + ν )! � − 2 d j n W n ( ν ; 2 k ) = ν ( x ) ν ! x d x x =0 ( − x 2 / 4) m � j ν ( x ) = ν ! m !( m + ν )! m ≥ 0

Moment Recursion For positive integers n 1 , n 2 , half-integer ν and nonnegative integer k k � k � ( k + ν )! ν ! � W n 1 + n 2 ( ν ; 2 k ) = ( k − j + ν )( j + ν )! W n 1 ( ν ; 2 j ) W n 2 ( ν ; 2( k − j )) j j =0

Moment Recursion For positive integers n 1 , n 2 , half-integer ν and nonnegative integer k k � k � ( k + ν )! ν ! � W n 1 + n 2 ( ν ; 2 k ) = ( k − j + ν )( j + ν )! W n 1 ( ν ; 2 j ) W n 2 ( ν ; 2( k − j )) j j =0 In particular when n 2 = 1 we have W n 2 ( ν, s ) = 1 we obtain the recursive relation k � k � ( k + ν )! ν ! � W n ( ν ; 2 k ) = ( k − j + ν )( j + ν )! W n − 1 ( ν ; 2 j ) j j =0