Short Walks and Ramble Integrals: the Arithmetic of Uniform - carma

158 Pages·2010·6.79 MB·English

Intro Combinatorics Analysis Probability OpenProblems
Short Walks and Ramble Integrals: the Arithmetic
of Uniform Random Walks
54th Annual AustMS Meeting
Jonathan M. Borwein Frsc Faaas Fbas Faa
Joint with Dirk Nuyens, Armin Straub, James Wan
& Wadim Zudilin Revised: 30/9/2010
Director,CARMA,theUniversityofNewcastle
September 30th 2010
JMB/JW ShortRandomWalks
Intro Combinatorics Analysis Probability OpenProblems
Outline
1 Introduction
2 Combinatorics
3 Analysis
4 Probability
5 Open Problems
JMB/JW ShortRandomWalks
Also random walks, random migrations, random flights.
•
Intro Combinatorics Analysis Probability OpenProblems
I. INTRODUCTION
An age old question: What is a walk?
•
JMB/JW ShortRandomWalks
Intro Combinatorics Analysis Probability OpenProblems
I. INTRODUCTION
An age old question: What is a walk?
•
Also random walks, random migrations, random flights.
•
JMB/JW ShortRandomWalks
Intro Combinatorics Analysis Probability OpenProblems
Abstract
Following Pearson in 1905, we study the expected distance of a
two-dimensional walk in the plane with n unit steps in random
directions — what Pearson called a random walk or a “ramble”.
While the statistics and large n behaviour are well
understood, the precise behaviour of the first few steps is
quite remarkable and less tractable.
Series evaluations and recursions are obtained making it possible to
explicitly determine this distance for small number of steps.
Hypergeometric and elliptic hyper-closed1 form expressions are
given for the densities and all the moments of a 2, 3 or 4-step walk.
Heavy use is made of analytic continuation of the integral (also of
modern special functions and computer algebra (CAS)).
1JMB & Crandall, “Closed forms: what they are and why they matter,”
Notices of the AMS, in press. See
http://www.carma.newcastle.edu.au/~jb616/closed-form.pdf
JMB/JW ShortRandomWalks
Intro Combinatorics Analysis Probability OpenProblems
“Birds and Frogs” (Freeman Dyson, NAMS 2010)
Some mathematicians are birds, others are frogs. Birds fly high in
the air and survey broad vistas of mathematics out to the far
horizon. They delight in concepts that unify our thinking and bring
together diverse problems from different parts of the landscape.
Frogs live in the mud below and see only the flowers that grow
nearby. They delight in the details of particular objects, and they
solve problems one at a time.
I happen to be a frog, but many of my best friends are birds. The
main theme of my talk tonight is this. Mathematics needs both
birds and frogs. Mathematics is rich and beautiful because birds
give it broad visions and frogs give it intricate details.
Mathematics is both great art and important science, because it
combines generality of concepts with depth of structures. It is
stupid to claim that birds are better than frogs because they see
farther, or that frogs are better than birds because they see deeper.
...
JMB/JW ShortRandomWalks
Intro Combinatorics Analysis Probability OpenProblems
“Experimental and Computational Mathematics”
Discussion. This article2 is one of our favourites.
Mathematics has frequently seen alternating
periods of theory building and periods of PsiPress iBook, 2010
pathology hunting. The first without the
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which no structures are to be discerned. ISBN 978-1-935638-05-6
In his wonderful Notices article Birds and Frogs Freeman Dyson
makes the same point forcibly and elegantly. In Dyson’s terms we
are unabashed frogs who consume the droppings of friendly birds
thereby enriching the pond’s nutrients for future visiting birds.
2“Strangeseriesevaluationsandhighprecisionfraud,”MAAMonthly,1992.
JMB/JW ShortRandomWalks
Intro Combinatorics Analysis Probability OpenProblems
Exploratory Experimentation and Computation in
Mathematics: ... and so to have ’Fun’
Numbers, symbols, and pictures
•
let us explore, refute and refine
conjectures (throughout this work).
Even to obtain secure knowledge Workshop
• Program
in areas where formal proof is out of
reach. See: 2010 Graduate Theme Program
5–16 July 2010, University of Queensland
Statistical Physics of Lattice Polymers
7–9 July 2010, University of Melbourne
Presentation: CoExmpplourtaattoioryn Einxp Neurimmbeenrt aTthioeonr aynd
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www.carma.newcastle.edu. E1x2a–1c4t Jluyly S20o1l0v, Uanbivleers itMy oof Qdueeelnss liannd Statistical Physics round close dates
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30 August–3 September 2010, Mission Beach, Queensland
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The AMSI Workshop on Network
(Notices, in press, with Bailey) M3 eNoavsemubreer m201e0,n Unti,v eMrstya onf Aadgelaeidmeent and Modelling
Riemannian and Differential Geometry
30 November–2 December 2010, La Trobe University
BioInfoSummer
29 November–3 December 2010, WEHI, University of Melbourne
For more information see www.amsi.org.au/events
JMB/JW ShortRandomWalks
Intro Combinatorics Analysis Probability OpenProblems
Random walk integrals — our starting point
For complex s
Definition
n s
Wn(s) := e2πxki dx
[0,1]n(cid:12) (cid:12)
(cid:90) (cid:12)(cid:88)k=1 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
W is analytic precisely for s > 2.
n
• (cid:60) −
Also, let W := W (1) denote the expectation.
n n
•
Simplest case (obvious for geometric reasons):
1
W (s) = e2πix sdx = 1.
1
0
(cid:90)
(cid:12) (cid:12)
(cid:12) (cid:12)
JMB/JW ShortRandomWalks
Mathematica 7 and Maple 13 ‘think’ the answer is 0.
•
There is always a 1-dimension reduction
•
n s
Wn(s) = e2πxki dx
(cid:90)[0,1]n(cid:12)k=1 (cid:12)
(cid:12)(cid:88) (cid:12)
(cid:12) n−1(cid:12) s
= (cid:12) 1+ e(cid:12)2πxki d(x1,...,xn−1)
(cid:90)[0,1]n−1(cid:12) k=1 (cid:12)
(cid:12) (cid:88) (cid:12)
(cid:12) (cid:12)
So W = 4 1/2cos(πx)dx(cid:12) = 4. (cid:12)
• 2 0 π
(cid:82)
Intro Combinatorics Analysis Probability OpenProblems
Second simplest case:
•
1 1
W = e2πix+e2πiy dxdy = ?
2
0 0
(cid:90) (cid:90)
(cid:12) (cid:12)
(cid:12) (cid:12)
JMB/JW ShortRandomWalks