Lecture 13

Drunk Man Walks

H. Risken, The Fokker-Planck Equation (Springer-Verlag Berlin 1989)

C. W. Gardiner, Handbook of Stochastic Methods (Springer Berlin 2004)

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Random Walks and Levy flights

Random Walk

The term “random walk” was first used by Karl Pearson in 1905. He proposed a

simple model for mosquito infestation in a forest: at each time step, a single

mosquito moves a fixed length at a randomly chosen angle. Pearson wanted to

know the mosquitos’ distribution after many steps.

The paper (a letter to Nature ) was answered by Lord Rayleigh, who had already

solved the problem in a more general form in the context of sound waves in

heterogeneous materials.

Actually, the the theory of random walks was developed a few years before

(1900) in the PhD thesis of a young economist: Louis Bachelier. He proposed

the random walk as the fundamental model for financial time series. Bachelier

was also the first to draw a connection between discrete random walks and the

continuous diffusion equation.

Curiously, in the same year of the paper of Pearson (1905) Albert Einstein

published his paper on Brownian motion which he modeled as a random walk,

driven by collisions with gas molecules. Einstein did not seems to be aware of

the related work of Rayleigh and Bachelier.

Smoluchowski in 1906 also published very similar ideas.

The simplest possible problem: 1 dimensional Random Walk

0 1 2 3-4 -3 -2 -1

The “walker” starts at position x = 0 at the step t = 0; at each

time-step the walker can go either forward or backward of one

position with equal probabilities 1/2.

We ask the probability P(x,t) to find the walker at the position x

a the time step t.

!

p(x, t) =t!

(t + x) /2( )! (t " x) /2( )!1

2

#

$ % &

' (

t

~t>>1

2

) te

"x 2

2t

probability of any given sequence of t steps

Number of sequences that take to x in t steps

(t+x)/2 steps taken in the positive

direction

(t-x)/2 steps taken in the negative

direction

Stochastic process - normal noise (finite variance) 1 dimension

!

x(t +1) = x(t) +"(t)

Central limit theorem (lecture 2)

The sum of independent identically-distributed variables with

finite variance will tend to be normally distributed. [lecture 2]

!

x(t) ="=1

t

# $(" )

!

p(x, t) ~t>>1

1

2" t #2exp $

x2

2t #2

%

&

' '

(

)

* *

Average traveled

distance:

!

xt= " = 0

!

x2

t

" xt

2

= #2 t $ t!

" = 0( )

Stochastic process - normal noise (finite variance) D dimensions

!

r r (t +1) =

r r (t) +

r " (t)

iid D-dimesional vectors

isotropic (uniformly distributed in the direction)

with moduli distributed accordingly to a givenprobability density function p(|!|).

!

p(r r ,t +1) =

0

"

# p(r $ )p(

r r %

r $ ,t)dD r

$

!

p(r r ,t +1) =

0

"

# p(r $ ) p(

r r ,t) %

r $

r & p(

r r ,t) +

1

2

r $

r &

r & p(

r r ,t)

r $ + ....

'

( ) *

+ , d

D r $

!

p(r r ,t +1) " p(

r r ,t) =

#2

2D$2

p(r r ,t) + ....

!

"p

"t=#2

2D$2p

!

p(",t) =2D

D / 2"D#1

$(D/2) 2 % 2 t( )D / 2exp

#D"2

2 %2 t

&

'

( (

)

*

+ +

!

" =r r

Random Walk - any dimension

D =1 D =2

D =3 D =10

D =100P(") at t = 0,1,..,9

for <!> =0 and <!2> =1

p(",t)

p(",t)

p(",t)

P(",t)

P(",t)

"

!

" =# (D+1)

2( )# D

2( )2 $2

Dt

!

"2 = #2 t"

""

"

This is not a sphere.

Is this a sphere?

...further jumps…

Stochastic process with large noise fluctuations (non-finite

variance)

!

x(t +1) = x(t) +"(t)

We already discussed the 1 dimensional case in the context

of the central limit theorem and stable distributions [Lecture 3].

!

p(x, t)"a(t)

x#

!

p(")#1

"$!

f (x,",#,c,µ) = dk$%

+%

& eikxexp ikµ $ ck

"$11$ i#

k

k'(" $1,k)

(

) *

+

, -

(

) * *

+

, - -

3.0

2.5

2.0

1.5

Super diffusive behavior

Probability of a jump larger or equal to Lmax

in t steps I have a finite probability of a jump equal to Lmax

if:

!

Pr(L " Lmax ) = p(#)d#Lmax

$

% ~ 1L

max

&'1

!

t Pr(L " L

max) ~ 1

!

Lmax~ t

1/"#1

Mean Square Displacement:

!

x2~ t

1/ 0.9= t

1.11

!

x ~ t1/ 0.9

= t1.11!

" =1.9 Large jumps

dominate the

behavior!

x(t +1) = x(t) +"(t)

!

p(")#1

"$

!

x2

= t "2 = t "2p(")d"Lmin

Lmax

# ~ tLmax3$%

~ t2 /(%$1)

!

" >1

!

1<" < 3

!

x ~ Lmax

!

x2~ L

max

2

!

x " t = t0.5

!

x2

= "2 t ~ t 0.5

Diffusive behavior (‘jumps’ with finite variance)

Super-diffusive behavior

!

x2~ t

1/1.5= t

0.66

!

x ~ t1/1.5

= t0.66

!

x2~ t

1/ 2= t

0.5

!

x ~ t1/ 2

= t0.5

!

x2~ t

1/ 0.9= t

1.11

!

x ~ t1/ 0.9

= t1.11

!

" = 3

!

" = 2.5

!

" =1.9

Scale free and Self similarity

1010

105

108

104

106

103

104

102

!

"x ~ "t( )2

!

"x ~ "t( )1/#

!

" = 0.5

Higher dimensions: Levi Flights

!

r r (t +1) =

r r (t) +

r " (t)

!

" =1.1

!

p(r " ) ~

1

"d +#

!

ˆ p (r k ) ~ exp("c k

#)

Sub-diffusive behaviors

!

x(t + ") = x(t) +#(t)

!

"2with finite, but

with power law distributed waiting times:

!

p(") ~1

"#

In n steps the mean square displacement will grow as

the total time elapsed is

!

x2

= n "2

!

x2~ t

"#1sub-diffusive!

!

t ~ n "

!

" > 2

!

t ~ n

!

1<" < 2

!

t ~ "max~ n

1/(#$1)

!

x2~ t diffusive

(same reasoning as for Lmax in previous slide)

Case

Case

Random walk on graphs

!

P( j, t +1) = pstay ( j)P( j, t) + pout ( j "1)P( j "1,t) + pin ( j +1)P( j +1,t)

The probability to be at time t+1 at a given geodesic distance j form the

starting point is given by the probability that at time t the walker is

probability to be at distance j and stay there

probability to be at distance j-1 and move forward probability to be at distance j+1 and move inward

!

P(0,t +1) = pin (1)P(1,t)

!

P( j,0) = " j ,0

with

and

Which is the probability to find the walker at distance j after t steps?

!

P( j, t +1) = pstay ( j)P( j, t) + pout ( j "1)P( j "1,t) + pin ( j +1)P( j +1,t)

Random walk on graphs - continuous limit

the equation for P(j,t) can be re-written as

!

P( j, t +1) " P( j, t) = 1

2pin ( j +1) + pout ( j +1)[ ]P( j +1,t)

+1

2pin ( j "1) + pout ( j "1)[ ]P( j "1,t)

+ pin ( j) + pout ( j)[ ]P( j, t)

+1

2pin ( j +1) " pout ( j +1)[ ]P( j +1,t)

-1

2pin ( j "1) " pout ( j "1)[ ]P( j "1,t)

the continuous limit (j #"; t # $) gives (Fokker-Plank Equation)

!

"P(#,$ )

"t=" 2

"#2pout (#) + pin (#)

2

%

& ' (

) * P(#,$) +

"

"#pout (#) + pin (#)[ ]P(#,$)

Random walk on graphs - shell map

Each one of the probabilities

!

pin ( j)

!

pout ( j)

!

pstay ( j)

is proportional to the relative number of paths that take the

walker inwards, outwards or within the “shell” j

j

J+1

J-1

square lattice (j > 1): - inward 4(j - 1);

- outward 4(j+1);

- stay 0

For a broad class of graphs holds (j>>1):

!

pout ( j) + pin ( j) ~ Const

!

pout ( j) " pin ( j)#V ( j +1) "V ( j)

V ( j)

Random walk on graphs - continuous asymptotic solutions

!

pout (") + pin (") ~ Const =# 2

D

!

pout (") # pin (") =$ 2

DV (")

%V (")

%"

Finite dimensions:

!

V (") ~ "D#1

!

P(",t) =2D

D / 2"D#1

$(D/2) 2% 2t( )D / 2exp

#D"2

2% 2t

&

' (

)

* +

Hyperbolic spaces:

!

V (") ~ exp(")

!

pout (") # pin (") ~ Const

which leads to an equation of the form

!

"P(#,$ )

"t= %D

1

"

"#P(#,$ ) + D

2

" 2

"#2P(#,$ )

and the solution for large times is a density wave which

moves ballistically outwards:<"> ~ t , < "2> ~ t2 and < "2> - < ">2 ~ t

T. Aste, "Random walks on disordered networks'', Phys. Rev. E 55 (1997), p.6233-6236.

!

pout (") # pin (") ~(D#1)$ 2

D"D#1"D#2

~1

"

Will the walker ever return to the origin?

Will the walker ever return to the origin?

The mean time spent in the origin is:

!

P(0,t)t= 0

"

#

!

" =1#1

P(0,t)t= 0

$

%

The probability to return to the origin is (Polya theorem)

!

P(0,t) =1

2"( )D

ˆ P (r k ,t)d

Dr k

#$

+$

%

!

ˆ P (r k ,t) = ˆ p (

r k )[ ]

t

!

t= 0

"

# P(0,t) =1

2$( )D

ˆ p (r k )[ ]

t

t= 0

"

# dDr k

%"

+"

& =1

2$( )D

1

1% ˆ p (r k )

dDr k

%"

+"

&

Which can be computed using the characteristic function

!

1

1" p(r k )

dDr k

"#

+#

$ ~1

k%

dDr k ~

"#

+#

$k

D"1

k%

d k"#

+#

$

!

p(r k ) ~ exp("c k

#) ~ 1" c k

#

!

" =

1 for D # 2

<1 for D > 2

$ % &

1 for D # '

<1 for D > '

$ % &

$

%

( (

&

( (

Finite

variance

Power Law,

Non defined

variance

! < 2

He might never get back home…

Recurrent " = 1(he always get back)

Transient " < 1(he might never get back)

D =1P(",t)

"

D =2P(",t)

"

D =100P(") at t =

0,1,..,9for <!> =0

and <!2> =1

P(",t)

"

Are sharks intelligent mathematicians?

G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, H. E. Stanley, Levy flight search

patterns of wandering albatrosses, Nature 381 (1996) 413 - 415.A.M. Reynolds, Cooperative random Levy flight searches and the flight patterns of honeybees, Physics Letters A 354 (2005)

384-388.

“maximally searchable graphs” (lecture 9)

!

ps(i, j)" di, j#$

We want to be able to navigate efficiently with

local information only (greedy algorithm).

Let us put the vertices on a lattice and choose

shortcuts between lattice vertices with

probability

with di,j the distance between the vertices on

the lattice.

It results that any target vertex from any

random starting vertex is found

within a time ~ (Log V )2 only when:

!

" = space dimension

J. M. Kleinberg, Navigation in asmall world, Nature, 406 (2000),

p. 845.

Shortfin Mako Shark 53802 1507012 26

Jul 2007 to 23 Mar 2008

Shortfin Mako Shark 68509 1507007

13 Jul 2007 to 9 Mar 2008

Shortfin Mako Shark 53802

1507012 26 Jul 2007 to 23 Mar

2008

Two dimensions:

!

ps(r)" r#2

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Supplemetary material

Weierstrass Random WalkThe “walker” starts at position x = 0 at step

t = 0; at each time-step the walker can go

either forward or backward bn positions

with probability ~ pn.

p

pn

!=b

!=bn

!

p(n) ~ pn

!

ˆ p (k) = p(")#$

+$

% eik"

= (1# p) pm

cos(kbm

)m= 0

$

&k'$~ exp(#c(() k

()

!

" = ±bn

!

n =log(" )

b

!

p(") ~1

"1+#

!

p(") =(1# p)

2m= 0

$

% pn &(" # bn ) + &(" + bn )[ ]

!

" = #log p( )log(b)

!

P(x, t) ~1

x1+"

!

p(") ~1

"1+#

Levy stable

Characteristic function

!

p(x, t) = d"1...

#$

+$

% d"t

#$

+$

% p("1)...p("t )&("1 + ...+"t # x)

!

ˆ p (k, t) = dxeikx

"#

+#

$ d%1...

"#

+#

$ d%t

"#

+#

$ p(%1)...p(%t )&(%

1+ ...+%t " x)

!

ˆ p (k, t) = ˆ p (k)[ ]t

~ exp "Deff t k#( )

!

" ˆ P (k, t)

"t= #Deff k

$ ˆ P (k, t)

Diffusive behavior % = 2

!

"P(x, t)

"t= Deff

" 2 ˆ P (k,t)

"x2

Fractional diffusion behavior

!

"P(x, t)

"t= Deff

"# ˆ P (k,t)

"x#

!

"#

"x#:= $

1

2%dk

$&

+&

' k#e$ikx Fractional calculus

Non-equally distributed time-steps can lead to the same

kind of fat-tailed jump probabilities.

solution

Probabulity to return to the origin, on graphs…

!

P( j, t +1) = pstay ( j)P( j, t) + pout ( j "1)P( j "1,t) + pin ( j +1)P( j +1,t)

From the above relation one can calculate the mean time

spent in the origin:

!

t= 0

"

# P( j, t) =1

# of paths between shell j and shell j +1j= 0

"

#

T. Aste, "Random walks on disordered networks'', Phys. Rev. E 55 (1997), p.6233-6236.

!

(# of paths between shell j and shell j +1 ~k

2

j

kj

V ( j) ~

V ( j) for k 2 finite

V ( j)[ ]2

"#1 scale free with 1 <" < 2

V ( j)[ ]"

"#1 scale free with 2 $" < 3

%

&

' '

(

' '

!

" =1#1

P(0,t)t= 0

$

%~

1 for D & 2 and finite k2

1 for scale free with 2 &' < 3 and D < 2 - (3 -')2

1 for scale free with 1<' < 2 and D < 2 - 1'

< 1 otherwhise

(

)

* *

+

* * !

V ( j) ~ jD"1

Brownian motion is a fractal with

dimension 4/3

G.F. Lawler, W. Werner (1999), Intersection exponents for planar Brownian motion, Ann. Probab. 27, 1601-1642.