Random Walks and Surfaces Generated by Random Permutations

of Natural Series

Gleb OSHANINTheoretical Condensed Matter Physics

University Paris 6/CNRSFrance

Isaac Newton Institute Workshop, June 2006

Outlook

RaphaelMyself

I. Random Walk Generated by Permutations of 1,2,3, …, n+1

(with R. Voituriez)

Northern face of Peak Oshanin, 6320 m Pamir

II. Statistics of Peaks in Surfaces Generated by Random

Permutations of Natural Series

(with F.Hivert, S.Nechaev and O.Vasilyev)

♠ Spades < ♣ Clubs < ♦ Diamonds < ♥ Hearts Convention:

2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < 10 < J < Q < K < A

Who won and how much when the game is over?

(52! =80658175170943878571660636856403766975289505440883277824000000000000answers (not necessarily different) on this question)

Random Walk Generated by Random Permutations of Natural Series

(with R. Voituriez)

= {1,2,3, …, n+1} – random unconstrained permutation of [n+1]

In two line notation

( )1 2 3 … n+1

1, 2, 3, …, n+1

time l, (l = 1,2,3, …., n+1)

random number

X

- at l=0 - the walker is at the origin- at l=1 – the walker is moved to the right if 1 < 2 (permutation rise, ↑) to the left if 1 > 2 (permutation descent, ↓)- at l=2 – the walker is moved to the right if 2 < 3 (permutation rise, ↑) to the left if 2 > 3 (permutation descent, ↓)

and etc up to time l=n+1

where is the walker at time l=n+1 ?

Trajectory:

1

1)(, 1

1

)(kkk

l

kk

nl signssX

I. Probability Distribution Function of the end-point P(Xn=X)?

Let N↑ (N↓) denote the number of “rises” (“descents”) in a given permutation i.e. number of k-s at whichk < k+1 (k > k+1).

Evidently, Xn = N↑ - N↓, and since N↑ + N↓= n

Xn = 2N↑ - n

Eulerian number:

N

r

nnr

r rNCN

n

0

12 )1()1(1

determines a total number of permutations of [n+1] having exactly N↑ rises

The PDF of the end-point:

2

1

)!1(2

)1(1)( nX

n

nXXP

nX

n

)cos()

)sin((

)1(1)(

0

2 Xkk

kdkXXP n

nX

n

n

X

nXXP n 2

3exp

2

3)(

22/1

Integral representation of the PDF:

Looks almost like the PDF of standard n-n 1D RW except for the integration limits and the kernel

Asymptotic limit n → ∞:

Lattice Green Function:

00 ))sin(sin(

)cos())sin(sin(1)(),(

kzk

Xkkz

zXXPzzXG n

n

n

0 )cos(1

)cos(1

kz

Xkstandard n-n 1D RW result

Hence, using permutations as the generator of RW leads to conventional diffusive behabior at long times! <Xn

2> → n/3 - two-thirds of the diffusion coefficient disappear somewhere (walker steps at each tick of the clock, stops nowhere and rises and descents are equiprobable).Hence, there should be something non-trivial with the transition probabilities – correlations in the “rise-and-descent” sequences.

II. Correlations in rise-and-descent sequences.

1

1 121

)2(22

1 12

),(2)(n

j

n

jjn jjCnNNX

)(,),()(11121 121

)2(21

)2(jjjjj signsssjjCjjmC

Inverse problem: Given the PDF and hence, the moments, to determine correlations in the rise-and-descent sequences

Second moment of the PDF

Pair correlations in the r&d sequence

Their generating functions:

0

2)2( )(n

nn zXzX

1

21)2()2( )()(

m

mzjjmCzC

)()1(

2

)1()( )2(

22)2( zC

z

z

z

zzX

Relation between them:

20

2)2(

)1(3

)23()(

z

zzzXzX

n

nn

From the PDF we get:

3)()(

1

)2()2( zzmCzC

m

m

Hence:

0

3/1)()2( mCand

m = 1

m > 1

4/1

6/1

4

)(1)()(

)2(

,,

mCmpmp

4/1

3/1

4

)(1)()(

)2(

,,

mCmpmp

Probability of having two rises (descents) at distance m:

Probability of having a rises and a descents at distance m:

Pair correlations extend to nn only!

rises “repel” each other

rises “attract” descents

squared mean density

squared mean density

m > 1

m = 1

m = 1

m > 1

Fourth moment of the PDF

)2()1()1(16)2()1(3 )2(4 nmCnnnnnX n

3

1

)4()2( )4()()2(!4)3()1()2)3((12n

m

nmCmnnmCnn

where C(4)(m) is the fourth-order correlation function of the form (all other vanish):

211)4(

1111)( mjmjjj ssssmC

)()1(

!4

)1(15

803515)( )4(

2

3

3

32)4( zC

z

z

z

zzzzX

Relation between the generating functions:

3

432)4(

)1(15

)848803515()(

z

zzzzzzX

...)(9

1

15

2

)1(45

)6()( 432)4(

zzzzz

zzzC

9/1

15/2)()4( mC

if m = 1

if m > 1

III. Probabilities of rise-and-descent sequences of length 3 and 4.

2/16/1

8/1)(, mp

2/16/1

24/1)(, mp

6/16/1

120/1)(, mp

6/16/1

20/1)(, mp

3/16/1

30/1)()( ,, mpmp

2, 3/1

120/11)( mp

2, 3/1

15/2)( mp

m = 1

m = 1

m = 1

m = 1

m = 1

m = 1

m = 1

m > 1 m > 1

m > 1

m > 1

m > 1

m > 1

m > 1

All configurations have different weights

3/16/1

40/3)()( ,, mpmp

IV. Reconstructing the PGRW trajectories of length 4.

A set of all possible PGRW trajectories for n=4. Numbers above the solid arcs with arrows indicate the corresponding transition probabilities. Dashed lines with arrows connect the trajectories for different l.Transition probabilities clearly depend not only on the number of previous turns to the left (right) but also on their order.

A Non-Markovian Random Walk!

Theorems:

-The probability P(Yl=X) that the trajectory Yl of an auxiliary process appears at site X at time l is exactly the same as the probability P(Xl

(l)=X) that random walk generated by permutations of [l+1] has its end-point at site X (Eulerian).

-The probability P(Xl(n)=X) that at any intermediate step l, l=1,2,3, …, n-1, the PGRW trajectory

appears at the site X obeys

V. A deeper look on the PGRW trajectories.

The idea is to build recursively an auxiliary Markovian stochastic process Y l which is distributed exact-ly as Xl

(n) (similarly to Hammersley’s analysis of the longest increasing subsequence problem).

Yl is a random walk on a line of integers defined as follows:- At each time step l we define a real-valued random variable xl+1, uniformly distributed in [0,1].- At each moment l compare xl+1 and xl; if xl+1 > xl, a walker is moved one step to the right; otherwise, to the left.

Trajectory Yl:

l

kkkl xxsignY

11 )(

- The joint process (xl+1,Yl) and therefore Yl, are Markovian, since they depend only on (xl,Yl-1).

- Yl is a sum of correlated random variables - one has to be cautious with central limit theorems

)()()( )()( XXPXYPXXP lll

nl

VI. Even more deeper look on the PGRW : Measure of different trajectories

},...,,,,{)( nlX

1

1

)(lx

ll dxI

Each given PGRW trajectory is uniquely defined by the sequence of rises and descent of the corresponding permutation π of [n+1]. And vice versa!

We introduce two integral operators, and

and a polynomial Q defined as

, 1, )()(n

1l)(

lXIxQ n

l

The probability measure of a given trajectory Xl(n) obeys

1

0

)( )(}{)(

dxxXP Q nlX

nl

Example: },,,,{)5( lX

1202412840

311)(

5431

5

1 1

0

1

4321

41

2

3

)5(

xxxxdxdxdxdxdxx

xx x

x

xX IIIIIQ

l

720

19}{ )5( lXP

l = 1, 2, 3, 4, …,n

1

0

)(lx

ll dxI

VII. Distribution of the number of U-turns of the PGRW trajectories.

Left U-turn: ↑↓ - permutation peak (πj < πj+1 > πj+2)

Right U-turn: ↓↑ - permutation through (πj > πj+1 < πj+2)

Number of U-turns:(both left and right)

(shows how scrambled the trajectories are)

1

1)(,)1(

2

11

1

11 jjj

n

jjj signsssN

We calculate the characteristic function of N (funny 1d Ising model):

1

11]

2exp[)

2

)1(exp(]exp[)(

n

jjjn ss

iknikikNkZ

]2/[

0,

1 )2

tanh()1()2

1()(

n

l

lnl

lnik

n

ikW

ekZ

l

j

mjjj

l

l

jj

llmmmm

l

jj

nlj

lj

B

mmmm

m

m

ln

W1

2211

321

1

...321

, ))!22(

)14(4(

!!...!!

)!(12

321

12/12

2/1

22

]12

1coth1

1[

)1(

4)(),(

z

ee

e

ezkZzzkZ ik

ik

ik

iknn

n

Generating function of the characteristic function

3

)1(2

nN

Moments of the PDF

)4(36

)127()3(

15

)3()1(

12

)275( 222

n

nnn

nn

nnN

Asymptotic n → ∞ behavior of the PDF of the number of U-turns

n

nN

nnNP

163

245

exp5

4

3),(

2

2/1

VIII. Distribution of the distance between nearest right U-turns.

decays with l much faster than for Polya RW

IX. Diffusion limit

Using the equivalence between the processes Yl and Xl(n) , we derive the following master equation

)1()1(2

)()1(

)1(2

)4()( 1

YYPl

YlYYP

l

YlYYP lll

Introducing spatial variable y=aY (“a” has a dimension of length) and t=τ n (“τ” has a dimension of time) we turn to the limit a, τ → 0 keeping the ratio D=a2/2 τ fixed. We find a Fokker-Planck-type equation for diffusion with a negative drift term (which similarly to the Ornstein-Uhlenbeck process is proportional to “y” but decreases as 1/t) – random walk in a well

)()()(2

2

YPt

y

yYP

yDYP

t

Solution of this equation is

Dt

y

DtYP

4

3exp

4

3)(

22/1

and coincides with our previous result obtained for the discrete time and space PGRW for D=1/2.

Local Extrema (peaks) of Surfaces Generated by Random Permutations

(with F.Hivert, O.Vasilyev and S.Nechaev)

I. One-Dimensional Systems

The probability P(M,L) of having M peaks in on a chain of length L can be determined exactly

Lk

r

rM

k

kMML

rMr

L

kM

kMLkL

MLLMP )(

1)1(

)1(

)1()12()1(

)2(

2),(

00

12

First three central moments of P(M,L):

LM3

1 LMM

45

222 LMM945

2)( 3

In the asymptotic limit L → ∞ the PDF P(M,L) converges to a Gaussian distribution

L

LM

LLMP

4

)31

(45exp

5

2

3),(

22/1

II. Two-Dimensional Systems

First three central moments of P(M,L):

LM5

1 LMM

225

13222 LMM32175

512)( 3

Expanding P(M,L) into the Edgeworth series (cumulant expansion) we show that in L → ∞ limit the normalized PDF converges to a Gaussian distribution

)1

()(1

12

exp2

1),(

2/12/1

22/1

2/12 Lxf

L

xL

M

MMxP

)3(6

1)( 3

2/32

32/1 xx

M

MLxf

is independent of L

L=NxN

III. Instead of conclusions – current work

Partition function of a 2D model:

MzZ z – activity, M – number of peaks in a given permutation

B.Derrida (personal communication, unpublished) observed numericaly that for a very similar model (not integers but numbers uniformly distributed in [0,1]) there is a sign of something which looks like a phase transition at z ≈ 5.9.

Why it may happen?

Peaks can not occupy nn sites – on a square lattice they are hard-squares – nn peaks have an infinite “repulsion”

nnn peaks “attract” each other

p=(1/5)2 p=2/45 p=1/20

Liquid of peaks → Solid of peaks transition

Common number (less than the least, depletion force)

Two common numbersSquared probability of having an isolated peak