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Random Walks and Surfaces
Generated by Random Permutations
of Natural Series
Gleb OSHANIN
Theoretical Condensed Matter Physics
University Paris 6/CNRS
France
Isaac Newton Institute Workshop, June 2006
Outlook
Myself
I. Random Walk Generated by
Permutations of
1,2,3, …, n+1
(with R. Voituriez)
Raphael
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)
Convention:
♠ Spades < ♣ Clubs < ♦ Diamonds < ♥ Hearts
2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < 10 < J < Q < K < A
Who won and how much when the game is over?
(52! =80658175170943878571660636856403766975289505440883277824000000000000
answers (not necessarily different) on this question)
Random Walk Generated by Random Permutations of Natural Series
(with R. Voituriez)
p = {p1,p2,p3, …, pn+1} – random unconstrained permutation of [n+1]
In two line notation
1 2 3
…
( p , p , p , …,
1
2
3
n+1
pn+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 p1 < p2 (permutation rise, ↑)
to the left if p1 > p2 (permutation descent, ↓)
- at l=2 – the walker is moved to the right if p2 < p3 (permutation rise, ↑)
to the left if p2 > p3 (permutation descent, ↓)
and etc up to time l=n+1
where is the walker at time l=n+1 ?
 1
  sk , sk  sign (p k 1  p k )  
k 1
 1
l
Trajectory:
X
(n)
l
I. Probability Distribution Function of the end-point P(Xn=X)?
Let N↑ (N↓) denote the number of “rises” (“descents”) in a given permutation p,
i.e. number of k-s at which pk < pk+1 (pk > pk+1).
Evidently, Xn = N↑ - N↓, and since N↑ + N↓= n
Xn = 2N↑ - n
Eulerian number:
N
n 1
  (1) r Crn 2 ( N  1  r ) n1
N
r 0
determines a total number of permutations p of [n+1] having exactly N↑ rises
1  (1) 
 X) 
X n
The PDF of the end-point:
P( X n
2( n  1)!
n 1
X n
2
1  (1) 
 X) 
X n 
Integral representation of the PDF:
P( X n
p
 dk (
0
sin( k ) n 2
) cos( Xk )
k
Looks almost like the PDF of standard n-n 1D RW except for the integration limits and the kernel
Lattice Green Function:

p
standard n-n
1D RW result
1 sin( z sin( k )) cos( Xk )
G ( X , z )   z n P( X n  X )  
pz 0 sin( k  z sin( k ))
n 0
1/ 2
Asymptotic limit n → ∞:
 3 
P( X n  X )  

2
p
n



1
p
cos( Xk )
p 0 1  z cos(k )
 3X 2 

exp  
 2n 
Hence, using permutations as the generator of RW leads to conventional diffusive
behabior at long times!
<Xn2> → 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.
Inverse problem: Given the PDF and hence, the moments, to determine correlations
in the rise-and-descent sequences
n 1
 X  ( N   N  )  n  2
Second moment
of the PDF
2
n
2
n
C
j1 1 j2  j1 1
( 2)
( j1 , j2 )
Pair correlations in the r&d sequence
C ( 2) (m  j1  j2 )  C ( 2) ( j1 , j2 )  s j1 s j2 , s j1  sign (p j1 1  p j1 )
Their generating functions:

X
( 2)
( z)    X  z
n 0
2
n

n
Relation between them:
C ( z )    C ( 2) (m  j1  j2 )  z m
( 2)
m 1
X ( 2) ( z ) 
z
2z
( 2)

C
( z)
2
2
(1  z ) (1  z )

From the PDF we get:
X
( 2)
( z )    X n2  z n 
n 0
z (3  2 z )
3(1  z ) 2

Hence:
C ( z )    C ( 2 ) ( m)  z m  
( 2)
m 1
and
 1 / 3
C ( m)  
0
( 2)
m=1
z
3
Pair correlations extend to nn only!
m>1
Probability of having two rises (descents) at distance m:
1  C ( 2) (m) 1 / 6 m = 1
p, (m)  p, (m) 

4
1 / 4 m > 1
rises “repel” each other
squared mean density
Probability of having a rises and a descents at distance m:
1  C ( 2) (m) 1 / 3 m = 1
p, (m)  p, (m) 

4
1 / 4 m > 1
rises “attract” descents
squared mean density
Fourth moment of the PDF
 X n4  n  3n(n  1) (n  2)  16(n  1)C ( 2) (m  1) (n  2) 
n 3
 12(n(n  3)  2)C (m  1) (n  3)  4! (n  2  m)C ( 4 ) (m) (n  4)
( 2)
m 1
where C(4)(m) is the fourth-order correlation function of the form (all other vanish):
C ( 4) (m)  s j1 s j1 1s j1 m1s j1 m2 
Relation between the generating functions: X
X
( 4)
( 4)
z (15  35z  80 z 2  48z 3  8z 4 )
( z) 
15(1  z )3
C ( 4) ( z ) 
z (6  z )
2
1
 z  ( z 2  z 3  z 4  ...)
45(1  z ) 15
9
15 z  35z 2  80 z 3
4! z 3
( 4)
( z) 

C
( z)
3
2
15(1  z )
(1  z )
2 / 15
C (m)  
1 / 9
( 4)
if m = 1
if m > 1
III. Probabilities of rise-and-descent sequences of length 3 and 4.
1 / 8
p , ( m )  
1 / 6  1 / 2
1 / 120
p , ( m )  
1 / 6  1 / 6
m=1
m>1
m=1
m>1
1 / 24
p , ( m )  
1 / 6  1 / 2
1 / 20
p , ( m )  
1 / 6  1 / 6
m=1
m>1
m=1
m>1
m=1
1 / 30
3 / 40
p , ( m )  p , ( m )  
p , ( m )  p , ( m )  
1 / 6  1 / 3
1 / 6  1 / 3 m > 1
11 / 120
p , ( m )  
2
1 / 3
m=1
m>1
2 / 15
p , ( m )  
2
1 / 3
All configurations have different weights
m=1
m>1
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!
V. A deeper look on the PGRW trajectories.
The idea is to build recursively an auxiliary Markovian stochastic process Yl which is distributed exactly 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.
l
Trajectory Yl:
Yl   sign ( xk 1  xk )
k 1
- 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
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
P( X l( n )  X )  P(Yl  X )  P( X l(l )  X )
VI. Even more deeper look on the PGRW : Measure of different trajectories
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!
X l( n )  { ,,,,..., }
l = 1, 2, 3, 4, …,n
1
 dx
Il (  ) 
We introduce two integral operators,
and
l
Il (  ) 
xl 1
 dx
l
0
xl 1
n
Q
and a polynomial Q defined as
X l( n )
( x )   I l (  )  1, ,
l 1
1
The probability measure of a given trajectory Xl(n) obeys
P{ X l( n ) }   Q
0
Example:
X l( n )
( x)dx
X l(5)  {, , , , }
1
1
x2
1
1
3 x x3 x 4 x5
Q X l(5) ( x)  I  I  I  I  I  1   dx1  dx2  dx3  dx4  dx5 1  40  8  12  24  120
x
x1
0
x3
x4
P{ X l(5) } 
19
720
VII. Distribution of the number of U-turns of the PGRW trajectories.
(shows how scrambled the trajectories are)
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)
 1
1 n 1
N   (1  s j s j 1 ), s j  sign (p j 1  p j )  
2 j 1
 1
We calculate the characteristic function of N (funny 1d Ising model):
ik (n  1)
ik
Z n (k )  exp[ ikN ]  exp(
)  exp[ 
2
2
n 1
s s
j 1
j
j 1
]
1  eik n1 [ n / 2]
ik
Z n (k )  (
)  (1)l Wl ,n tanh( )l
2
2
l 0
l
( m j )!
 n  2l  1
l
 l

4 j 1 (4 j 1  1) B2 j  2 m j
j 1
Wl ,n 
(
)




m
m
!
m
!
m
!...
m
!
(
2
j

2
)!
j
m1  2 m2  3 m3 ...  lml l  
l j 1
 1 2 3
j

1


Generating function of the characteristic function
 1  eik
4
n
Z (k , z )   z Z n (k ) 
[
ik 2 
z (1  e )  1  eik
n2

1/ 2





coth  1  e 2ik


1/ 2
z
1
  1]
2
Moments of the PDF
 N 
2(n  1)
3
(5n 2  7n  2)
(n  3)
(n 2  7n  12)
 N 
 (n  1) 
 (n  3) 
 (n  4)
12
15
36
2
Asymptotic n → ∞ behavior of the PDF of the number of U-turns
2


2



45 N  n  
1/ 2
3 5 
3  

P( N , n)    exp   

4  pn 
16n






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
P(Yl 1  Y ) 
(l  Y  4)
(l  Y )
P(Yl  Y  1) 
P(Yl  Y  1)
2(l  1)
2(l  1)
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
 y

P(Y )  D 2 P(Y )   P(Y ) 
t
y
y  t

Solution of this equation is
1/ 2
 3 
P(Y )  

4
p
Dt


 3y2 

exp  
4
Dt


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
2 L 2 M 1 M
( L  M  1  k ) k
M k
r  L  1
L


P ( M , L) 
(

1
)
(
L

2
k

1
)
(

1
)
(
M

r
)

 r 
( L  2 M ) k  0
( M  k  1) r 0


First three central moments of P(M,L):
1
 M  L
3
 M 2    M 2 
2
L
45
 ( M   M )3  
In the asymptotic limit L → ∞ the PDF P(M,L) converges to a Gaussian distribution
1/ 2
3 5 
P ( M , L)   
2  pL 
1 2

 45( M  L) 
3

exp  
4L






2
L
945
II. Two-Dimensional Systems
First three central moments of P(M,L):
 M 
1
L
5
 M 2    M 2   2 
13
L
225
 ( M   M )3 
512
L
32175
Expanding P(M,L) into the Edgeworth series (cumulant expansion) we show that in L → ∞ limit
the normalized PDF converges to a Gaussian distribution
 x 2 
M M 
1
1 
 1 


P( x 
,
L
)

exp

1

f
(
x
)


(
)



1/ 2
1/ 2 
 2
 M 2 1/ 2
2
p
L
L





3
1/ 2  M  1
3
is independent of L
f ( x)  L
(
x
 3x)
2
3/ 2
M  6
1/ 2
L=NxN
III. Instead of conclusions – current work
Partition function of a 2D model:
Z   zM
p
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
Squared probability of having
an isolated peak
p=2/45
Common number (less than
the least, depletion force)
Liquid of peaks → Solid of peaks
transition
p=1/20
Two common numbers