Random Walks and Surfaces Generated by Random Permutations of Natural Series

1. 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

2. Outlook Northern face of Peak Oshanin, 6320 m Pamir Raphael Myself I. Random Walk Generated by Permutations of 1,2,3, …, n+1 II. Statistics of Peaks in Surfaces Generated by Random Permutations of Natural Series (with R. Voituriez) (with F.Hivert, S.Nechaev and O.Vasilyev)

3. 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)

4. 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 … n+1 ( ) time l, (l = 1,2,3, …., n+1) p1, p2, p3, …, pn+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 ?

5. Trajectory: 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: determines a total number of permutations p of [n+1] having exactly N↑ rises

6. The PDF of the end-point: Integral representation of the PDF: Looks almost like the PDF of standard n-n 1D RW except for the integration limits and the kernel Lattice Green Function: standard n-n 1D RW result Asymptotic limit n → ∞: 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.

7. 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 Second moment of the PDF Pair correlationsin the r&d sequence Their generating functions: Relation between them:

8. From the PDF we get: Hence: m = 1 Pair correlations extend to nn only! and m > 1 Probability of having two rises (descents) at distance m: rises “repel” each other m = 1 m > 1 squared mean density Probability of having a rises and a descents at distance m: rises “attract” descents m = 1 m > 1 squared mean density

9. Fourth moment of the PDF where C(4)(m) is the fourth-order correlation function of the form (all other vanish): Relation between the generating functions: if m = 1 if m > 1

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

11. 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!

12. V. A deeper look on the PGRW trajectories. The idea is to build recursively an auxiliary Markovian stochastic process Yl 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 comparexl+1 andxl; ifxl+1 >xl, a walker is moved one step to the right; otherwise, to the left. Trajectory Yl: - 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

13. 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! l = 1, 2, 3, 4, …,n and We introduce two integral operators, and a polynomial Q defined as The probability measure of a given trajectory Xl(n) obeys Example:

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

15. Generating function of the characteristic function Moments of the PDF Asymptotic n → ∞ behavior of the PDF of the number of U-turns

16. VIII. Distribution of the distance between nearest right U-turns. decays with l much faster than for Polya RW

17. IX. Diffusion limit Using the equivalence between the processes Yl and Xl(n) , we derive the following master equation 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 Solution of this equation is and coincides with our previous result obtained for the discrete time and space PGRW for D=1/2.

18. Local Extrema (peaks) of Surfaces Generated by Random Permutations (with F.Hivert, O.Vasilyev and S.Nechaev)

19. I. One-Dimensional Systems The probability P(M,L) of having M peaks in on a chain of length L can be determined exactly First three central moments of P(M,L): In the asymptotic limit L → ∞ the PDF P(M,L) converges to a Gaussian distribution

20. II. Two-Dimensional Systems First three central moments of P(M,L): Expanding P(M,L) into the Edgeworth series (cumulant expansion) we show that in L → ∞ limit the normalized PDF converges to a Gaussian distribution is independent of L L=NxN

21. III. Instead of conclusions – current work z – activity, M – number of peaks in a given permutation Partition function of a 2D model: 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=2/45 p=1/20 p=(1/5)2 Two common numbers Common number (less than the least, depletion force) Squared probability of having an isolated peak Liquid of peaks → Solid of peaks transition