1. Mixing Times of Self-Organizing Lists and Biased Permutations Sarah Miracle Georgia Institute of Technology

2. Joint work with.... Prateek BhaktaDana RandallAmanda Streib

3. Biased Card Shuffling 5n173...ij n-126 - pick a pair of adjacent cards uniformly at random - put j ahead of i with probability p j,i = 1- p i,j Converges to: π(σ) = Π / Z i σ(j) p ij p ji This is related to the “Move-Ahead-One Algorithm” for self-organizing lists.

4. What is already known? - proved this conjecture for n ≤ 4 Q: If the {p ij } are positively biased, is M nn always rapidly mixing? If p i,j ≥ ½ ∀ i < j we say the chain is positively biased. Conjecture (Fill): If {p ij } satisfy a “regularity” condition, then the spectral gap is minimized when p ij = 1/2 ∀ i,j Uniform bias: If p i,j = ½ ∀ i, j then M nn mixes in θ(n 3 log n) [Wilson] Constant bias: If p i,j = p > ½ ∀ i < j, then M nn mixes in θ(n 2 ) time [Benjamini, Berger, Hoffman, Mossel]

5. What is already known? Uniform bias: If p i,j = ½ ∀ i, j then M nn mixes in θ(n 3 log n) [Wilson] Constant bias: If p i,j = p > ½ ∀ i < j, then M nn mixes in θ(n 2 ) time [Benjamini, Berger, Hoffman, Mossel] Linear extensions of a partial order: If p i,j = ½ or 1 ∀ i < j, then M nn mixes in O(n 3 log n) time [Bubley, Dyer] M nn is fast for two new classes: “Choose your weapon” and “League hierarchies” [Bhakta, M., Randall, Streib] M nn can be slow even when the chain is positively biased [Bhakta, M., Randall, Streib]

6. Talk Outline 1.Background 2.New Classes of Bias where M nn is fast Choose your Weapon League Hierarchies 3.M nn can be slow ✔

7. The mixing time Definition: The total variation distance is ||P t,π|| = max __ ∑ |P t (x,y) - π(x)|. x  Ω y   Ω 2 1 A Markov chain is rapidly mixing if  (  ) is poly(n, log(  -1 )). (or polynomially mixing) Definition: Given , the mixing time is  = min { t: ||P t’,π|| < , t’ ≥ t }. A A Markov chain is slowly mixing if  (  ) is at least exp(n).

8. Choose your weapon Thm 1: Let p i,j = r i ∀ i < j. Then M NN is rapidly mixing. Proof sketch: A. Define auxiliary Markov chain M’ B. Show M’ is rapidly mixing C. Compare the mixing times of M NN and M’ M’ can swap pairs that are not nearest neighbors -Maintains the same stationary distribution -Allowed moves are based on inversion tables Given r 1, …, r n-1, r i ≥ 1/2.

9. Inversion Tables 3323000 Inversion Table I σ : 5631274 Permutation σ: I σ (i) = # elements j > i appearing before i in σ The map I is a bijection from S n to T = { (x 1,x 2,...,x n ): 0 ≤ x i ≤ n-i }. I σ (1) I σ (2)

10. Inversion Tables 3323000 Inversion Table I σ : 5631274 Permutation σ: M’ on Inversion Tables - choose a column i uniformly - w.p. r i : subtract 1 from x i (if x i >0) - w.p. 1- r i : add 1 to x i (if x i <n-i) M’ is just a product of n independent biased random walks M’ is rapidly mixing. 

11. Choose your weapon Thm 1: Let p i,j = r i ∀ i < j. Then M NN is rapidly mixing. Proof sketch: A. Define auxiliary Markov chain M’ B. Show M’ is rapidly mixing C. Compare the mixing times of M NN and M’ Given r 1, …, r n-1, r i ≥ 1/2. ✔ ✔

12. (Diaconis,Saloff-Coste’93) Unknown (M nn ) P Known (M’) P _ w z For each edge (x,y)  P, make a path  x,y using edges in P. Let  (z,w) be the set of paths  x,y using (z,w) _ x y Thm: Gap(P) ≥ Gap(P). _ 1 A A = max { ∑ |  x, y |π(x)P(x,y) } 1 Q(e) e   xy e _  Comparison Theorem

13. Inversion Tables 3323000 Inversion Table I σ : 5631274 Permutation σ: I σ (i) = # elements j > i appearing before i in σ M ’ on Inversion Tables - choose a column i uniformly - w.p. r i : subtract 1 from x i (if x i > 0) - w.p. 1- r i : add 1 to x i (if x i < n-i) - swap element i with the first j>i to the left w.p. r i - swap element i with the first j>i to the right w.p. 1-r i M ’ on Permutations - choose a card i uniformly

14. Want stationary dist.: 5631274 Permutation σ: 5621374 Permutation τ: π(σ) = Π / Z i σ(j) p ji p ij P ’ (σ,τ) P ’ (τ,σ) π(τ) π(σ) = p 1,2 p 3,2 p 3,1 p 2,1 p 2,3 p 1,3 = Comparing M’ and M nn - swap element i with the first j>i to the left w.p. r i - swap element i with the first j>i to the right w.p. 1-r i M ’ on Permutations - choose a card i uniformly P ’ (σ,τ) = r 2 = p 2,3 p 3,2 p 2,3 =

15. 5631274 Permutation σ: 5621374 Permutation τ: Comparing M’ and M nn 56132745612374

16. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing.

17. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing. A-League B-League

18. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing. 1 st Tier 2 nd Tier

19. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing. Proof sketch: A. Define auxiliary Markov chain M’ B. Show M’ is rapidly mixing C. Compare the mixing times of M NN and M’ M’ can swap pairs that are not nearest neighbors -Maintains the same stationary distribution -Allowed moves are based on the binary tree T

20. League Hierarchy Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing.

21. Thm 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing.

22. Thm 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing.

23. Thm 2: Proof sketch Let T be a binary tree with leaves labeled {1,…,n}. Given q v ≥ 1/2 for each internal vertex v. Thm 2: Let p i,j = q i^j for all i < j. Then M NN is rapidly mixing.

24. Thm 2: Proof sketch Markov chain M’ allows a transposition if it corresponds to an ASEP move on one of the internal vertices. q 2^4 1 - q 2^4

25. Want stationary distribution: π(σ) = Π / Z i σ(j) p ji p ij P ’ (σ,τ) P ’ (τ,σ) π(τ) π(σ) = p 1,5 p 3,5 p 6,8 p 6,9 p 1,6 p 3,6 p 5,8 p 6,9 p 5,6 p 5,1 p 5,3 p 8,6 p 9,6 p 6,1 p 6,3 p 8,5 p 9,6 p 6,5 = What about the Stationary Distribution? P ’ (σ,τ) =1 – v 5^6 = p 6,5 p 5,6 p 6,5 = 9385742 Permutation τ: 16 9386742 Permutation σ: 15

26. Thm 2: Proof sketch Markov chain M’ allows a transposition if it corresponds to an ASEP move on one of the internal vertices. Each ASEP is rapidly mixing M’ is rapidly mixing. M NN is also rapidly mixing if {p} is weakly regular. i.e., for all i, p i,j i. (by comparison) 

27. 6 5 8 9 2 10 3 4 1 7 Permutation σ:Permutation τ: Comparing M’ and M nn 6 7 8 9 2 10 3 4 1 5 6 5 2 8 9 3 10 4 1 7 6 2 8 9 3 10 4 1 5 76 2 8 9 3 10 4 1 7 5 6 7 2 8 9 3 10 4 1 5

28. So M NN is rapidly mixing when Choose your weapon: p i,j = r i ≥ 1/2 ∀ i < j Tree hierarchies: p i,j = q i^j > 1/2 ∀ i < j

29. Talk Outline 1.Background 2.New Classes of Bias where M nn is fast Choose your Weapon League Hierarchies 3.M nn can be slow

30. But…..M NN can be slow Thm 3: There are examples of positively biased {p ij } for which M NN is slowly mixing. 1. Reduce to biased lattice paths 13n2... 2 n +1 n 2 always in order { 1 if i < j ≤ 2 n or < i < j 2 n p ij = Permutation σ: 16278395410 1 ’ s and 0 ’ s 2 n 2 n 1010010110

31. Slow mixing example Thm 3: There are examples of positively biased {p} for which M NN is slowly mixing. 1. Reduce to biased lattice paths { 1 if i < j ≤ 2 n or < i < j 2 n p ij = 1/2+1/n 2 if i+(n-j+1) < M Each choice of p ij where i ≤ < j 2 n determines the bias on square (i, n-j+1) (“fluctuating bias”) 1/2 + 1/n 2 M= n 2/3 1- δ otherwise 1-δ 2. Define bias on individual cells (non-uniform growth proc.) [Greenberg, Pascoe, Randall]

32. Slow mixing example Thm 3: There are examples of positively biased {p} for which M NN is slowly mixing. 1. Reduce to biased lattice paths 2. Define bias on individual cells 3. Show that there is a “bad cut” in the state space Implies that M NN can take exp. time to reach stationarity. S S  Therefore biased card shuffling can be slow too!

33. Open Problems 1.Is M NN always rapidly mixing when {p i,j } are positively biased and satisfy a regularity condition? (i.e., p i,j is decreasing in i and j) 2. When does bias speed up or slow down a chain?

34. Thank you!