Probabilistic Juggling

Transcription

1 Probabilistic Juggling Arvind Ayyer (joint work with Jeremie Bouttier, Sylvie Corteel and Francois Nunzi) arxiv:1402:3752 Conference on STOCHASTIC SYSTEMS AND APPLICATIONS (aka Borkarfest) Indian Institute of Science Bangalore September 10, 2014

2 Outline 1 Solitary infallible juggler 2 Markov chain on set partitions 3 Superhuman juggler 4 Errant juggler with a partner

3 Multivariate Juggling Markov Chain (MJMC) Data: Siteswap notation h: the maximum height the juggler can throw. l: the number of balls he juggles. k = h l: the number of empty spaces he has to throw the balls in x a probability distribution on {0,..., k}. State space: St h,k {, } h Configurations B = (b 1, b 2,..., b h ) St h,k

4 Example: h = 8, k = l = 4

5 Example: h = 8, k = l = 4

6 Example: h = 8, k = l = 4

7 Example: h = 8, k = l = 4

8 Example: h = 8, k = l = 4 x 0 x 1 x 2 x 3 x 4

9 History 1994 Juggling sequences Buhler, Eisenbud, Graham and Wright (AMM) Connections to other combinatorial structures 2005 Juggling probabilities Warrington (AMM) x i uniform 2012 Juggler s exclusion process on Z, Leskelä and Varpanen

10 MJMC on St 4,2 x 2 1 x 0 x 1 x 0 x 1 x 2 x 1 1 x 2 1

11 MJMC on St 4,2 Basis: (gravity ) (,,,,, ) Transition matrix: x 0 x 1 x x x 1 x x 0 0 x 1 0 x Unnormalised Stationary distribution: 1 x 1 + x 2 x 2 (x 1 + x 2 ) 2 x 2 (x 1 + x 2 ) x 2 2

12 Results about the MJMC Proposition (A, Bouttier, Corteel, Nunzi, 14) If x i > 0 for all i {0,..., k}, then the MJMC is irreducible and aperiodic. Warrington s proof of this was incomplete!

13 Results about the MJMC For B St h,k, define e i (B) = #{j > i : b j = } and (B) = i {1,...,h} b i = (1 + e i (B)) Theorem (Warrington 05) If x is uniform, the stationary probability distribution of B St h,k is π(b) = (B) { h+1 k+1 }

14 Stirling Numbers of the Second Kind { n } j is the number of ways to partition an n-set into j parts. { } { } { } n + 1 n n = j + ; j j j 1 { } n = δ n,0. 0 For example { 4 2} = 7, since {1, 2, 3, 4} can be partitioned as 123 4, 3 124, 2 134, 1 234, 12 34, 13 24, The stationary probability distribution on St 4,2 becomes

15 Stationary distribution of the MJMC Theorem (A, Bouttier, Corteel, Nunzi, 14) The stationary distribution π of the MJMC is given by π(b) = 1 Z h,k i {1,...,h} b i = ( xei (B) + + x k ), where E i (B) = #{j < i b j = }, and Z h,k Z h,k (x 0,..., x k ) is the normalisation factor. Back to example (x 0 + x 1 + x 2 ) 2 (x 0 + x 1 + x 2 )(x 1 + x 2 ) (x 0 + x 1 + x 2 )x 2 (x 1 + x 2 ) 2 x 2 (x 1 + x 2 ) x 2 2

16 Normalisation factor Z h,k Let y m = k x j for m = 0,..., k. j=m Let h n (z 0,..., z k ) be the complete homogeneous symmetric polynomial of degree n, e.g., h 2 (z 0, z 1, z 2 ) = z z 0 z 1 + z 0 z 2 + z z 1 z 2 + z 2 2. Lemma (A, Bouttier, Corteel, Nunzi, 14) The normalisation factor can be written as Z h,k = h l (y 0, y 1,..., y k ).

17 Special cases Corollary (A, Bouttier, Corteel, Nunzi, 14) { } h + 1 Z h,k (1, 1,..., 1, 1) =, k + 1 { } h + 1 Z h,k (q k, q k 1,..., q, 1) = k + 1, q Z h,k (1, q,..., q k 1, q k ) = q k(h k) { h + 1 k + 1 }, 1/q Z h,k ((1 q), (1 q)q,..., (1 q)q k 1, q k ) = ( ) h. k q

18 Enriched Multivariate Juggling Markov Chain (EMJMC) Data: H = h + 1: Cardinality of the set {1,..., H} K = k + 1: Number of parts of the H-set x a probability distribution on {0,..., K 1}. State space: S(H, K) σ S(H, K) written in increasing order of block maxima, e.g., S(7, 3).

19 Example: ?

20 Example: ?

21 Example: ?

22 Example: ?

23 Example: ?

24 Example: ?

25 Example: ?

26 Example: ? x 0 x 1 x 2 x3 x 4 9

27 Results about the EMJMC Proposition (A, Bouttier, Corteel, Nunzi, 14) If x i > 0 for all i {0,..., k}, then the EMJMC is irreducible and aperiodic.

28 Arches (s, t) is an arch of σ S(H, K) if s and t are nearest neighbours in a block. C σ (s, t) be the number of blocks containing at least one element in {s, s + 1,..., t 1, t}. Example with S(7, 3)

29 Results about the EMJMC Theorem (A, Bouttier, Corteel, Nunzi, 14) The stationary distribution π of the EMJMC is given by ˆπ(σ) = 1 Ẑ H,K (s,t) arch of σ x K Cσ(s,t) where Ẑ H,K = Z h,k (x 0,..., x k ) is the normalisation factor. Corollary (Warrington, 05) When x is uniform, the stationary distribution on S(H, K) is uniform.

30 Lumping Proof for EMJMC by verifying the master equation Proof for MJMC by lumping/projection from EMJMC as follows:

31 Lumping Proof for EMJMC by verifying the master equation Proof for MJMC by lumping/projection from EMJMC as follows: Define ψ : S(H, K) St h,k so that b i = iff i is a block maximum in σ.

32 Full chain on S(4, 2) Red arrows and arches have probability x 0 Green arrows and arches have probability x 1 Gray arrows have probability x 0 + x 1 = ψ

33 Unbounded Multivariate Juggling Markov Chain (UMJMC) l balls, which can be thrown to arbitrary heights.

34 Unbounded Multivariate Juggling Markov Chain (UMJMC) l balls, which can be thrown to arbitrary heights. State space St (l) {, } N x a probability distribution on N

35 Unbounded Multivariate Juggling Markov Chain (UMJMC) l balls, which can be thrown to arbitrary heights. State space St (l) {, } N x a probability distribution on N Formally, states are infinite words in and with exactly l occurences of. T i (A) St (l) is the word obtained by replacing the (i + 1)-th occurrence of in A by. The transition probability from A = a 1 a 2 a 3 to B is 1 if a 1 = and B = a 2 a 3, P A,B = x i if a 1 = and B = T i (a 2 a 3 ), 0 otherwise.

36 Results about UMJMC Proposition (A, Bouttier, Corteel, Nunzi, 14) If x 0 and infinitely many x i s are nonzero, then the UMJMC is irreducible and aperiodic. Theorem (A, Bouttier, Corteel, Nunzi, 14) The unique invariant measure (up to constant of proportionality) of the UMJMC is given by w(b) = i N, b i = y Ei (B) where B = b 1 b 2 b 3 St (l), E i (B) = #{j < i b j = } and y m = j=m x j.

37 More about the invariant measure Theorem The invariant measure of the UMJMC is finite if and only if i x i <, i=0 in which case its total mass reads Z (l) = h l (y 0, y 1, y 2,...). Further, the UMJMC is positive recurrent if and only if the above holds. In that case, there is a unique stationary probability distribution, and the chain started from any initial state converges to it in total variation as time tends to infinity.

38 Remarks Both the solitary and superhuman juggling chains can be interpreted in terms of natural Markov chains on integer partitions. The superhuman juggling chain can also be naturally generalised to have an infinite number of balls (IMJMC). The conditions for the existence of a probability measure and positive recurrence in the IMJMC are identical to the UMJMC.

39 Multivariate Annihilation Juggling Markov Chain (MAJMC) Data: h: the maximum height the juggler can throw. z a probability distribution on {0,..., h}, with a z 0. State space: St h = {, } h

40 Example: h = 8

41 Example: h = 8

42 Example: h = 8 z 1 z 2 z 3 z 4 z 5 a + z 6 + z 7 + z 8

43 MAJMC for h = 2 Transition matrix in the basis (,,, ) z 1 0 z 2 + a 0 P = z 1 0 z 2 + a 0, 0 z 1 z 2 a 0 z 1 z 2 a Stationary probability distribution ( z 2 1, z 1 (z 2 + a), (z 1 + z 2 ) (z 2 + a), a (z 2 + a) )

44 Results for the MAJMC Theorem (A, Bouttier, Corteel, Nunzi, 14) The stationary probability distribution of the annihilation model is Π(B) = h ( z1 + + z ei (B)+1) k (z j z h + a), i=1 b i = j=1 where e i (B) = #{j : i < j h, b j = }. Moreover, Π(B) = (z z h + a) h = 1 B

45 Convergence to stationarity Theorem For any initial probability distribution η over St h, the distribution at time h is equal to the stationary distribution, namely ηp h = Π. In particular, the only eigenvalues of the transition matrix P are 1 (with multiplicity 1) and 0.

46 Thank you for your attention!