Domino shuffling and TASEP. Xufan Zhang. Brown Discrete Math Seminar, March 2018

Transcription

1 Domino shuffling and TASEP Xufan Zhang Brown Discrete Math Seminar, March 2018

2 Table of Contents 1 Domino tilings in an Aztec diamond 2 Domino shuffling 3 Totally asymmetric simple exclusion process(tasep) 4 Directed last passage percolation 5 Hydrodynamic limits

3 Domino tilings in an Aztec diamond Tile the region exactly by 1 2 and 2 1 dominoes. Depending on the color, there are four types of dominoes: North West South East

4 Domino tilings in an Aztec diamond

5 Domino tilings in an Aztec diamond Given the region, we put a probability measure on all possible tilings, where the probability of each tiling is a #Vertical Tiles, where a > 0.

6 Domino tilings in an Aztec diamond Given the region, we put a probability measure on all possible tilings, where the probability of each tiling is a #Vertical Tiles, where a > 0. When a = 1, all possible tilings have equal probability.

7 Domino tilings in an Aztec diamond A random sample with size= 100, a = 1, tilted 45 degrees

8 Domino tilings in an Aztec diamond A random sample with size= 100, a = 0.5, tilted 45 degrees

9 Domino tilings in an Aztec diamond Theorem (Arctic ellipse theorem) When the size of the Aztec diamond grows, with probability going to 1, the boundary of the unfrozen region approximates an ellipse x 2 + y 2 a 2 = 1 when appropriately zoomed out.

10 Domino tilings in an Aztec diamond Theorem (Arctic ellipse theorem) When the size of the Aztec diamond grows, with probability going to 1, the boundary of the unfrozen region approximates an ellipse x 2 + y 2 a 2 = 1 when appropriately zoomed out. When a = 1, this was proved by Jockusch, Propp and Shor in 1995 using TASEP. For general a using similar method, follows from Seppäläinen s approach.

11 Domino tilings in an Aztec diamond Theorem (Arctic ellipse theorem) When the size of the Aztec diamond grows, with probability going to 1, the boundary of the unfrozen region approximates an ellipse x 2 + y 2 a 2 = 1 when appropriately zoomed out. When a = 1, this was proved by Jockusch, Propp and Shor in 1995 using TASEP. For general a using similar method, follows from Seppäläinen s approach. This is also a result of the more general theory on the limit shape of dimer models, eg. by Cohn, Kenyon and Propp using variational principle.

12 Domino shuffling Domino shuffling is a discrete time dynamics on domino tilings. Recall the domino types: North West South East Roughly speaking, each domino attempts to slide 1 unit towards the direction according to its type, unless two collide, in which case both are destroyed. Then we fill the empty space randomly with 2 2 blocks.

13 Domino shuffling Take an Aztec diamond tiling as an example:

14 Domino shuffling Step 1: Destroy the colliding pairs

15 Domino shuffling Step 2: Slide the rest

16 Domino shuffling and TASEP Domino shuffling Step 3: Now we get a partial tiling in an Aztec diamond of rank 1 larger. The empty space can be uniquely divided into 2 2 a2 blocks. For each block, with probability 1+a 2 fill with and with probability 1 1+a2 fill with

17 Domino shuffling Step 3: In this way the checkerboard coloring is preserved.

18 Domino shuffling This shuffling algorithm generates random tilings with exactly the desired probability measure.

19 Domino shuffling This shuffling algorithm generates random tilings with exactly the desired probability measure. To prove by induction, assume every tiling of Aztec diamond of size n 1 has probability a #Vertical Tiles.

20 Domino shuffling This shuffling algorithm generates random tilings with exactly the desired probability measure. To prove by induction, assume every tiling of Aztec diamond of size n 1 has probability a #Vertical Tiles. For any given tiling of Aztec diamond of size n, we can see exactly how many 2 2 blocks were added in step 3, say i, and suppose j of them were filled vertically.

21 Domino shuffling This shuffling algorithm generates random tilings with exactly the desired probability measure. To prove by induction, assume every tiling of Aztec diamond of size n 1 has probability a #Vertical Tiles. For any given tiling of Aztec diamond of size n, we can see exactly how many 2 2 blocks were added in step 3, say i, and suppose j of them were filled vertically. Because the area of an Aztec diamond of size n is 2n(n + 1), there were i (2n(n + 1) 2(n 1)n)/4 = i n blocks destroyed in step 1.

22 Domino shuffling Say there were k vertical tiles during step 2 (sliding), then the total probability of the predecessors is a k (1 + a 2 ) i n, because each destroyed block could have been vertical or horizontal.

23 Domino shuffling Say there were k vertical tiles during step 2 (sliding), then the total probability of the predecessors is a k (1 + a 2 ) i n, because each destroyed block could have been vertical or horizontal. The transition probability from any possible predecessor ( ) i j ( ) 1 a 2 j to the current tiling is all. 1 + a a 2

24 Domino shuffling Say there were k vertical tiles during step 2 (sliding), then the total probability of the predecessors is a k (1 + a 2 ) i n, because each destroyed block could have been vertical or horizontal. The transition probability from any possible predecessor ( ) i j ( ) 1 a 2 j to the current tiling is all. 1 + a a 2 Therefore the resulted probability of the given tiling is ( ) i j ( ) 1 a a k (1 + a 2 ) i n 2 j 1 + a a 2 a k+2j where k + 2j is the number of vertical tiles in the given tiling.

25 Domino shuffling This shuffling dynamics can also be performed on domino tilings on a torus, cylinder or the plane.

26 Domino shuffling This shuffling dynamics can also be performed on domino tilings on a torus, cylinder or the plane. If we draw the following arrows in the West, East and North dominoes: We see paths formed by dominoes.

27 Domino shuffling

28 Domino shuffling These paths have some nice properties under shuffling:

29 Domino shuffling These paths have some nice properties under shuffling: 1 There is a bijection between the paths before and after the shuffle(except Aztec diamond).

30 Domino shuffling These paths have some nice properties under shuffling: 1 There is a bijection between the paths before and after the shuffle(except Aztec diamond). 2 On a cylinder or torus, the homology of each path does not change.

31 Domino shuffling These paths have some nice properties under shuffling: 1 There is a bijection between the paths before and after the shuffle(except Aztec diamond). 2 On a cylinder or torus, the homology of each path does not change. 3 The evolution of the southmost path (if exists) is not affected by what is above it.

32 Domino shuffling

33 Domino shuffling By adding two auxiliary dominoes to the Aztec diamond, there is always a lowest path in the Aztec diamond connecting them:

34 Domino shuffling In fact the boundary of the south frozen region in the Aztec diamond can be obtained by looking at the evolution of a single path in the plane under shuffling.

35 Totally asymmetric simple exclusion process(tasep) Consider the following model. Some balls move on Z N or Z lattice, such that each lattice site has at most 1 ball. The rule is such that, at each second t Z, every ball with an empty right neighbor will move there with probability β independently. β β This is called the discrete time totally asymmetric simple exclusion process (TASEP) with parallel updates.

36 Totally asymmetric simple exclusion process(tasep) Going back to domino shuffling of a single path on a cylinder or the plane, if we add a particle at each West and North domino, and look at their vertical projections before and after a shuffle, they exactly perform a TASEP with β = 1 1+a 2.

37 Domino shuffling and TASEP Totally asymmetric simple exclusion process(tasep)

38 Totally asymmetric simple exclusion process(tasep) Furthermore, this projection is almost reversible, modulo a local ambiguity.

39 Totally asymmetric simple exclusion process(tasep) So the problem of deciding the boundary of the frozen region becomes the problem of predicting the evolution of a TASEP starting from the (1, 0)-Riemann initial condition. 0

40 Totally asymmetric simple exclusion process(tasep) So the problem of deciding the boundary of the frozen region becomes the problem of predicting the evolution of a TASEP starting from the (1, 0)-Riemann initial condition. 0 In particular, if σ(i, t) {0, 1} denotes whether site i has a ball at time t, we want to find a deterministic function φ(x, t) such that 1 lim n n ny i=nx σ(i, nt) = φ(y, t) φ(x, t)

41 Totally asymmetric simple exclusion process(tasep) Some properties about the TASEP: Proposition The unique stationary measure µ N,M on Z N with M particles is µ N,M (σ) (1 β) G(σ) where G(σ) is the number of 01 pairs (or equivalently 10 pairs) in σ.

42 Totally asymmetric simple exclusion process(tasep) Proof. Each state σ has 2 G(σ) predecessors. After one second, the probability of σ is (1 β) G(η) (1 β) G(η) J(η σ) β J(η σ) η σ where J(η σ) is the number of hops. This is just (1 β) G(σ) (1 β) G(σ) J(η σ) β J(η σ) η σ =(1 β) G(σ) (β + 1 β) G(σ) = (1 β) G(σ)

43 Totally asymmetric simple exclusion process(tasep) Using saddle point analysis, we deduce the following Proposition For each p (0, 1), the sequence (µ N, pn ) converges weakly to a measure µ p on {0, 1} Z as N.

44 Totally asymmetric simple exclusion process(tasep) Using saddle point analysis, we deduce the following Proposition For each p (0, 1), the sequence (µ N, pn ) converges weakly to a measure µ p on {0, 1} Z as N. µ p defines a Markov process, where each site only depends on its left neighbor, with transition probabilities q ij = P µp [σ(x + 1) = j σ(x) = i] satisfying pq 11 + (1 p)q 01 = p q 10 + q 11 = 1 q 00 + q 01 = 1 q 00 q 11 = (1 β)q 10 q 01

45 Totally asymmetric simple exclusion process(tasep) Since weak convergence means convergence on finite cylindrical sets, and the TASEP dynamics is local, Corollary µ p described above are stationary measures of TASEP on Z.

46 Totally asymmetric simple exclusion process(tasep) Since weak convergence means convergence on finite cylindrical sets, and the TASEP dynamics is local, Corollary µ p described above are stationary measures of TASEP on Z. In fact, they are extremal in the space of ergodic and stationary measures.

47 Totally asymmetric simple exclusion process(tasep) Since weak convergence means convergence on finite cylindrical sets, and the TASEP dynamics is local, Corollary µ p described above are stationary measures of TASEP on Z. In fact, they are extremal in the space of ergodic and stationary measures. When β 1 2, a good coupling exists, which can be used to show that these are the only extremal measures. This led to the first proof of arctic circle theorem.

48 Totally asymmetric simple exclusion process(tasep) Since weak convergence means convergence on finite cylindrical sets, and the TASEP dynamics is local, Corollary µ p described above are stationary measures of TASEP on Z. In fact, they are extremal in the space of ergodic and stationary measures. When β 1, a good coupling exists, which can be used 2 to show that these are the only extremal measures. This led to the first proof of arctic circle theorem. Conservation of local equilibrium can also be proved in this case, in the sense that a traveller with constant speed will see a measure converging to one of the extremal measures.

49 Totally asymmetric simple exclusion process(tasep) A common crucial ingredient in hydrodynamic limit theory is attractiveness, which means there exists a coupling such that if one configuration completely dominates another, then this dominance is preserved under the dynamics.

50 Totally asymmetric simple exclusion process(tasep) A common crucial ingredient in hydrodynamic limit theory is attractiveness, which means there exists a coupling such that if one configuration completely dominates another, then this dominance is preserved under the dynamics. For TASEP, if we couple particles at the same location, it is not attractive.

51 Totally asymmetric simple exclusion process(tasep) When β 1, we can couple diagonally, 2 β β 1-2β

52 Totally asymmetric simple exclusion process(tasep) When β 1, we can couple diagonally, 2 β β 1-2β When β > 1, attractiveness is just impossible. 2

53 Totally asymmetric simple exclusion process(tasep) Seppäläinen s approach is to look at a height process h(x, t) defined by TASEP, where each particle represents a step-up. 0

54 Totally asymmetric simple exclusion process(tasep) Seppäläinen s approach is to look at a height process h(x, t) defined by TASEP, where each particle represents a step-up. 0 This is similar to the domino path representation. But TASEP knowledge is still useful later.

55 Totally asymmetric simple exclusion process(tasep) This height process is attractive for all β, using the trivial coupling

56 Totally asymmetric simple exclusion process(tasep) Furthermore, this height process satisfies an enveloping property, meaning if a height function h(x) is the upper envelope of a family of height functions g i (x), then this remains true under coupling. then That is, if h(x, 0) = sup {g i (x, 0)} i h(x, t) = sup {g i (x, t)} ( ) i

57 Totally asymmetric simple exclusion process(tasep) Proof by induction: Suppose site x is asked to jump at time t. Case 1: If h(x, t) decreases by 1, then ( ) remains true due to attractiveness;

58 Totally asymmetric simple exclusion process(tasep) Proof by induction: Suppose site x is asked to jump at time t. Case 1: If h(x, t) decreases by 1, then ( ) remains true due to attractiveness; Case 2: If h(x, t) cannot decrease because h(x, t 1) = h(x 1, t 1), then the induction hypothesis implies some g i agrees with h on sites x, x 1 at time t 1, so g i (x, t) = h(x, t).

59 Totally asymmetric simple exclusion process(tasep) Proof by induction: Suppose site x is asked to jump at time t. Case 1: If h(x, t) decreases by 1, then ( ) remains true due to attractiveness; Case 2: If h(x, t) cannot decrease because h(x, t 1) = h(x 1, t 1), then the induction hypothesis implies some g i agrees with h on sites x, x 1 at time t 1, so g i (x, t) = h(x, t). Case 3: h(x, t) cannot decrease because h(x, t 1) = h(x + 1, t 1), similar to case 2.

60 Totally asymmetric simple exclusion process(tasep) Any legal initial height function h(x, 0) is the upper envelope of the following height function g 0 (x) and its translations O which is exactly the (1, 0)-Riemann initial condition.

61 Directed last passage percolation The goal is to deduce the hydrodynamic limit of g(x, t) given g(x, 0) = g 0 (x). Same as the growing frozen region of Aztec diamond, consider the region above the height process as a growing region.

62 Directed last passage percolation The goal is to deduce the hydrodynamic limit of g(x, t) given g(x, 0) = g 0 (x). Same as the growing frozen region of Aztec diamond, consider the region above the height process as a growing region. A square can grow iff its left neighbor and upper right neighbor have grown.

63 Directed last passage percolation Let T (x, y) denote the first time box (x, y) joins the region, then T (x, y) satisfies the recurrence relation T (x, y) = max {T (x 1, y), T (x + 1, y + 1)} + ω(x, y) where ω(x, y) s are iid geometrically distributed random variables P(ω(x, y) = k) = (1 β) k 1 β defined on all boxes not initially in the growing region.

64 Directed last passage percolation Let T (x, y) denote the first time box (x, y) joins the region, then T (x, y) satisfies the recurrence relation T (x, y) = max {T (x 1, y), T (x + 1, y + 1)} + ω(x, y) where ω(x, y) s are iid geometrically distributed random variables P(ω(x, y) = k) = (1 β) k 1 β defined on all boxes not initially in the growing region. Then the height process is the level curve of T (x, y), i.e. g(x, t) = max {y : T (x, y) t}

65 Directed last passage percolation This is called a directed last passage percolation, where the allowed edge are (x 1, y) (x, y) and (x + 1, y + 1) (x, y).

66 Directed last passage percolation This is called a directed last passage percolation, where the allowed edge are (x 1, y) (x, y) and (x + 1, y + 1) (x, y). More generally, define T ((u, v), (z, w)) = max π Π (x,y) π ω(x, y) where Π is the collection of all legal paths from (u, v) to (z, w), minus (u, v).

67 Directed last passage percolation This is called a directed last passage percolation, where the allowed edge are (x 1, y) (x, y) and (x + 1, y + 1) (x, y). More generally, define T ((u, v), (z, w)) = max π Π (x,y) π ω(x, y) where Π is the collection of all legal paths from (u, v) to (z, w), minus (u, v). In particular, T (x, y) = T ((0, 0), (x, y)).

68 Directed last passage percolation By definition, these passage times satisfy superadditivity: T ((u, v), (x, y)) T ((u, v), (z, w)) + T ((z, w), (x, y))

69 Directed last passage percolation By definition, these passage times satisfy superadditivity: T ((u, v), (x, y)) T ((u, v), (z, w)) + T ((z, w), (x, y)) Theorem (Kingman s superadditive ergodic theorem) If {X n } is a sequence of random variables, τ is an ergodic transformation on the common probability space, and X n+m X n + X m τ n, then is constant almost surely. X n lim n n

70 Directed last passage percolation By definition, these passage times satisfy superadditivity: T ((u, v), (x, y)) T ((u, v), (z, w)) + T ((z, w), (x, y)) Theorem (Kingman s superadditive ergodic theorem) If {X n } is a sequence of random variables, τ is an ergodic transformation on the common probability space, and X n+m X n + X m τ n, then is constant almost surely. X n lim n n For example, this implies the strong law of large numbers for iid random variables.

71 Directed last passage percolation In this case, this theorem will imply that for all real (x, y), lim n is a constant almost surely. 1 T ( nx, ny ) n

72 Directed last passage percolation In this case, this theorem will imply that for all real (x, y), lim n is a constant almost surely. 1 T ( nx, ny ) n This limit is a function of x and y, homogeneuous of degree 1 and concave. Recall that the height process g(x, t) is the level curve of T (x, y), so 1 ( x ) lim n n g ( nx, nt ) = tγ t for some concave γ, which is the hydrodynamic limit we are after.

73 Hydrodynamic limits Recall that any legal initial height function h(x, 0) is the upper envelope of the following height function g 0 (x) and its translations O This is like the (1, 0)-Riemann initial condition.

74 Hydrodynamic limits Suppose we know ahead all the randomness of h(x, t), which has Z N copies of iid Bernoulli random variables.

75 Hydrodynamic limits Suppose we know ahead all the randomness of h(x, t), which has Z N copies of iid Bernoulli random variables. Let g z (y, t) be the height evolution that starts with initial condition g 0, but uses the Bernoulli random variable at (z + y, t). Then the enveloping property implies that h(x, t) = sup {h(z, 0) + g z (x z, t)} z

76 Hydrodynamic limits Suppose we know ahead all the randomness of h(x, t), which has Z N copies of iid Bernoulli random variables. Let g z (y, t) be the height evolution that starts with initial condition g 0, but uses the Bernoulli random variable at (z + y, t). Then the enveloping property implies that h(x, t) = sup {h(z, 0) + g z (x z, t)} z This is a microscopic Hopf-Lax formula, a type of phenomena observed first by Aldous and Diaconis when analyzing the longest increasing subsequence in a random permutation.

77 Hydrodynamic limits Given and we also know that h(x, t) = sup {h(z, 0) + g z (x z, t)}, z 1 ( x lim n n g ( nx, nt ) = tγ t for some unknown concave function γ. So suppose we can pass the limit, { ( )} 1 1 x z lim h ( nx, nt ) = sup lim h ( nz, 0) + tγ n n z n n t )

78 Hydrodynamic limits lim n { 1 h ( nx, nt ) = sup lim n z n 1 h ( nz, 0) + tγ n This would make sense if the deterministic/random initial height function h(z, 0) satisfies that lim n 1 for some deterministic φ 0, in which case lim n 1 h ( nx, nt ) = sup n z { φ 0 (z) + tγ ( )} x z n h ( nz, 0) = φ 0(x) t ( )} x z t

79 Hydrodynamic limits lim n { 1 h ( nx, nt ) = sup lim n z n 1 h ( nz, 0) + tγ n This would make sense if the deterministic/random initial height function h(z, 0) satisfies that lim n 1 for some deterministic φ 0, in which case lim n 1 h ( nx, nt ) = sup n z { φ 0 (z) + tγ Passing the limit is due to linear propagation. ( )} x z n h ( nz, 0) = φ 0(x) t ( )} x z t

80 Hydrodynamic limits For Hamilton-Jacobi equations φ t + H(Dφ) = 0 with initial data φ(x, 0), and strictly concave H, its unique viscosity solution is given by the Hopf-Lax formula { ( )} x z φ(x, t) = sup φ(z, 0) + tl z R t d where L is the Legendre transform of H, L(v) = inf p {pv H(p)}.

81 Hydrodynamic limits Modulo some technicality, conclude that the height process has hydrodynamic limit described by a H-J equation φ t + H(φ x ) = 0, and γ is the Legendre transform of H.

82 Hydrodynamic limits Modulo some technicality, conclude that the height process has hydrodynamic limit described by a H-J equation φ t + H(φ x ) = 0, and γ is the Legendre transform of H. If the random initial height function is a TASEP extremal measure µ p, then the PDE initial condition φ 0 (x) = lim n 1 h( nx, 0) is just a straight line y = px. n

83 Hydrodynamic limits Modulo some technicality, conclude that the height process has hydrodynamic limit described by a H-J equation φ t + H(φ x ) = 0, and γ is the Legendre transform of H. If the random initial height function is a TASEP extremal measure µ p, then the PDE initial condition φ 0 (x) = lim n 1 h( nx, 0) is just a straight line y = px. n Under the PDE, this line just moves down at speed H(p). But it is also the average speed of particles in the TASEP, which has to be βpq 10. Take a Legendre transform to finally obtain γ.

84 Hydrodynamic limits x x β γ(x) = x 1 β(1 β)(β x + 2 ) β < x < β 2 2β 0 x β When β = 0.6,

85 Hydrodynamic limits Advantages: 1 Allows general initial conditions 2 No assumption of uniqueness of extremal measures 3 A priori works for any dimension d

86 Hydrodynamic limits Advantages: 1 Allows general initial conditions 2 No assumption of uniqueness of extremal measures 3 A priori works for any dimension d Limitations: 1 Does not give local equilibrium 2 Requires enveloping property 3 Requires convex/concave speed function

87 Hydrodynamic limits Some further developments In 2001, Rezakanlou provided a still state-of-the-art scheme that works for any dimension, without assuming enveloping property and concavity, but does not produce deterministic limit, and so far no actual application has been done for d > 1.

88 Hydrodynamic limits Some further developments In 2001, Rezakanlou provided a still state-of-the-art scheme that works for any dimension, without assuming enveloping property and concavity, but does not produce deterministic limit, and so far no actual application has been done for d > 1. In parallel, since 2000s, Bahadoran et al. have developed a very robust scheme for a wide range of processes, without assuming concavity or invariant measures, but only works for d = 1.

89 Hydrodynamic limits Some further developments In 2001, Rezakanlou provided a still state-of-the-art scheme that works for any dimension, without assuming enveloping property and concavity, but does not produce deterministic limit, and so far no actual application has been done for d > 1. In parallel, since 2000s, Bahadoran et al. have developed a very robust scheme for a wide range of processes, without assuming concavity or invariant measures, but only works for d = 1. In 2017, Legras and Toninelli proved hydrodynamic limit for a 2d Lozenge tiling dynamics, either up to the first shock time or when the initial condition is convex.

90 Hydrodynamic limits One can consider a two dimensional analogue using the height function of a domino tiling:

91 Hydrodynamic limits Under the domino shuffling, the height function has a local update rule

92 Hydrodynamic limits Conjecture At a = 1, the height function h(x, y, t) has a hydrodynamic limit where for all time. φ t + H(Dφ) = 0 ( H(z, w) = 2 ( cos πz ) ( π arcsin 2 cos πw 2 2 ))

93 Domino shuffling and TASEP Hydrodynamic limits Starting from the following periodic initial condition:

94 Domino shuffling and TASEP Hydrodynamic limits Domino shuffling on grid at t = 0.25 (rescaled):

95 Domino shuffling and TASEP Hydrodynamic limits PDE solved numerically at t = 0.25:

96 Hydrodynamic limits The speed function H is neither convex nor concave, and there is no enveloping property. Shock develops very quickly.

97 Hydrodynamic limits Further questions 1 What is the family of PDEs that we can obtain from dimer models? 2 Is there a probabilistic interpretation of Evans general formula for H-J PDEs? 3 Is it possible to define/show conservation of local equilibrium?

98 Hydrodynamic limits Thank you!

99 Hydrodynamic limits References I C. Bahadoran, H. Guiol, K. Ravishankar, E. Saada, et al. Euler hydrodynamics for attractive particle systems in random environment. In Annales de l Institut Henri Poincaré, Probabilités et Statistiques, volume 50, pages Institut Henri Poincaré, W. Jockusch, J. Propp, and P. Shor. Random domino tilings and the arctic circle theorem. arxiv:math/ v1 [math.co].

100 Hydrodynamic limits References II M. Legras and F. L. Toninelli. Hydrodynamic limit and viscosity solutions for a 2d growth process in the anisotropic kpz class. arxiv preprint arxiv: , F. Rezakhanlou. Continuum limit for some growth models ii. Annals of probability, pages , T. Seppäläinen. Translation Invariant Exclusion Processes (Book in Progress). ajo.pdf Accessed: