Domino tilings, non-intersecting Random Motions and Critical Processes

Transcription

1 Domino tilings, non-intersecting Random Motions and Critical Processes Pierre van Moerbeke Université de Louvain, Belgium & Brandeis University, Massachusetts Institute of Physics, University of Edinburgh (Dec 2011)

2 1. The Airy process 2. The Pearcey process 3. The Tacnode process 4. Domino tilings of Aztec Diamonds 5. Domino tilings of Double Aztec Diamonds

3 I. The Airy process A(τ) Non-intersecting Brownian motions leaving from 0 at t = 0 returning to 0 at t = 1 (Equivalent with Dyson s Brownian motions) Let n : Infinite-dimensional diffusion! (Prähofer-Spohn 02, Johansson 03, 05, Tracy-Widom 04, Adler-PvM 05),

4

5 Look through an Airy microscope at the fluctuations of the paths about any point on the curve, for very large n: (x, t) = ( 2n 2 cosh σ, e σ ) 2 cosh σ { curve: x = } 2nt(1 t) Markov cloud A(τ) in the neighborhood of this point (x, t) curve : Pick times 0 < τ 1 <... < τ m < 1 and points ξ 1,..., ξ m R. lim n P Br m k=1 all x i e σ+τ kn 13 2 cosh(σ + τ k n 13 ) 2n + ξ k 2n 16 2 cosh(σ + τ k n 13 ) =: det ( ) I (χ (ξi, ) KA (τ i, ξ i ; τ j, ξ j )χ (ξj, ) ) 1 i,j m, independent of σ!!! = P (A(τ 1 ) ξ 1,..., A(τ m ) ξ m ) (Universality!) Airy process

6 with kernel: (matrix Fredholm determinant!!) K A (τ i, ξ i ; τ j, ξ j ) := 1 (2πi) 2 du dv e1 3 U 3 ξ i U e 1 3 V 3 ξ j V 1 (V + τ j ) (U + τ i ) 1 j ξ i)2 (τ j >τ i ) 4(τ e (ξ j τ i ) 1 2 (τ j τ i )(ξ j +ξ i ) (τ j τ i ) 3 4π(τj τ i ) In particular, for τ i = τ j = τ, we have the Airy kernel K A (τ, x; τ, y) = K Ai (x, y) = = ( 1 2πi 0 ) 2 du dv V U Ai(x + λ)ai(y + λ)dλ e 1 3 U 3 xu e 1 3 V 3 yv Ai(x) = 1 π cos ( t 3 + xt ) dt, satisfying d2 y xy = 0, dx2

7 P(A(τ) x) = e x (α x)g2 (α)dα =: F(x) (T-W distribution 95) (time-independent: stationary process)

8 P(A(τ) x) = e x (α x)g2 (α)dα =: F(x) (T-W distribution 95) (time-independent: stationary process) Q(s; x, y) := log P (A(τ 1 ) x + y, A(τ 2 ) x y), s = τ 2 τ 1 > 0 satisfies 3 Q 2s s x y = ( s 2 2 y y ) ( 2 ) Q x y 2 2 Q x 2 + { 2 } Q x y, 2 Q x 2. x with initial condition: for s = τ 2 τ 1, {, } x is a Wronskian with regard to x

9 2. The Pearcey process P(τ) Non-intersecting Brownian motions: leaving from 0 at t = 0 [(1 p)n] paths forced to end at 0 [pn] right paths forced to end at a n at t = 1 a > 0, 0 < p < 1 Let n : Pastur 72, Brézin-Hikami 96-98, Johansson 01, Bleher-Kuijlaars I, II, III 04, Tracy-Widom 05, Okounkov-Reshetikhin 05, Adler-PvM 05, Adler- Orantin-PvM 08

10 cusp x x 0 = 2 ( ) t t at 2q 1 x 0 = at 0 q+1 t 0 = ( a(q 3 1) q+1 ) 2. convenient parametrization: p = q 3

11 Look through the Pearcey microscope at the fluctuations of the paths about the cusp, for very large n: ( t t0 cusp : x x 0 = 2 3 ) 32 New process P(τ) in the neighborhood of the cusp : [ξ 1, ξ 2 ] R ( lim n P(n) Br (all x j t 0 + (c 0 µ) 2 2τ ) n 12 x 0 n 12 + c 0 Aτ + c 0 µ [ξ 1, ξ 2 ] c ) n 14 = det ( I K P) [ξ 1,ξ 2 ] (independent of pn and a: universality!), =: P P (P(τ) [ξ 1, ξ 2 ] = ) (Pearcey process) (via steepest descent method) µ 4 = q2 q + 1, c 0 := at 0 q q 2 q + 1 q + 1, A = q (2q 1)t 0 q

12 with Pearcey kernel: K P (τ; ξ, η) = 1 (2πi) 2 i i du contour Γ = 0 Γ dv 1 U V e U4 4 +τ U2 2 ηu e V 4 4 +τ V 2 2 ξv Then Q(t; x, y) = log P P (P(t) [x, y] = ) satisfies a PDE: 3 Q t (E 2t t ) 2 2 Q 1 2 { 2 Q, Q t } = 0. := x + y, E := x x + y y (Adler-PvM 06, Adler-Orantin-PvM 08, Adler-Cafasso-PvM)

13 3. The Tacnode process

14 Universality: three models, leading to the Tacnode process: (1) continuous time and discrete space process: (random walks) Mark Adler, Patrik Ferrari & PvM : Non-intersecting random walks in the neighborhood of a symmetric tacnode, Annals of Prob 2011 (arxiv: ) (2) discrete time and discrete space process (Domino tilings): Mark Adler, Kurt Johansson & PvM: Double Aztec Diamonds (to appear) (3) continuous time and continuous space process Delvaux-Kuijlaars-Zhang: Critical behavior of non-intersecting Brownian motions at a Tacnode (arxiv: ) K. Johansson: Non-colliding Brownian Motions and the extended tacnode process (arxiv: )

15 4. Domino tilings of Aztec Diamonds

16 An Aztec diamond (size n):

17 Domino tilings of an Aztec diamond : 4 types of situations for the domino s! Kasteleyn 1961, Elkies, Kuperberg, Larsen, Propp 92 Cohn, Elkies, Propp 96, Johansson 00, 03, 05 M. Fulmek, C. Krattenthaler 00, Johansson-Nordenstam 06

18 Arctic circle for a = 1.

19 Associated paths: above the top curve X n (t) for n t n only (North) (South) (West) (East) The curves never penetrate North domino s

20 Associated point process: red lines Interlacing red dots on the Interlacing red dots

21 A weight on domino s = a probability on domino tilings : put the weight 0 < a 1 on vertical dominoes put the weight 1 on horizontal dominoes, Define: P(domino tiling T ) = a #vertical domino s in T a #vertical domino s in T (1) all possible tilings T Let n :

22 Frozen North Frozen West y Disordered region x Frozen East Frozen South Macroscopically: arctic Ellipse (N. Elkies, Kuperberg, Larsen and Propp 92, W. Jockush J. Propp and P. Shor 98) x 2 p + y2 q = 1, with q = a and p = 1 q = a 1 a + a 1 Fluctuations about the generic points of the ellipse: Airy process (Johansson 05) (for a = 1) X n (2 16 n 23 t) n n 13 A(t) t 2 a + a 1,

23 Frozen North Frozen West y Disordered region x Frozen East Frozen South Fluctuations about the generic points of the ellipse: Airy process (Johansson 05) (for a = 1) X n (2 16 n 23 t) n n 13 A(t) t 2 Fluctuations near the points of osculation of the ellipse with the square: GUE-minor process (Johansson-Nordenstam 06): Spectrum of consecutive principal minors of GUE-matrices!

24 Associated point process: red dots on the red lines Interlacing red dots for n spectrum of consecutive principal minors of GUE-matrices

25 Limits to the Airy process and to the GUE-minor process: For the Aztec diamond, (Johansson 05, Johansson & Nordenstam 06) P ( finding red dots at position xj along line r j for j = 1,..., m ) = det ( K n (r i, x i ; r j, x j ) ) 1 i,j m where (Krawtchouk kernel) (for a = 1) K n (2r, x; 2s, y) = 1 (2πi) 2 Γ dz γ dw z w w y 1 z x (1 w) s (1 + w 1 )n s (1 z) r (1 + 1, for r s z )n r Limit to Airy process: lim n n13 K n (cn τ 1 n 23, c n ξ 1 n 13 ; cn τ 2 n 23, c n ξ 2 n 13 ) = K A (τ 1, ξ 1 ; τ 2, ξ 2 ) Limit to GUE-minor process: ( [ g n (r, ξ) n n n g n (s, η) 2 K n 2r, 2 + ξ 2 lim n ] ; 2s, [ n n 2 + η 2 ]) = K GUEminor (r, ξ; s, η)

26 Johansson & Nordenstam prove: (r, s Z 0 ) K GUEminor (r, x; s, y) := 1 j= j=0 (s + j)! (r + j)! h r+j(x)h s+j (y)e (x2 +y 2 )2 (s + j)! (r + j)! h r+j(x)h s+j (y)e (x2 +y 2 )2 for r s for r > s P n i=1 {spectrum of i i minor} E i = E i R = det [I (χ Ei (x i )K GUEminor ] (i, x i ; j, x j )χ Ej (x j )) 1 i,j n

27 5. Domino tilings of Double Aztec Diamonds

28 Double Aztec diamond (Adler-Johansson-PvM 11) Diamond A o 1 o 1 n Y = n + m o n ( m, m + o n 1 2 ) i 2m+1 i 2m+1 i 4 i 4 i 3 (0, 0) i 3 2m + 1 inliers i 2 i 2 i 1 i 1 (m, m 1 2 ) o n+1 o n+1 o 2n o 2n Diamond B Figure 1: Double Aztec diamond of type (n, m) = (7, 2) with #{inliers} = M = 2m + 1 = 5.

29 A weight on domino s, a probability on domino tilings and on nonintersecting random walks: put the weight 0 < a < 1 on vertical dominoes put the weight 1 on horizontal dominoes, Define: P(domino tiling T ) = a #vertical domino s in T a #vertical domino s in T (2) all possible tilings T

30 Cover randomly with domino s, leading to non-intersecting paths: n-1 n n height and level lines for n= n n n n n n n n n n n n n n n n n n n+2 n n+3 n n+4 n n+5 n n 2n 2n n n n+1 h h h h h h h h h = North h+1 h+1 h+1 = South h h h h h+1 h = East h h+1 h+1 h+1 h+1 h+1 = West

31 h h h h h h h h h = North h+1 h+1 h+1 = South h h h h h+1 h = East h h+1 h+1 h+1 h+1 h+1 = West Figure 4. Height function on domino s and level-lines. Domino-tiling Random Surface (piecewise-linear) The curves are level curves for this Random Surface Implies fixed heights along the boundary

32 n-1 n n height and level lines for n= n n n n n n n n n n n n n n n n n n n+2 n n+3 n n+4 n n+5 n n 2n 2n n n n+1

33 Question: For an axis Y 2r going through black squares only, and an interval [k, l] Z, P(height function is flat along the interval [k, l]) =? P(domino s are pointing to the left of Y 2r along the points of interval [k, l]) = P(no dots along the points of interval [k, l]) =? h h h h h h h h h = North h+1 h+1 h+1 = South h h h h h+1 h = East h h+1 h+1 h+1 h+1 h+1 = West Figure 4. Height function on domino s and level-lines.

34 How does one turn this into a determinantal process? In other terms: How does one turn this into non-intersecting random walks, with synchronized time? (i) Complete the double Aztec diamond with South domino s (in a trivial way). (South) (ii) Then add dot-particles and circle-particles according to the recipe: West East South

35 C 2 D 2 C 3 D 3 C 4 D 4 C 5 D r C r+1 D 6 C 7 D 7 C 8 D 8 O 1 O 2 O 3 O 4 O 5 O 6 O n Y = 0 O n+1 O n+2 O n+3 O n+4 Y A 1 B 1 A 2 B 2 A 3 B 3 A 4 B 4 A 5 B r A r+1 B 6 A 7 B 7 O n+5 O n+6 O 2n l Y 0 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 2r 1 Y 2r Y 2r+1 Y 12 Y 13 Y 14 Figure 8. Outlier non-intersecting domino paths, circles and dots. The coordinate Y 2r (resp. Y 2r 1 ) records the location of the dots (resp. circles), obtained by moving the dots (resp. circles) in between Y 2r and Y 2r 1 to the axis Y 2r (resp. Y 2r 1 ). Gaps at 4, 2, 0, 4, 6.

36 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 2r 1 Y 2r Y 11 Y 12 Y 13 Y 14 P ({non-intersecting random walks} [k, l] = along Y 2r ) =?

37 A little preview: Let n : ( r 2, 1 + r 2 ) Diamond A 1 2 (p r 2, q + r 2 ) ( 1 r 2, r 2 ) ( r 2, r 2 ) (0, 0) Diamond B 2r (p + r 2, q r 2 ) ( r 2, r 2 ) (1 + r 2, r 2 ) 2(1 r) q = a a+a 1, p = a 1 a+a 1 ( r 2, 1 r 2 ) r = 2 pq = 2 a+a 1 Figure 10: The macroscopic description of the double Aztec diamonds and their frozen region outside the ellipses.

38 In order to compute the kernel for the determinantal process, it is easier to first consider Inliers, instead of the walks before (outliers). Therefore remember the height function and introduce a dual heigth: h h h h h h h h h = North h+1 h = South = East h+1 h+1 h+1 h h h+1 h+1 h+1 h+1 h h = West h h+1 Figure 4. Height function h on domino s and level line. h + 1 h + 1 h + 1 = North = South h h h h h h h h h h + 1 h + 1 h + 1 h = East h + 1 h + 1 h h + 1 h h h h = West Figure 5. Dual height function h on domino s and level line.

39 2m+1 2m+1 2m 2m+1 2m+1 2m+1 2m+1 2m+1 2m+1 2m+1 2m+1 2m+1 2m+1 2m+1 0 2m+1 2m+1 2m 2m 2m. 2m m m+1 2m Y 2r 1 Y 2r Y 2r+1

40 x B A B A B A B A B A B A B A B

41 Figure 17: Lattice paths: superimposing in- and outliers.

42 B B B B B B B B x } dot-to-circle -steps A A A A : ψa 2j,2j+1 (x, y) := A A { a x y, if y x 0 0 otherwise } = Γ 0,a dz 2πiz z x y 1 a z circle-to-dot -steps } Then define in general: where : ψ 2j+1,2j+2 (x, y) := a, if y x = 1 1, if y x = 0 0, otherwise = Γ 0 { ψr,s = ψ r,r+1... ψ s 1,s, if s > r = 0, if s r. f(x) g(x) := x Z f(x)g(x) dz 2πiz zx y (1 + az).

43 Lindström-Gessel-Viennot (Karlin-McGregor) P(r; x 1,..., x 2m+1 ) = 1 Z n,m det(ψ 0,r ( m + i 1, x j )) 1 i,j 2m+1 det(ψ r,2n+1 (x j, m + i 1)) 1 i,j = det ( K n,m (r, x i ; r, y j ) ) 1 i,j m Then (Eynard-Mehta formula) K n,m (r, x; s, y) = 2m+1 i,j=1 ψ r,2n+1 (x, m + i 1)([A 1 ] ij )ψ 0,s ( m + j 1, y) 1 r<s ψ r,s (x, y),

44 Lindström-Gessel-Viennot (Karlin-McGregor) P(r; x 1,..., x 2m+1 ) = 1 Z n,m det(ψ 0,r ( m + i 1, x j )) 1 i,j 2m+1 det(ψ r,2n+1 (x j, m + i 1)) 1 i,j Then (Eynard-Mehta formula) K n,m (r, x; s, y) = 2m+1 i,j=1 ψ r,2n+1 (x, m + i 1)([A 1 ] ij )ψ 0,s ( m + j 1, y) 1 r<s ψ r,s (x, y), with the entries of the (2m + 1) (2m + 1) matrix A, A ij = ψ 0,2n+1 ( m + i 1, m + j 1) = f i g j, defined in terms of the functions f j (y) = ψ 0,n ( m + j 1, y) = g k (x) = ψ n,2n+1 (x, m + k 1) = Γ 0,a Γ 0,a dz z j 1 z m+y ρ L a {}} (z) { ) n2 ( 1 + az 2πiz 1 a z dz z m+x (1 + az) n2 2πiz z k 1 ( ) 1 a n2+1, z } {{ } ρ R a (z)

45 A A A A A A A B B B B B B B B x P(red dots at x 1,..., x k, along vertical line Y n ) = det(k n,m (n, x i ; n, x j )) 1 i,j k Consider the kernel at a single (half-way even) time n: where K n,m (n, x; n, y) := K n,m (r, x; s, y) = r=s=n A kl = f k g l, 2m+1 i,j=1 g i (x)([a 1 ] ij )f j (y). Find linear combinations f i = 1 j i c ij f j and g i = 1 j i c ij g j such that A kl = δ k,l

46 Kernel (inlier paths): Expression in terms of orthogonal polynomials P k and P k on the circle for the weight ρ L a (z)ρ R a (z): x B A B A B A B A B A B A B A B K n,m (n, x; n, y) = = 2m+1 k,l=1 2m+1 i,j=1 = 1 (2πi) 2 g k (x)([ã 1 ] kl ) f l (y) with à = 1, g i (x) f j (y) dw γ r2 w ρr a (w) dz (z)wx+m γ r1 z ρl a z y+m 2m k=0 (inlier paths) P k (z) P k (w 1 ). Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y2r 1 Y2r Y11 Y12 Y13 Y14 Kernel (outlier paths) K n,m (n, x; n, y) = δ x,y K n,m (n, x; n, y). Kernel for the outlier paths at different levels: (Extended kernel) K n,m (2r, x; 2s, y) (outlier paths) = 1 s<r ψ (2s 2r) (x, y) + ψ n 2r (x, ) K n,m (n, ; n, ) ψ (2s n) (, y),

47 Then the kernel for the Double Aztec Diamond reads: K n,m (2r, x; 2s, y) = {Kernel for the single Aztec diamond} + ( 1) x y k=2m+1 b x,r (k)[(1 K) 1 a y,s ](k)

48 SCALING: n 2 2t = t, m = + σρt13 a + a 1 x = 2a 2 θτ 1 t 23 + ξ 1 ρt 13, y = 2a 2 θτ 2 t 23 + ξ 2 ρt 13 r = t + (1 + a 2 )θτ 1 t 23, s = t + (1 + a 2 )θτ 2 t 23, Given the weight 0 < a < 1 on vertical dominoes, define: v 0 := 1 a 1 + a < 0, A3 := a(1 + a)5 (1 a)(1 + a 2 ), ρ := Av 0 > 0, θ := ρ(a + a 1 ) > 0,

49 lim t ( v 0) y x+r s ( 1) y x K n,m (2r, x; 2s, y)ρt 13 = K tac (τ 1, ξ 1 ; τ 2, ξ 2 ) For τ 2 > τ 1, one has the tacnode process, depending on a pressure σ: K tac (τ 1, ξ 1 ; τ 2, ξ 2 ) = K A (τ 1, ξ 1 ; τ 2, ξ 2 ) + g(τ 1, ξ 1 ) g(τ 2, ξ 2 ) 213 σ ( (1 KAi ) 1 ) σ Aτ 1 ξ1 σ (λ)a ( τ 2 ) ξ 2 σ (λ)dλ. where A τ ξ (κ) := Aiτ (ξ κ) 0 Aiτ ( ξ β)ai(κ + β)dβ Ai (s) (x) := e sx+2 3 s3 Ai(x + s 2 ). (Adler-Ferrari-PvM 09, Adler-Johansson-PvM 11)

50 Same tacnode process for two groups of non-intersecting Brownian motions

51 Same tacnode process for two groups of non-intersecting random walkers x(t) on Z with exponential holding time t 0 }{{} 2m + 1 gaps t

52 OPEN PROBLEMS: Find PDE s for the transition probability for the tacnode process? Limiting behavior for more general dimer models? Can one find other singularities at the macroscopic level, leading to other statistical fluctuations (and other kernels)? Relation with the KPZ equation?

53 Thank you!