Domino tilings, non-intersecting
Random Motions
and the tacnode Process
Pierre van Moerbeke
Université de Louvain, Belgium & Brandeis University, Massachusetts
Mathematics Institute, University of Warwick (March 2012)
JOINT WORK with
MARK ADLER & KURT JOHANSSON
Three models, leading to Tacnode process: (Universality!)
(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:1007.1163)
(2) continuous time and continuous space process
K. Johansson: Non-colliding Brownian Motions and the extended tacnode
process (arXiv:1105.4027 )
Delvaux-Kuijlaars-Zhang: Critical behavior of non-intersecting Brownian
motions at a Tacnode (arXiv:1009.2457 )
(3) discrete time and discrete space process (Domino
tilings of Double Aztec Diamonds):
Mark Adler, Kurt Johansson & PvM: Double Aztec Diamonds and the
Tacnode Process (arXiv:1112.5532)(to appear 2012)
1. Domino tilings of Double Aztec Diamonds
Domino tilings of an Aztec diamond :
//////
//////
////// //////
////// //////
////// ////// //////
////// ////// //////
////// ////// ////// //////
////// ////// ////// //////
////// ////// ////// ////// //////
////// ////// ////// ////// //////
////// ////// ////// ////// ////// //////
////// ////// ////// ////// ////// //////
////// ////// ////// ////// ////// ////// //////
////// ////// ////// ////// ////// ////// //////
////// ////// ////// ////// ////// ////// //////
////// ////// ////// ////// ////// ////// //////
////// ////// ////// ////// ////// //////
////// ////// ////// ////// ////// //////
////// ////// ////// ////// //////
////// ////// ////// ////// //////
////// ////// ////// //////
////// ////// ////// //////
////// ////// //////
////// ////// //////
////// //////
////// //////
//////
//////
4 types of coverings by 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
Double Aztec diamond (Adler-Johansson-PvM ’11)
Diamond A
•
Y =n+m
o1 //////
o1
////// //////
////// ////// //////
////// ////// ////// //////
////// ////// ////// ////// //////
////// ////// ////// ////// ////// //////
on
on //////
////// ////// //
//// m +////12//) ////// //////
(−m,
•
i2m+1 • //////
////// ////// ////// ////// ////// ////// i2m+1
i4 • //////
////// ////// ////// ////// ////// ////// i4
i3 • //////
////// //////(0, 0)•////// ////// ////// ////// i3
i2 • //////
////// ////// ////// ////// //////
////// i2
2m + 1
i1 • //////
////// ////// ////// ////// ////// ////// i1
inliers
• −m − 1 )
on+1
on+1 //////
////// ////// ////// (m, /
///// 2 ////// //////
////// ////// ////// ////// ////// //////
////// ////// ////// ////// //////
////// ////// ////// //////
////// ////// //////
////// //////
o2n
o2n //////
Diamond B
n
Figure 1: Double Aztec diamond of type (n, m) = (7, 2) with #{inliers} =
M = 2m + 1 = 5.
Random cover with domino’s: two groups of non-intersecting paths
0
0
0
height and level lines for n=7
////// 1 1 1
//////2 2 ////// 1 2 2
2
2
//////3 3//////2 2 //////2
3
3
3
3
//////4 4 //////3 3 //////3 3 //////3
4
4
4
4
.
//////5 5 //////4 4//////4 4 //////4 4 //////4
5
5
5
5 ..
//////6 6 //////5 5 //////5 5 //////4 5 //////5 5 //////5
6
6
n-1
n-1
//////6 6 //////6 6 //////6 6//////5 5 //////5 5 //////5 6 //////6
n
n
//////6 6 //////6 6 //////6 6 //////6 6 //////6 6 //////6 7 ////// n n
n
n
//////6 6 //////6 6//////6 7 //////7 7 //////7 7 //////7 8 ////// n n
n
n
//////6 6 //////6 7 //////7 7 //////7 7 //////7 8 //////8 8 ////// n n
n
n
//////7 7//////7 7 //////7 8 //////8 8 //////8 8 //////8 8 ////// n n
n
n
//////7 7 //////7 8 //////8 9 //////8 8 //////8 9 //////8 8////// n
8
n
//////8 8 //////8 8 //////8 9 //////9 9 //////9 9 //////9 9 //////8
n+1
n+1
//////9 9 //////9 9 //////9 10//////9 10/////
/
/
/
/
/
/
/
10 10
9
n+2
n+2
//////
/////
/ 10 ////10
// 10 /////
/ 10/////
/ 10 /////
/
10
10
10
10
n+3
n+3
/////
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
11 11 11
11 11 11 11
n+4
n+4
//////
/////12
/ 12 /////
/
/
/
/
/
/
/
12 12 12
n+5
n+5
/////
/ 13/////
/
13
13
...
. . . //////
//////
1
1
2n
2n
2n
h
h
h
h
h
////// = North
h
h
h
h
//////
h+1 h+1
h
= South
h+1
h+1
h
h
////// h
h
h+1
= East
h+1
h
h+1
////// = West
h+1
h+1
h
h
h
h
h
////// = North
h
h
h
h
//////
h+1 h+1
h
= South
h+1
h+1
h
h
////// h
h
h+1
= East
h+1
h
h+1
////// = West
h+1
h+1
Figure 4. Height function on domino’s and level-lines.
Domino-tiling ⇔ Random Surface
(piecewise-linear)
The level curves for this Random Surface give
non-intersecting paths
Implies Fixed boundary condition!
0
0
0
height and level lines for n=7
////// 1 1 1
//////2 2 ////// 1 2 2
2
2
//////3 3//////2 2 //////2
3
3
3
3
//////4 4 //////3 3 //////3 3 //////3
4
4
4
4
.
//////5 5 //////4 4//////4 4 //////4 4 //////4
5
5
5
5 ..
•/
/////6 6 //////5 5 //////5 5 //////4 5 //////5 5 //////5
6
6
n-1
n-1
//////6 6 //////6 6 //////6 6//////5 5 //////5 5 //////5 6 //////6
n
n
//////6 6 //////6 6 //////6 6 //////6 6 //////6 6 //////6 7 ////// n n
n
n
//////6 6 //////6 6//////6 7 //////7 7 //////7 7 //////7 8 ////// n n
n
n
//////6 6 //////6 7 //////7 7 //////7 7 //////7 8 //////8 8 ////// n n
n
n
//////7 7//////7 7 //////7 8 //////8 8 //////8 8 //////8 8 ////// n n
n
n
//////7 7 //////7 8 //////8 9 //////8 8 //////8 9 //////8 8////// n
8
n
//////8 8 //////8 8 //////8 9 //////9 9 //////9 9 //////9 9 //////8 n+1
n+1
//////9 9 //////9 9 //////9 10//////9 10/////
/ 10 //////9
10
n+2
n+2
//////
/////
/ 10 ////10
// 10 /////
/ 10/////
/ 10 /////
/
10
10
10
10
n+3
n+3
/////
/ 11 ////11
// 11/////
/ 11/////
/
11
11
11
Y2s−1
n+4
n+4
//////
/////12
/ 12 /////
/ 12/////
/
12
12
n+5
n+5
/////
/
/
/
/
/
/
/
13 13 13
...
.
. . //////
//////
Y2r
2n
2n
2n
1
1
A weight on domino’s and 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
X
a#vertical domino’s in T
all possible tilings T
(1)
Question:
For an axis Y2r going through black squares only, and an interval [k, `] ⊂ Z,
P(height function is flat along the interval [k, `]) =?
⇐⇒
P(no dots along the points of interval [k, `]) =?
⇐⇒
P(domino’s are pointing to the left of Y2r along the points of interval [k, `]) =
0
0
0
height and level lines for n=7
////// 1 1 1
//////2 2 ////// 1 2 2
2
2
//////3 3//////2 2 //////2
3
3
3
3
//////4 4 //////3 3 //////3 3 //////3
4
4
4
4
.
//////5 5 //////4 4//////4 4 //////4 4 //////4
5
5
5 5 ..
•/
/////6 6 //////5 5 //////5 5 //////4 5 //////5 5 //////5 6
n-1 6
n-1
//////6 6 //////6 6 //////6 6//////5 5 //////5 5 //////5 6 //////6
n
n
//////6 6 //////6 6 //////6 6 //////6 6 //////6 6 //////6 7 ////// n n
n
n
//////6 6 //////6 6//////6 7 //////7 7 //////7 7 //////7 8 ////// n n
n
n
//////6 6 //////6 7 //////7 7 //////7 7 //////7 8 //////8 8 ////// n n
n
n
//////7 7//////7 7 //////7 8 //////8 8 //////8 8 //////8 8 ////// n n
n
n
//////7 7 //////7 8 //////8 9 //////8 8 //////8 9 //////8 8////// n
8
n
//////8 8 //////8 8 //////8 9 //////9 9 //////9 9 //////9 9 //////8 n+1
n+1
//////9 9 //////9 9 //////9 10//////9 10/////
/ 10 //////9
10
n+2
n+2
//////
/////10
/ 10 ////10
// 10 /////10
/ 10/////
/ 10/////
/
10
10
n+3
n+3
/////11
/ 11 ////11
// 11/////11/ 11/////
/
11
Y2s−1
n+4
n+4
//////
/////12
/ 12 /////12
/ 12/////
/
12
n+5
n+5
/////13/ 13/////
/
13
...
. . . //////
//////
Y2r
2n 2n 2n
1
1
2. A determinantal process
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?
By completing the double Aztec diamond with South domino’s (in a trivial
way).
//////
(South)
C2
D2
C3
D3
C4
D4
C5
Dr Cr+1 D6
//////
C7
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
Y
//////
//////
//////
`
B1
Y0
//////
//////
B2
Y1
Y2
//////
//////
B3
Y3
//////
//////
//////
A3
//////
//////
//////
//////
A2
//////
//////
//////
//////
//////
//////
//////
A1
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
Y =0
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
//////
D8
//////
//////
//////
C8
//////
//////
//////
D7
//////
//////
//////
A4
Y4
//////
B4
Y5
//////
//////
A5
Y6
//////
//////
//////
Br Ar+1 B6
Y7
A7
//////
B7
Y8 Y2r−1 Y2r Y2r+1 Y1
Y1
Y2
Y3
Y4
Y5
Y6
Y7
Y8
Y2r−1
Y2r
Y11
Y12
Y13
Y14
P ({non-intersecting random walks} ∩ [k, `] = ∅ along Y2r ) =?
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+1
////// h
h
h
h
h
h
h
h+1
h
////// = North ///•
/// ◦ = South
///•/// = West
= East
h
h
h
h+1 h+1
h+1
h+1
h+1
h+1
h+1
Figure 4. Height function h on domino’s and level line.
h̃ + 1
h̃ + 1 h̃ + 1 h̃ + 1
h̃
◦ ///•/// = North
h̃
h̃
h̃
h̃
//////
h̃
h̃
h̃
h̃ + 1
= South
h̃
h̃
h̃ + 1
///•/// h̃
◦ = East
h̃
h̃ + 1
h̃
h̃
h̃ + 1
h̃ + 1
////// = West
h̃
Figure 5. Dual height function h̃ on domino’s and level line.
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+1
2m+1
////// ////// ////// ////// ////// ////// //////
2m+1
2m+1
////// ////// ////// ////// ////// ////// ////// 2m 2m
2m
2m
////// ////// ////// ////// ////// ////// ////// . .
.
2m-1
////// ////// ////// ////// ////// ////// ////// 2 2
2m-2
////// ////// ////// ////// ////// ////// ////// 1 1
..
1
.
////// ////// ////// ////// ////// ////// ////// 0
0
////// ////// ////// ////// ////// ////// //////
0
0
0
////// ////// ////// ////// ////// //////
0
0
0
//////
////// ////// ////// ////// //////
0
0
0
////// ////// ////// //////
0
0
//////
////// ////// //////
0
0
////// //////
0
0
//////
//////
2m+1
0
0
0
Y2s−1
Y2r
x
B
A
B
A
B
A
B
A
B
A
B
A
B
A
B
Figure 17: Lattice paths: superimposing in- and outliers.
3. The kernel for the inlier determinantal process
x
)
dot-to-circle
B
A
B
A
-steps
B
A
B
A
B
A
: ψ2j,2j+1(x, y) :=
B
A
B
A
B
( x−y
)
a
, if y − x ≤ 0
0 otherwise
)
circle-to-dot
-steps
: ψ2j+1,2j+2(x, y) :=
1, if y−x = 0



(
Then define in general:


a, if y−x = 1 





0, otherwise



dz z x−y
=
a
Γ0,a 2πiz 1 − z
=
I
dz x−y
z
(1 + az).
Γ0 2πiz
I
ψr,s = ψr,r+1 ∗ . . . ∗ ψs−1,s, if s > r
= 0, if s ≤ r.
where
f (x) ∗ g(x) :=
X
x∈Z
f (x)g(x)
• Lindström-Gessel-Viennot / Karlin-McGregor
P(r; x1, . . . , x2m+1)
1
det(ψ0,r (−m + i − 1, xj ))1≤i,j≤2m+1 det(ψr,2n+1(xj , −m + i − 1))1≤i,j≤
=
Zn,m
• Via the Eynard-Mehta formula: Expression in terms of orthogonal
polynomials on the circle (inliers)
Kn,m(r, x; s, y)
=
2m+1
X
i,j=1
ψr,2n+1(x, −m + i − 1)([A−1]ij )ψ0,s(−m + j − 1, y) − 1r<sψr,s(x, y),
dw R
dz L
wx+m
1
ρa (w)
ρa (z) y+m
=
2
(2πi) γr2 w
z
γr1 z
I
I
2m
X
k=0
|
Pk (z)Pbk (w−1) , for r = s = n
{z
}
bi-orthogonal pol on
R
circle for ρL
a (z)ρa (z)
ρL
a (z) =
1 + az
1 − az
!n/2
ρL
a (z)
R
and ρa (z) =
1 − az
4. The kernel for the outlier determinantal process
• Kernel for the dot-determinantal process (outlier paths) (Borodin
’00)
e doubleAztec(n, x; n, y) = δ
K
x,y − Kn,m (n, x; n, y).
n,m
• Extended kernel (outlier paths) :
e doubleAztec(2r, x; 2s, y)
K
n,m
e doubleAztec(n, · ; n, ◦) ∗ ψ
= −1s<r ψ(2s−2r)(x, y) + ψn−2r (x, ·) ∗ K
n,m
(2s−n) (◦, y),
with
dz
ψ2k (x, y) :=
Γ0,a 2πiz
I
1 + az
1 − az
!k
Two approaches:
(1) Work immediately with
2m
X
k=0
Pk (w)Pbk (z −1)
(2) Or use the Christoffel-Darboux formula for bi-orthonormal polynomials
on the circle,
2m
X
k=0
Pk (w)P̂k (z −1)
z −2m−1P2m+1(z)w2m+1P̂2m+1(w−1) − P̂2m+1(z −1)P2m+1(w)
=
.
1−w
z
Using approach (1), the kernel for the Double Aztec Diamond reads:
e DoubleAztec(2r, x; 2s, y)
(−1)x−y K
n,m
SingleAztec
= Kn+1
+
∞
X
k=2m+1

P
2(n + 1 − r), m + 1 − x; 2(n + 1 − s), m + 1 − y
b−x,r (k)[(1 − K)−1a−y,s](k)
s n
\
i=1

o
the line Y2ri has a gap ⊃ [ki, `i] 
= det 1 − χ[ki,`i]K̃DoubleAztec
(2ri, xi; 2rj , xj )χ[kj ,`j ]
.
n,m
1≤i,j≤s
h
i
Using approach (1), the kernel for the Double Aztec Diamond reads:
e DoubleAztec(2r, x; 2s, y)
(−1)x−y K
n,m
SingleAztec
= Kn+1
+
∞
X
k=2m+1
2(n + 1 − r), m + 1 − x; 2(n + 1 − s), m + 1 − y
b−x,r (k)[(1 − K)−1a−y,s](k)
I
I
(−1)k−x
dv v x+m (1 + av)s(1 − av )n−s+1
ax,s(k) :=
du
2
(2πi)
ϕa(2n; u)
γr1
γr2 u − v uk+1
v`
ϕa(2n; v)
(−1)`−y
dv
du
by,r (`) :=
a
(2πi)2 γr1
γr2 u − v uy+m+1 (1 + au)r (1 − u )n−r+1
I
I
(−1)k+`
dv
u`
ϕa(2n, u)
du
Kk,` :=
(2πi)2 γr1
γr2 v − u v k+1 ϕa (2n, v)
I
I
with ϕa(2n; z) := (1 + az)n(1 − az )n+1
5. Determinantal point process for single Aztec diamond: Interlacing red dots on the red lines (Johansson
’05)
//////
Interlacing red dots for n → ∞ '
////// ////// spectrum of consecutive principal
////// • ////// ////// minors of GUE-matrices
////// • ////// ////// //////
•/
///// • ////// • ////// ////// //////
////// ////// • ////// ////// ////// • //////
////// ////// • ////// • ////// ////// • ////// //////
////// ////// ////// • ////// ////// ////// //////
////// ////// ////// • ////// ////// • //////
////// ////// ////// ////// • //////
////// ////// ////// //////
////// ////// ////// •
////// //////
//////
In the limit for 0 < a < 1:
Frozen North
y
Disordered region
Frozen West
x
Frozen South
Frozen East
6. Limiting behavior for the double Aztec diamond: the
Tacnode Process, for n → ∞.
Bringing Arctic ellipses together (tacnode):
(x ± 2r )2
p
+
(y ± 2r )2
q
a
a−1
with q =
and p = 1 − q =
,
−1
−1
a+a
a+a
= 1,
,1 + r
)
(r
2
2
Diamond A
√
12
(p − r
,q + r
)
2
2
, r)
(− r
2 2 •
−1 − r
, r)
2 2
Diamond B
(0, 0)
•
√
2r
,−r
)
•( r
2
2
(p + r
,q − r
)
2
2
(1 + r
,−r
)
2
2
√
2(1 − r)
q=
(r
,1 − r
)
2
2
a
,
a+a−1
p=
√
r = 2 pq =
a−1
a+a−1
2
a+a−1
SCALING: From old to new variables: ξi, τi
(r, x; s, y) 7→ (τ1, ξ1; τ2, ξ2)
2m
2
−2/3
=
+
σρt
n
a + a−1
n = 2t,
x
−2/3 ,
= a2θτ1t−1/3 + 1
ξ
ρt
1
2
n
y
−2/3
= a2θτ2t−1/3 + 1
ξ
ρt
2
2
n
r
1
= + 2θ (1 + a2)τ1t−1/3,
n
2
s
1
= + 2θ (1 + a2)τ2t−1/3,
n
2
Given the weight 0 < a < 1 on vertical dominoes, define:
1−a
v0 := −
< 0,
1+a
5
a(1
+
a)
A3 :=
,
(1 − a)(1 + a2)
ρ := −Av0 > 0, θ :=
q
ρ(a + a−1) >
e DoubleAztec(2r, x; 2s, y)ρt1/3 = Ktac(τ , ξ ; τ , ξ )
lim (−v0)y−x+r−s(−1)y−xK
1 1 2 2
n,m
t→∞
For τ2 > τ1, one has the tacnode process:
Ktac(τ1, ξ1; τ2, ξ2) = KAiryProcess(τ2, σ − ξ2 + τ22; τ1, σ − ξ1 + τ12)
Z ∞
g(τ1, ξ1) 1/3
2
+
g(τ2, ξ2)
σ̃
(−τ2 )
−1 τ1
(1 − KAi)σ̃ Aξ −σ (λ)Aξ −σ
(λ)dλ.
1
2
where
Aτξ (κ) := Ai(τ )(ξ + 21/3κ) −
Z ∞
τ x+ 2
τ3
(τ
)
3
Ai (x) := e
Ai(x + τ 2).
0
Ai(τ )(−ξ + 21/3β)Ai(κ + β)dβ