Warwick, March 19-23th 2012
Non-intersecting random walk in the
neighborhood of a symmetric tacnode
Patrik L. Ferrari
with Mark Adler and Pierre van Moerbeke (arXiv:1007.1163)
and Bálint Vet® (arXiv:1112.5002)
http://www-wt.iam.uni-bonn.de/∼ferrari
Outline
From Brownian Bridge to Tacnode process
Idea of the construction, key steps
Introduction Airy Pearcey Tacnode Method
1
Brownian Bridge
Let
X(t)
2
a Brownian Bridge with
X(−N ) = X(N ) = 0.
O(N 1/2 )
a ements
−N
N
Macroscopic shape
h(α) :=
E(X(αN ))
= 0.
N
Fluctuations scale
p
Var(X(αN )) =
Introduction Airy Pearcey Tacnode Method
p
N α(1 − α).
Brownian Bridge
Let
X(t)
2
a Brownian Bridge with
X(−N ) = X(N ) = 0.
O(N 1/2 )
a ements
−N
N
Distribution law
X(αN )
p
= N (0, 1) :
N α(1 − α)
Local process: Brownian Motion
Introduction Airy Pearcey Tacnode Method
Gaussian RW
m Brownian Bridges
Let
3
X1 (t) ≥ X2 (t) ≥ · · · ≥ Xm (t) be m non-intersecting
Xk (−N ) = Xk (N ) = 0, k = 1, . . . , m.
Brownian Bridges with
O(N 1/2 )
a ements
−N
N
Macroscopic shape
h1 (α) := lim
N →∞
E(X1 (αN ))
= 0.
N
Fluctuations scale
p
Var(X1 (αN )) = O(N 1/2 ).
Introduction Airy Pearcey Tacnode Method
m Brownian Bridges
Let
3
X1 (t) ≥ X2 (t) ≥ · · · ≥ Xm (t) be m non-intersecting
Xk (−N ) = Xk (N ) = 0, k = 1, . . . , m.
Brownian Bridges with
O(N 1/2 )
a ements
−N
N
Distribution law
X1 (αN )
= Largest
N →∞ N 1/2
lim
eigenvalue of a
Local process: Dyson's Brownian Motion
Introduction Airy Pearcey Tacnode Method
m×m
GUE matrix
Airy2 process
4
m = N non-intersecting
Xk (−N ) = Xk (N ) = 0.
Consider now
Brownian Bridges with
N 1/3
4N
N 2/3
ag repla ements
Macroscopic shape:
h(α) := limN →∞
Introduction Airy Pearcey Tacnode Method
E(X1 (αN ))
N
√
= 2 1 − α2 .
Airy2 process
4
m = N non-intersecting
Xk (−N ) = Xk (N ) = 0.
Consider now
Brownian Bridges with
Fluctuations scale
p
Var(X1 (αN )) = O(N 1/3 ).
Distribution law: Tracy-Widom
lim P
N →∞
F2
X1 (αN ) − N h(α)
≤s
σ(α)N 1/3
= F2 (s)
F2 discovered by Tracy, Widom '94
Local process: Airy2
X1 (αN + τ c(α)N 2/3 ) − N h(α + τ c(α)N −1/3 )
= A2 (τ )
N →∞
σ(α)N 1/3
lim
Airy2 process discovered in a stochastic growth model in the
Kardar-Parisi-Zhang universality class
Measured experimentally (liquid crytals)
Introduction Airy Pearcey Tacnode Method
Prähofer, Spohn '02
Takeuchi, Sano '10
Point processes point of view: edge
Consider the point process
Scaling at the edge:
ξ(t, x) =
t = τ N 2/3
and
5
PN
k=1 δXk (t) (x).
x = 2N + sN 1/3
lim ρ(n) (s1 , τ1 ; . . . ; sn , τn ) = det[KAi (si , τi ; sj , τj )]1≤i,j≤n
N →∞
where
KAi
is the extended Airy kernel
KAi (s1 , τ1 ; s2 , τ2 )
Z
Z
3
2
2
−1
ez /3+τ1 z +(τ1 −s1 )z
1
=
dw dz w3 /3+τ w2 +(τ 2 −s )w
2
2
2
(2πi) h
z−w
2
e
i
1[τ1 > τ2 ] −(s1 −s2 +τ22 −τ12 )2 /4(τ1 −τ2 )
e
−p
4π(τ1 − τ2 )
Introduction Airy Pearcey Tacnode Method
Point processes point of view: bulk
Scaling in the bulk
lim ρ(n) (x1 , t1 ; . . . ; xn , tn ) = det[KSine (xi , ti ; xj , tj )]1≤i,j≤n
N →∞
where
KSine
is the extended Sine kernel. In particular,
KSine ((x, 0), (y, 0)) =
Introduction Airy Pearcey Tacnode Method
sin(π(x − y))
.
π(x − y)
6
Pearcey process
7
Now consider
2N
non-intersecting Brownian Bridges with
xk (−N ) = 0,
xk (N ) = bN , 1 ≤ k ≤ N,
xk (N ) = −bN , N + 1 ≤ k ≤ 2N.
A2
rag repla ements
1 ≤ k ≤ 2N
Sine
Pear ey
N 1/4
A2
N 1/2
Macroscopic shape of
t = a(b)N :
XN
and
a cusp!
Introduction Airy Pearcey Tacnode Method
XN +1
come together, say at
Pearcey process
Now consider
7
2N
non-intersecting Brownian Bridges with
xk (−N ) = 0,
xk (N ) = bN , 1 ≤ k ≤ N,
1 ≤ k ≤ 2N
xk (N ) = −bN , N + 1 ≤ k ≤ 2N.
Scaling at the cusp: Density of BB is
like Brownian motions:
t = a(b)N + τ c(b)N 1/2 ,
O(N −1/4 )
x = sσ(b)N 1/4
Rescaled point process is determinantal with (as
n-point
and locally
correlation given by the Pearcey kernel.
N → ∞)
Tracy, Widom '05
Introduction Airy Pearcey Tacnode Method
Pearcey process: the kernel
Consider the point process
Scaling at the edge:
x=
8
ξ(t, x) =
PN
k=1 δXk (t) (x).
t = a(b)N + τ c(b)N 1/2
and
sσ(b)N 1/4
lim ρ(n) (s1 , τ1 ; . . . ; sn , τn ) = det[KP (si , τi ; sj , τj )]1≤i,j≤n
N →∞
where
KP
is the extended Pearcey kernel
KP (s1 , τ1 ; s2 , τ2 )
Z
Z
4
2
ez /4−τ1 z /2+s1 z
−1
1
d
w
d
z
=
4 /4−τ w 2 /2+s w
2
w
2
2
(2πi) iR
z−w
e
×
1[τ2 > τ1 ] −(s1 −s2 )2 /2(τ2 −τ1 )
−p
e
2π(τ2 − τ1 )
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process
Now consider
2N
non-intersecting Brownian Bridges with
xk (±N ) = bN , 1 ≤ k ≤ N,
Macroscopic shape:
±a(b)N ⇒
9
xk (±N ) = −bN , N + 1 ≤ k ≤ 2N.
0 < b < 2,
there are two cusps around
Pearcey process in scale
(N 1/2 , N 1/4 ).
A2
Sine
Pear ey
epla ements
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process
Now consider
2N
non-intersecting Brownian Bridges with
xk (±N ) = bN , 1 ≤ k ≤ N,
xk (±N ) = −bN , N + 1 ≤ k ≤ 2N.
b > 2, there are two (asymptotically)
independent blocks of N non-intersecting Brownian Bridges:
2/3 , N 1/3 ).
Airy2 processes in the scale (N
Macroscopic shape:
Introduction Airy Pearcey Tacnode Method
9
Symmetric tacnode process
Now consider
2N
9
non-intersecting Brownian Bridges with
xk (±N ) = bN , 1 ≤ k ≤ N,
Macroscopic shape:
xk (±N ) = −bN , N + 1 ≤ k ≤ 2N.
b = 2 + o(1),
the two cusps joints into a
tacnode.
Ta node
epla ements
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process
Q: Is the tacnode process like two Pearcey joining together or like
two Airy2 processe colliding?
Introduction Airy Pearcey Tacnode Method
10
Symmetric tacnode process
10
Q: Is the tacnode process like two Pearcey joining together or like
two Airy2 processe colliding?
A: Like two Airy2 processes touching. Take
σ
bN = 2N + σN 1/3 .
modulate the strength of interaction between the two set
of Brownian Bridges.
Under Airy "microscope":
t = τ N 2/3
and
x = sN 1/3
we
obtain that the limit process has determinantal correlations
governed by the symmetric tacnode kernel
Kσ
Simplied expressions in the recent work by
Adler, Johansson, van Moerbeke:
Introduction Airy Pearcey Tacnode Method
arXiv:1112.5532
Symmetric tacnode process
One-time kernel at
11
τ = 0:
Kσ (s1 , s2 ) = C(s1 − s2 )
s1 u
Z
Z
u3
du
e
dv e 3 −σu
es1 v (1 − P̂(u))(1 − P̂(−v))
+
+
v3
es2 v
es2 u
u−v
δ+iR 2πi −δ+iR 2πi
e 3 −σv
Z
Z
u3
dv
e 3 −σu
es1 u
du
es1 v (1 − P̂(u))Q̂(−v)
+
+
v3
s
v
e2
es2 u
u−v
2δ+iR 2πi δ+iR 2πi
e− 3 −σv
Z
Z
u3
du
dv e− 3 −σu
es1 v (1 − P̂(−v))Q̂(u)
es1 u
+
+
v3
s
v
e2
es2 u
u−v
−δ+iR 2πi −2δ+iR 2πi
e 3 −σv
Z
Z
u3
du
dv e− 3 −σu
es1 u
es1 v Q̂(−v)Q̂(u)
+
+
.
v3
es2 v
es2 u
u−v
−δ+iR 2πi δ+iR 2πi
e− 3 −σv
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process
12
where
∞
, with Q(κ) := [(1 − KAi )−1
(σ̃,∞) Ai](κ)
Z ∞
Z ∞
1/3
P̂(u) := −
dκ e−κu2
dµ Q(µ)Ai(µ + κ), with σ̃ := 22/3 σ
0
σ̃
Z ∞
−1/3
C(s1 ) := 2
dκ Q(κ) Ai(κ − 2−1/3 s1 ) + Q(κ + 2−1/3 |s1 |)
σ̃
Z ∞
−
dλ Q(λ)KAi (κ, λ − 2−1/3 |s1 |)
Q̂(u) :=
Z
σ̃
1/3
dκ Q(κ)eκu2
σ̃
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process
13
Alternative formulation:

A(s1 − γ)A(s2 − γ) + A(−s1 − γ)A(−s2 − γ)







∞
−A(s1 − γ)B(s2 − γ) − B(−s1 − γ)A(−s2 − γ) 


dγ 

0
−B(s1 − γ)A(s2 − γ) − A(−s1 − γ)B(−s2 − γ) 




−B(s1 + γ)B(s2 + γ) − B(−s1 + γ)B(−s2 + γ)
Z
Kσ (s1 ; s2 ) = C(s1 −s2 )+
where
Z ∞ Z ∞
A(s1 ) := Ai(σ − s1 ) +
dκ
dµ Q(κ)Ai(κ + µ)Ai(21/3 µ + σ − s1 )
σ̃
0
Z ∞
1/3
B(s1 ) :=
dκ Q(κ)Ai(2 κ + s1 − σ)
σ̃
Introduction Airy Pearcey Tacnode Method
The tacnode process: dierent approaches
14
(1) Discrete (space) continuous time random walks
Adler, Ferrari, van Moerbeke:
arXiv:1007.1163
(2) Brownian Bridges, not only symmetric, use Riemann-Hilbert
Delvaux, Kuijlaars, Zhang:
(3) Brownian Bridges, symmetric
arXiv:1009.2457
Johansson: arXiv:1105.4027
(4) Brownian Bridges, asymptotics for asymmetric (= generic)
case, based on (3)
Ferrari, Vet®: arXiv:1112.5002
(5) Space and time discrete: double Aztec diamond, symmetric,
based on the approaches of (1) and (3)
Adler, Johansson, van Moerbeke:
arXiv:1112.5532
For the equivalence of formulations of (1) and (3), see (4).
(3), (5): see next two talks
Introduction Airy Pearcey Tacnode Method
The asymmetric tacnode
15
Asymmetric tacnode: two-parameter (ratio of curvatures
strength of interaction
σ)
λ
and
family process with asymptotic kernel
obtained in
Ferrari, Vet®: arXiv:1112.5002
Remark: the Airy2 and Pearcy Process have no free parameters.
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process: our approach
Discrete model: consider
time random walks from
16
∞ many non-intersecting
−1, −2, . . ..
continuous
This is the multilayer polynuclear growth (PNG) model where
the Airy2 process was discovered (top layer).
Prähofer,Spohn'02
x=−t
x=t
h0
h−1
h−2
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process: our approach
Discrete model: consider
time random walks from
∞ many non-intersecting
−1, −2, . . ..
16
continuous
Idea: Take two multilayer PNG models, one with top layer
starting at
starting at
−m − 1,
m + 1.
the other up-side down with lower layer
Introduction Airy Pearcey Tacnode Method
Symmetric tacnode process: our approach
2m + 1
Step 1: Let us start with only
k ∈ Im = {−m, . . . , m}, starting
−m, −m + 1, . . . , m − 1, m.
This point process
with kernel
em.
K
η̃(x) :=
walkers
x̃k (t),
and leaving at
P
k∈Im δxk (0),x is determinantal
How to get the kernel?
Introduction Airy Pearcey Tacnode Method
17
Symmetric tacnode process: our approach
In Step 1, to get the kernel of the
2m + 1
random walks we
use:
(a) Karlin-Mc Gregor formula
(b) Orthogonalize in the Fourier representation
(c) Transform a Toeplitz determinant into a Fredholm determinant
by the Borodin-Okounkov Formula
Introduction Airy Pearcey Tacnode Method
18
Symmetric tacnode process: our approach
19
One-time kernel (t1 = t2 = 0).
e m (x, y) =
K
+
+
Vm
I
(2πi)2
Vm
2πi
dw
Γ0
Γ0,z
I
(2πi)2
I
Vm
eT (z−z
I
dz
I
dw
dz
Γ0
dz
Γ0
eT (w−w
z x−y+1
−1 )
eT (w−w
Γ0,w
1
−1 )
eT (z−z
H2m+1 (z
−1 )
−1 )
−1
wy−m−1 H2m+1 (w)H2m+1 (z −1 )
z x−m
z−w
wy+m
H2m+1 (z)H2m+1 (w−1 )
z x+m+1
w−z
)H2m+1 (z),
with Vm := 1/(H2m+1 (0)H2m+2 (0)). The function Hn is itself the Fredholm determinant
Hn (z
−1
) := det(1 − K(z
−1
))`2 ({2m+1,2m+2,...})
of the kernel
K(z
−1
)k,` :=
(−1)k+`
(2πi)2
I
I
du
Γ0
dv
Γ0,u
Introduction Airy Pearcey Tacnode Method
u`
v k+1
1
u − z e2T (u−u
v−u v−z
−1 )
−1 )
e2T (v−v
,
Symmetric tacnode process: our approach
Step 2: Particle-hole transformation
Introduction Airy Pearcey Tacnode Method
20
Symmetric tacnode process: our approach
Step 2: Particle-hole transformation
Introduction Airy Pearcey Tacnode Method
20
Symmetric tacnode process: our approach
Step 2: Particle-hole transformation
Introduction Airy Pearcey Tacnode Method
20
Symmetric tacnode process: our approach
Step 2: Particle-hole transformation
The point process
with kernel
η̃(x) :=
Pm
k=−m δx̃k (0),x is determinantal
em
K
We want to characterize the complementary point process,
X
η(x) :=
δxk (0),x .
k∈{−m,...,m}
/
It is determinantal with kernel
em
Km = 1 − K
Borodin, Olshanski, Okounkov '00.
Introduction Airy Pearcey Tacnode Method
21
Symmetric tacnode process: our approach
22
One-time kernel (t1 = t2 = 0).
Km (x, y) = −
−
Vm
I
(2πi)2
Vm
− 1[x6=y]
dw
Γ0
Γ0,z
I
Γ0
Vm
2πi
dz
dz
Γ0
−1 )
eT (w−w
Γ0,w
eT (z−z
1
I
−1 )
eT (w−w
I
dw
(2πi)2
eT (z−z
I
dz
z x−y+1
wy−m−1 H2m+1 (w)H2m+1 (z −1 )
z x−m
z−w
−1 )
−1 )
H2m+1 (z
wy+m
H2m+1 (z)H2m+1 (w−1 )
z x+m+1
w−z
−1
)H2m+1 (z),
with Vm := 1/(H2m+1 (0)H2m+2 (0)). The function Hn is itself the Fredholm determinant
Hn (z
−1
) := det(1 − K(z
−1
))`2 ({2m+1,2m+2,...})
of the kernel
K(z
−1
)k,` :=
(−1)k+`
(2πi)2
I
I
du
Γ0
dv
Γ0,u
Introduction Airy Pearcey Tacnode Method
u`
v k+1
1
u − z e2T (u−u
v−u v−z
−1 )
−1 )
e2T (v−v
,
Symmetric tacnode process: our approach
23
Step 3: Asymptotic analysis
(a) First we reshaped the kernel
have a limit as
T →∞
Km
so that all the terms would
from the naïve steepest descent
analysis, though not easy to carry out rigorously on the
complex integral representations.
(b) Rewrite all complex integrals in the kernel
Km
in terms of
Bessel function.
(c) Limit
T →∞
and get Airy functions instead
Rewriting of the Airy functions as complex integrals, leads to
the result of the naïve steepest descent analysis.
Introduction Airy Pearcey Tacnode Method
The tacnode
a ements
Introduction Airy Pearcey Tacnode Method
24
The asymmetric tacnode
Ferrari, Vet®: arXiv:1112.5002
Introduction Airy Pearcey Tacnode Method
25