Extremes of vicious walkers: application to the
directed polymer and KPZ interfaces
G. Schehr
Laboratoire de Physique Théorique et Modèles Statistiques
CNRS-Université Paris Sud-XI, Orsay
G. S., preprint arXiv:1203.1658
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
1 / 29
Extremes of vicious walkers: application to the
directed polymer and KPZ interfaces
G. Schehr
Laboratoire de Physique Théorique et Modèles Statistiques
CNRS-Université Paris Sud-XI, Orsay
Further references:
G. S., S. N. Majumdar, A. Comtet, J. Randon-Furling, Phys. Rev. Lett. 101, 150601 (2008)
J. Rambeau, G. S., Europhys. Lett. 91, 60006 (2010)
P. J. Forrester, S. N. Majumdar, G. S., Nucl. Phys. B 844, 500 (2011)
J. Rambeau, G. S., Phys. Rev. E 83, 061146 (2011)
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
1 / 29
Motivations: Directed Polymer in a Random Medium
DPRM with one free end (”point to line”)
ǫij i.i.d.’s
T
E(C) =
X
ij
hi,ji∈C
X
x
t
• Eopt ≡ Energy of the optimal polymer
•X
≡ Transverse coordinate of the optimal polymer
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
2 / 29
Motivations: Directed Polymer in a Random Medium
DPRM with one free end (”point to line”)
E (Eopt ) ∼ aT , E (X ) = 0
ǫij i.i.d.’s
Eopt − E (Eopt ) ∼ O(T 1/3 )
T
X ∼ O(T 2/3 )
X
t
x
Q: what is the joint pdf of
Eopt , X ?
• Eopt ≡ Energy of the optimal polymer
•X
≡ Transverse coordinate of the optimal polymer
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
3 / 29
Fluctuations in DPRM and the Airy2 process minus a
parabola
The "Airy2 process minus a parabola"
Y (u) = A2 (u) − u 2
where A2 (u) is the Airy2 process
Prähofer, Spohn, 02
Fluctuations in the DPRM
1
T−3
(Eopt (T ) − E (Eopt (T ))) = max Y (u) = M
u∈R
T →∞ e0
lim
and
2
T−3
lim
X = arg max Y (u) = T
u∈R
T →∞ ξ
Johansson, 03
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
4 / 29
Extreme statistics of Airy2 process minus a parabola
M = max A2 (u) − u 2 , T = arg max A2 (u) − u 2
u∈R
u∈R
Joint pdf P̂(m, t) of M, T
P̂(m, t) = 21/3 F1 (22/3 m)
Moreno Flores, Quastel, Remenik, arXiv:1106.2716
∞
Z
Z
dx
0
0
∞
dy Φ−t,m (21/3 x)ρ22/3 m (x, y ) Φt,m (21/3 y )
where
Φt,m (x) = 2ex t [tAi(t 2 + m + x) + Ai0 (t 2 + m + x)]
and
ρm (x, y ) = (I − Π0 Bm Π0 )−1 (x, y ) , Bm (x, y ) = Ai(x + y + m)
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
5 / 29
Extreme statistics of Airy2 process minus a parabola
M = max A2 (u) − u 2 , T = arg max A2 (u) − u 2
u∈R
u∈R
Joint pdf P̂(m, t) of M, T
P̂(m, t) = 21/3 F1 (22/3 m)
Moreno Flores, Quastel, Remenik, arXiv:1106.2716
∞
Z
Z
∞
dx
0
0
Marginal distribution P̂(t) of T
dy Φ−t,m (21/3 x)ρ22/3 m (x, y ) Φt,m (21/3 y )
Quastel, Remenik, arXiv:1203.2907
log P̂(t) = −ct 3 + o(t 3 ) , t → ∞ with
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
4
32
≤c≤
3
3
Warwick, March 20
5 / 29
Extreme statistics of Airy2 process minus a parabola
This work: joint pdf P̂(m, t) of M, T
P̂(m, t) =
π2
2
G. Schehr (LPTMS Orsay)
14
3
F1 (22/3 m)
Z
∞
G. S., arXiv:1203.1658
f (x, 24/3 t)f (x, −24/3 t) dx
22/3 m
Extremes of N vicious walkers
Warwick, March 20
6 / 29
Extreme statistics of Airy2 process minus a parabola
This work: joint pdf P̂(m, t) of M, T
P̂(m, t) =
π2
2
14
3
F1 (22/3 m)
Z
∞
G. S., arXiv:1203.1658
f (x, 24/3 t)f (x, −24/3 t) dx
22/3 m
where
13
f (x, t) = −
G. Schehr (LPTMS Orsay)
22
π2
∞
Z
2
ζΦ2 (ζ, x)e−tζ dζ
0
Extremes of N vicious walkers
Warwick, March 20
6 / 29
Extreme statistics of Airy2 process minus a parabola
This work: joint pdf P̂(m, t) of M, T
P̂(m, t) =
π2
2
14
3
F1 (22/3 m)
Z
∞
G. S., arXiv:1203.1658
f (x, 24/3 t)f (x, −24/3 t) dx
22/3 m
where
13
f (x, t) = −
∂
∂
Ψ = AΨ ,
Ψ = BΨ , Ψ =
∂ζ
∂x
|
{z
}
22
π2
∞
Z
2
ζΦ2 (ζ, x)e−tζ dζ
0
Φ1 (ζ, x)
Φ2 (ζ, x)
Lax Pair
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
6 / 29
Extreme statistics of Airy2 process minus a parabola
This work: joint pdf P̂(m, t) of M, T
P̂(m, t) =
π2
2
14
3
F1 (22/3 m)
Z
∞
G. S., arXiv:1203.1658
f (x, 24/3 t)f (x, −24/3 t) dx
22/3 m
where
13
f (x, t) = −
∂
∂
Ψ = AΨ ,
Ψ = BΨ , Ψ =
∂ζ
∂x
|
{z
}
22
π2
∞
Z
2
ζΦ2 (ζ, x)e−tζ dζ
0
Φ1 (ζ, x)
Φ2 (ζ, x)
Lax Pair
A(ζ, x) =
−4ζ 2
G. Schehr (LPTMS Orsay)
4ζq
− x − 2q 2 + 2q 0
4ζ 2 + x + 2q 2 + 2q 0
−4ζq
Extremes of N vicious walkers
, B(ζ, x) =
q
−ζ
Warwick, March 20
ζ
−q
6 / 29
Extreme statistics of Airy2 process minus a parabola
This work: joint pdf P̂(m, t) of M, T
P̂(m, t) =
π2
2
14
3
F1 (22/3 m)
Z
∞
G. S., arXiv:1203.1658
f (x, 24/3 t)f (x, −24/3 t) dx
22/3 m
where
13
f (x, t) = −
∂
∂
Ψ = AΨ ,
Ψ = BΨ , Ψ =
∂ζ
∂x
|
{z
}
22
π2
∞
Z
2
ζΦ2 (ζ, x)e−tζ dζ
0
Φ1 (ζ, x)
Φ2 (ζ, x)
, Φ2 (ζ, x) ∼ − sin
4ζ 3
+ xζ
3
!
, ζ1
Lax Pair
A(ζ, x) =
−4ζ 2
G. Schehr (LPTMS Orsay)
4ζq
− x − 2q 2 + 2q 0
4ζ 2 + x + 2q 2 + 2q 0
−4ζq
Extremes of N vicious walkers
, B(ζ, x) =
q
−ζ
Warwick, March 20
ζ
−q
6 / 29
Extreme statistics of Airy2 process minus a parabola
This work: joint pdf P̂(m, t) of M, T
P̂(m, t) =
π2
2
14
3
F1 (22/3 m)
Z
Marginal distribution P̂(t) of T
∞
G. S., arXiv:1203.1658
f (x, 24/3 t)f (x, −24/3 t) dx
22/3 m
G. S., arXiv:1203.1658
log P̂(t) = −ct 3 + o(t 3 ) , t → ∞ with c =
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
4
3
Warwick, March 20
7 / 29
Outline
1
Non-intersecting Brownian motions
2
Discrete orthogonal polynomials
3
Asymptotic analysis for large N: double scaling limit
4
Conclusion
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
8 / 29
Outline
1
Non-intersecting Brownian motions
2
Discrete orthogonal polynomials
3
Asymptotic analysis for large N: double scaling limit
4
Conclusion
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
9 / 29
Non-intersecting Brownian motions and Airy2
N non intersecting Brownian motions in one-dimension
xi(t)
x4(0)
x1 (t) < x2 (t) < ... < xN (t) ,
∀t
≥0
x3(0)
x2(0)
x1(0)
t
0
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
10 / 29
Non-intersecting Brownian motions and Airy2
N non intersecting Brownian motions in one-dimension
xi(t)
x1 (t) < x2 (t) < ... < xN (t) ,
∀t
≥0
N =4
0
1
watermelons
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
10 / 29
Non-intersecting Brownian motions and Airy2
N non intersecting Brownian motions in one-dimension
xi(t)
x1 (t) < x2 (t) < ... < xN (t) ,
∀t
N =4
0
≥0
1
watermelons
xi(t)
N =4
0
1
t
watermelons "with a wall"
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
10 / 29
Non-intersecting Brownian motions and Airy2
Large N limit
√ p
xN (τ ) ∼ 2 N τ (1 − τ )
0
lim
N→∞
1
h
√ i
1
2 xN ( 21 + u2 N − 3 ) − N
Prähofer & Spohn ’02
G. Schehr (LPTMS Orsay)
N
− 16
τ
d
= A2 (u) − u 2
A2 (u) ≡ Airy2 process
Extremes of N vicious walkers
Warwick, March 20
11 / 29
Non-intersecting Brownian motions and Airy2
Focus on non-intersecting excursions
√ p
xN (τ ) ∼ 2 2N τ (1 − τ )
N =4
xi(t)
−→
N1
0
1
t
0
1
τ
Expected: the fluctuations of the top path are insensitive to the
boundary, for N 1
lim
N→∞
h
√ i
1
α xN ( 21 + βuN − 3 ) − 2N
Tracy & Widom ’07
G. Schehr (LPTMS Orsay)
N
− 16
d
= A2 (u) − u 2
A2 (u) ≡ Airy2 process
Extremes of N vicious walkers
Warwick, March 20
12 / 29
Extreme of N non-intersecting excursions
xi(t)
N =4
h
0 < x1 (t) < x2 (t) < ... < xN (t)
HN = max xN (t)
t∈[0,1]
0
τh
1
t
TH = arg max xN (t)
t∈[0,1]
PN (h, τh ) ≡ joint probability distribution function of HN , TH
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
13 / 29
Extreme of N non-intersecting excursions
xi(t)
N =4
h
0 < x1 (t) < x2 (t) < ... < xN (t)
HN = max xN (t)
t∈[0,1]
τh
0
1
t
TH = arg max xN (t)
t∈[0,1]
PN (h, τh ) ≡ joint probability distribution function of HN , TH
HERE :
• Compute exactly PN (h, τh ) for any finite N
• Perform the large N limit to describe the extremes
of Airy2 minus a parabola
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
13 / 29
An exact expression for PN (h, τh )
PN (h, τh )
=
AN
hN(2N+1)+3
"
X
0 )∈ZN+1
(n1 ,··· ,nN ,nN
2
0 2
×∆N (n12 , . . . , nN−1
, nN
)e
G. Schehr (LPTMS Orsay)
0
2 0
(−1)nN +nN nN
nN
J. Rambeau, G. S. 10 & ’11
2
N−1
Y
2
2
)
, nN
ni2 ∆N (n12 , . . . , nN−1
i=1
i#
P 2 π2 h
2 N−1
0 2
2
− π2
+τh nN
ni − 2 (1−τh )nN
2h
2h
i=1
Extremes of N vicious walkers
Warwick, March 20
14 / 29
An exact expression for PN (h, τh )
PN (h, τh )
=
"
AN
hN(2N+1)+3
X
0 )∈ZN+1
(n1 ,··· ,nN ,nN
2
=
2
N 2 +N/2+1
G. Schehr (LPTMS Orsay)
0
2 0
(−1)nN +nN nN
nN
2
0
×∆N (n12 , . . . , nN−1
, nN
)e
AN
J. Rambeau, G. S. 10 & ’11
N π 2N
QN−1
j=0
2
2
N−1
Y
2
2
)
, nN
ni2 ∆N (n12 , . . . , nN−1
i=1
P 2 π2
2 N−1
− π2
ni − 2
2h
2h
i=1
h
i#
0 2
2
(1−τh )nN
+τh nN
+N+2
Γ(2 + j)Γ
3
2
+j
Extremes of N vicious walkers
Warwick, March 20
14 / 29
Outline
1
Non-intersecting Brownian motions
2
Discrete orthogonal polynomials
3
Asymptotic analysis for large N: double scaling limit
4
Conclusion
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
15 / 29
Introducing (discrete) orthogonal polynomials
"
PN (h, τh )
X
∝
0
2 0
nN
(−1)nN +nN nN
2
0 )∈ZN+1
(n1 ,··· ,nN ,nN
0 2
2
, nN
)e
×∆N (n12 , . . . , nN−1
G. Schehr (LPTMS Orsay)
N−1
Y
2
2
)
, nN
ni2 ∆N (n12 , . . . , nN−1
i=1
i#
P 2 π2 h
2 N−1
0 2
2
− π2
ni − 2 (1−τh )nN
+τh nN
2h
2h
i=1
Extremes of N vicious walkers
Warwick, March 20
16 / 29
Introducing (discrete) orthogonal polynomials
"
PN (h, τh )
X
∝
0
2 0
nN
(−1)nN +nN nN
2
0 )∈ZN+1
(n1 ,··· ,nN ,nN
N−1
Y
2
2
)
, nN
ni2 ∆N (n12 , . . . , nN−1
i=1
0 2
2
, nN
)e
×∆N (n12 , . . . , nN−1
i#
P 2 π2 h
2 N−1
0 2
2
− π2
ni − 2 (1−τh )nN
+τh nN
2h
2h
i=1
Discrete orthogonal polynomials
∞
X
pk (n)pk 0 (n)e
2
− π 2 n2
2h
= δk ,k 0 hk , pk (n) = nk + · · ·
n=−∞
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
16 / 29
Introducing (discrete) orthogonal polynomials
"
PN (h, τh )
X
∝
0
2 0
nN
(−1)nN +nN nN
2
0 )∈ZN+1
(n1 ,··· ,nN ,nN
N−1
Y
2
2
)
, nN
ni2 ∆N (n12 , . . . , nN−1
i=1
0 2
2
, nN
)e
×∆N (n12 , . . . , nN−1
i#
P 2 π2 h
2 N−1
0 2
2
− π2
ni − 2 (1−τh )nN
+τh nN
2h
2h
i=1
Discrete orthogonal polynomials
∞
X
pk (n)pk 0 (n)e
2
− π 2 n2
2h
= δk ,k 0 hk , pk (n) = nk + · · ·
n=−∞
After some manipulations...
PN (h, τh ) ∝
N
Y
h2j−1
(−1)n+m n m
n,m
j=1
|
X
{z
k =1
}
P[HN ≤ h] = FN (h)
G. Schehr (LPTMS Orsay)
h
i
N
X
p2k −1 (n)p2k −1 (m) − π22 (1−τh )n2 +τh m2
e 2h
h2k −1
P. J. Forrester, S. N. Majumdar, G. S. ’11
Extremes of N vicious walkers
Warwick, March 20
16 / 29
Intermezzo on the marginal distribution of HN
Cumulative distribution of the maximum HN
FN (h)
= P [xN (τ ) ≤ h , ∀ 0 ≤ τ ≤ 1]
Z 1
Z h
dτh
dx PN (x, τh )
=
0
G. Schehr (LPTMS Orsay)
0
Extremes of N vicious walkers
Warwick, March 20
17 / 29
Intermezzo on the marginal distribution of HN
Cumulative distribution of the maximum HN
FN (h)
= P [xN (τ ) ≤ h , ∀ 0 ≤ τ ≤ 1]
Z 1
Z h
=
dτh
dx PN (x, τh )
0
0
Exact result for finite N
FN (h) =
=
BN
h2N 2 +N
B̃N
h2N 2 +N
G. Schehr (LPTMS Orsay)
G. S, S. N. Majumdar, A. Comtet, J. Randon-Furling ’08
+∞
X
N
Y
n1 ,··· ,nN =0 i=1
N
Y
h2j−1
ni2
Y
1≤j<k ≤N
2
(nj2 − nk2 )2 e
− π2
PN
2h
i=1
ni2
see also T. Feierl ’08, M. Katori et al. ’08
j=1
Extremes of N vicious walkers
Warwick, March 20
17 / 29
Intermezzo on the marginal distribution of HN
Exact result for finite N
G. S, S. N. Majumdar, A. Comtet, J. Randon-Furling ’08
FN (h) =
B̃N
h2N 2 +N
N
Y
h2j−1
j=1
Large N analysis
√ FN (M) → F1 211/6 N 1/6 M − 2N Z
1 ∞
2
(s − t) q (s) − q(s) ds
F1 (t) = exp −
2 t
≡ Tracy-Widom distribution for β = 1
P. J. Forrester, S. N. Majumdar, G.S. ’11
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
18 / 29
Intermezzo on the marginal distribution of HN
Exact result for finite N
G. S, S. N. Majumdar, A. Comtet, J. Randon-Furling ’08
FN (h) =
B̃N
h2N 2 +N
N
Y
h2j−1
j=1
Large N analysis
√ FN (M) → F1 211/6 N 1/6 M − 2N Z
1 ∞
2
(s − t) q (s) − q(s) ds
F1 (t) = exp −
2 t
≡ Tracy-Widom distribution for β = 1
P. J. Forrester, S. N. Majumdar, G.S. ’11
A recent rigorous proof of this result:
G. Schehr (LPTMS Orsay)
K. Liechty arXiv:1111.4239
Extremes of N vicious walkers
Warwick, March 20
18 / 29
Back to discrete orthogonal polynomials
Discrete orthogonal polynomials
∞
X
pk (n)pk 0 (n)e
2
− π 2 n2
2h
= δk ,k 0 hk , pk (n) = nk + · · ·
n=−∞
After some manipulations...
PN (h, τh ) ∝
N
Y
h2j−1
(−1)n+m n m
n,m
j=1
{z
|
X
h
i
N
X
p2k −1 (n)p2k −1 (m) − π22 (1−τh )n2 +τh m2
e 2h
h2k −1
k =1
}
P[HN ≤ h] = FN (h)
P. J. Forrester, S. N. Majumdar, G. S. ’11
N
PN (h,
X
1
B̃N
G2k −1 (h, u)G2k −1 (h, −u)
+ u) = 3 FN (h)
2
h
k =1
G2k −1 (h, u) =
∞
X
(−1)n n ψ2k −1 (n)e
2
− uπ2 n2
2h
n=−∞
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
pk (n) − π2 n2
, ψk (n) = p e 4h2
hk
Warwick, March 20
19 / 29
Differential recursion relations (I)
∞
X
2
pk (n)pk 0 (n)e
n=−∞
− π 2 n2
2h
pk (n) − π2 n2
= δk ,k 0 hk , ψk (n) = √ e 4h2
hk
Three terms recursion relation
hk
,
hk −1
p
x ψk (x) = γk +1 ψk +1 (x) + γk ψk −1 (x) , γk = Rk
x pk (x) = pk +1 (x) + Rk pk −1 (x) , Rk =
A first differential recursion relation for G2k −1 (h, u)
G2k −1 (h, u) =
∞
X
(−1)n n ψ2k −1 (n)e
2
− uπ2 n2
2h
n=−∞
i
∂
π2 h
2
2
G2k −1 = − 2 γ2k γ2k +1 G2k +1 + (γ2k
+ γ2k
−1 )G2k −1 + γ2k −2 γ2k −1 G2k −3
∂u
2h
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
20 / 29
Differential recursion relations (II)
∞
X
2
pk (n)pk 0 (n)e
− π 2 n2
2h
n=−∞
pk (n) − π2 n2
= δk ,k 0 hk , ψk (n) = √ e 4h2
hk
A differential recursion relation for ψk (n)
−
h3 ∂
1
1
ψk = γk γk −1 ψk −2 − γk +2 γk +1 ψk +2
2π 2 ∂h
4
4
A second differential recursion relation for G2k −1 (h, u)
h3 ∂
G2k −1
− 2
2π ∂h
"
=
−
G. Schehr (LPTMS Orsay)
1
u
u 2
2
−
γ2k −1 γ2k −2 G2k −3 −
γ2k + γ2k
−1 G2k −1
4
2
2
#
1
u
+
γ2k γ2k +1 G2k +1
4
2
Extremes of N vicious walkers
Warwick, March 20
21 / 29
Outline
1
Non-intersecting Brownian motions
2
Discrete orthogonal polynomials
3
Asymptotic analysis for large N: double scaling limit
4
Conclusion
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
22 / 29
Double scaling limit
1
h → ∞ & N → ∞ , with N 6 (h −
√
2N) fixed
Analysis of the coefficients Rk = hk /hk −1 for large k
Rk =
h4
− (−1)k h10/3 f1 (xk ) + h8/3 f2 (xk ) + O(h2 ) , xk = h4/3
π2
f1 (x) = −
1−
k
h2
25/3
q(22/3 x) , q 00 (s) = sq(s) + 2q 3 (s) , q(s) ∼ Ai(s)
s→+∞
π2
Gross, Matytsin ’94
Analysis of G2k −1 (h, u) for large k
G2k −1 (h, u) = (−1)k h5/3 g1 (x2k , v ) + h−2/3 g2 (x2k , v ) + h−4/3 g3 (x2k , v ) + O(h−2 )
2k
x2k = h4/3 1 − 2
, v = u h2/3
h
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
23 / 29
Differential recursion relations in the dble scaling limit
i
π2 h
∂
2
2
+ γ2k
G2k −1 = − 2 γ2k γ2k +1 G2k +1 + (γ2k
−1 )G2k −1 + γ2k −2 γ2k −1 G2k −3
∂u
2h
G2k −1 (h, u) = (−1)k h5/3 g1 (x2k , v ) + O(h)
g1 (x, v ) = f (s = 22/3 x, w = 27/3 v )
where f (s, w) satisfies
G. S. ’12
2
∂
∂
2
0
f (s, w) =
− (q − q ) f (s, w)
∂w
∂s2
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
24 / 29
Differential recursion relations in the dble scaling limit
−
"
h3 ∂
G2k −1
2π 2 ∂h
1
u
u 2
2
−
γ2k −1 γ2k −2 G2k −3 −
γ2k + γ2k
−1 G2k −1
4
2
2
#
u
1
+
γ2k γ2k +1 G2k +1
4
2
=
−
G2k −1 (h, u) = (−1)k h5/3 g1 (x2k , v ) + O(h)
g1 (x, v ) = f (s = 22/3 x, w = 27/3 v )
where f (s, w) satisfies
∂3f
G. S. ’12
∂2f
i
h
i
∂f h
4 3 − 2w 2 −
6(q 2 − q 0 ) + s − f 3(q 2 − q 0 )0 + 2 − 2w(q 2 − q 0 ) = 0
∂s
∂s
∂s
with the asymptotic behavior (matching the regime h f (s, w) ∼
−211/3
π
ew
3
+ w4s
G. Schehr (LPTMS Orsay)
√
2N)
w
1
Ai(w 2 /28/3 + s/22/3 ) + 2/3 Ai0 (w 2 /28/3 + s/22/3 )
4
2
Extremes of N vicious walkers
Warwick, March 20
25 / 29
Solution in terms of a psi-function associated to P II
f (s, w) = −
∂
∂
Ψ = AΨ ,
Ψ = BΨ , Ψ =
∂ζ
∂x
{z
}
|
213/2
π2
Z
∞
2
ζΦ2 (ζ, s)e−wζ dζ
0
Φ1 (ζ, x)
Φ2 (ζ, x)
, Φ2 (ζ, x) ∼ − sin
4ζ 3
+ xζ
3
!
, ζ1
Lax Pair
A(ζ, x) =
−4ζ 2
4ζq
− x − 2q 2 + 2q 0
G. Schehr (LPTMS Orsay)
4ζ 2 + x + 2q 2 + 2q 0
−4ζq
Extremes of N vicious walkers
, B(ζ, x) =
q
−ζ
ζ
−q
Warwick, March 20
26 / 29
Large N limit and back to A2 (u) − u 2
Large N asymptotics for extremes of excursions
√
11
1 1
8
1
9
1
lim 2− 2 N − 2 PN ( 2N + 2− 6 s N − 6 , + 2− 3 w N − 3 ) = P(s, w)
2
Z ∞
π2
f (x, w)f (x, −w) dx
P(s, w) = 20 F1 (s)
s
23
13 Z
∞
2
22
ζΦ2 (ζ, x)e−wζ dζ
f (x, w) = − 2
π
0
N→∞
Results for Airy2 process minus a parabola
P̂(m, t) = 4 P(22/3 m , 24/3 t)
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
27 / 29
Outline
1
Non-intersecting Brownian motions
2
Discrete orthogonal polynomials
3
Asymptotic analysis for large N: double scaling limit
4
Conclusion
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
Warwick, March 20
28 / 29
Conclusion
Exact results for extreme statistics of N vicious walkers
Large N analysis using dble scaling limit of discrete orthogonal
polynomials
Connection between extreme statistics of A2 (u) − u 2 and Painlevé
Marginal distribution P̂(t) of T
G. S., arXiv:1203.1658
log P̂(t) = −c t 3 + o(t 3 ) , t → ∞ with c =
G. Schehr (LPTMS Orsay)
Extremes of N vicious walkers
4
3
Warwick, March 20
29 / 29