Random walks in cones: critical exponents

Random walks in cones: critical exponents
Lecture #2
Analytic and probabilistic tools for lattice path enumeration
Kilian Raschel
77th Séminaire Lotharingien de Combinatoire
September 13, 2016
Strobl, Austria
Introduction
Dimension 1: examples & limits
Central idea in dimension > 2: approximation by Brownian motion
Application #1: excursions
Application #2: walks with prescribed length
Random processes (RW & BM) in cones
First exit time from a cone C
C
x
-
Random processes (RW & BM) in cones
First exit time from a cone C
*
x
C
-
?
-
Random processes (RW & BM) in cones
First exit time from a cone C
*
x
@
I
@
C
-
?
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
Persistence probabilities ; total number of walks
. Px [τC > n] ∼ κ · V (x) · ρn · n−α
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
Persistence probabilities ; total number of walks
×
. Px [τC > n] ∼ κ · V (x) · ρn · n−α
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
Persistence probabilities ; total number of walks
××
. Px [τC > n] ∼ κ · V (x) · ρn · n−α
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
Persistence probabilities ; total number of walks
×× ×
. Px [τC > n] ∼ κ · V (x) · ρn · n−α
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
Persistence probabilities ; total number of walks
×× ×
. Px [τC > n] ∼ κ · V (x) · ρn · n−α
Local limit theorems ; excursions
. Px [τC > n, S(n) = y ] ∼ κ · V (x, y ) · ρn · n−α
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
Persistence probabilities ; total number of walks
×× ×
. Px [τC > n] ∼ κ · V (x) · ρn · n−α
Local limit theorems ; excursions
× ××
. Px [τC > n, S(n) = y ] ∼ κ · V (x, y ) · ρn · n−α
-
Random processes (RW & BM) in cones
First exit time from a cone C
. τC = inf{n ∈ N : S(n) ∈
/ C } (S RW)
. TC = inf{t ∈ R+ : B(t) ∈
/ C } (B BM)
*
x
@
I
@
C
-
?
Persistence probabilities ; total number of walks
×× ×
. Px [τC > n] ∼ κ · V (x) · ρn · n−α
Local limit theorems ; excursions
× ××
. Px [τC > n, S(n) = y ] ∼ κ · V (x, y ) · ρn · n−α
Aim of the talk: understanding the critical exponents α
-
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
where the X (i) are i.i.d.
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
where the X (i) are i.i.d. (e.g., uniform on a step set S ⊂ Zd )
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
where the X (i) are i.i.d. (e.g., uniform on a step set S ⊂ Zd )
. Example (Dyck paths): simple random walk X (i) ∈ {−1, 1}
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
-3
-2
-1
0
1
2
where the X (i) are i.i.d. (e.g., uniform on a step set S ⊂ Zd )
. Example (Dyck paths): simple random walk X (i) ∈ {−1, 1}
5
10
15
20
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
-25
-20
-15
-10
-5
0
5
where the X (i) are i.i.d. (e.g., uniform on a step set S ⊂ Zd )
. Example (Dyck paths): simple random walk X (i) ∈ {−1, 1}
50
100
150
200
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
-400
-300
-200
-100
0
100
where the X (i) are i.i.d. (e.g., uniform on a step set S ⊂ Zd )
. Example (Dyck paths): simple random walk X (i) ∈ {−1, 1}
50000
100000
150000
200000
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
where the X (i) are i.i.d. (e.g., uniform on a step set S ⊂ Zd )
. Example (Dyck paths): simple random walk X (i) ∈ {−1, 1}
Motivations
. Persistence probabilities in statistical physics
. Constructing processes conditioned never to leave cones
Definition of random walks & motivations
Random walk on Zd
. A random walk {S(n)}n>0 is
S(n) = x + X (1) + · · · + X (n),
where the X (i) are i.i.d. (e.g., uniform on a step set S ⊂ Zd )
. Example (Dyck paths): simple random walk X (i) ∈ {−1, 1}
Motivations
. Persistence probabilities in statistical physics
. Constructing processes conditioned never to leave cones
. Asymptotics of numbers of walks
. Transcendental nature of functions counting walks in cones
Alin Bostan’s course at AEC
. Important & combinatorial cones (quarter/half/slit plane,
orthants, Weyl chambers, etc.)
Introduction
Dimension 1: examples & limits
Central idea in dimension > 2: approximation by Brownian motion
Application #1: excursions
Application #2: walks with prescribed length
-3
-2
-1
0
1
2
Non-constrained walk with S = {−1, 1}
5
10
15
20
-3
-2
-1
0
1
2
Non-constrained walk with S = {−1, 1}
5
n
. #{x −→ Z} = 2n
10
15
20
Walk
Exponent 0
-3
-2
-1
0
1
2
Non-constrained walk with S = {−1, 1}
5
n
10
15
. #{x −→ Z} = 2n
Walk
r
n
2 2n
n
√ Bridge
. #{x −→ y } = n+(y −x) ∼
π n
2
20
Exponent 0
1
Exponent
2
-3
-2
-1
0
1
2
Non-constrained walk with S = {−1, 1}
5
n
10
15
20
. #{x −→ Z} = 2n
Walk
Exponent 0
r
n
2 2n
1
n
√ Bridge
. #{x −→ y } = n+(y −x) ∼
Exponent
π n
2
2
P 1
√ = ∞: recurrence of the simple random walk in Z
.
n
-3
-2
-1
0
1
2
Non-constrained walk with S = {−1, 1}
5
n
10
15
20
. #{x −→ Z} = 2n
Walk
Exponent 0
r
n
2 2n
1
n
√ Bridge
. #{x −→ y } = n+(y −x) ∼
Exponent
π n
2
2
P 1
√ = ∞: recurrence of the simple random walk in Z
.
n
q
. Constant π2 independent of x & y in the asymptotics
0
1
2
3
4
5
Constrained walk with S = {−1, 1} (Dyck paths)
5
10
15
20
0
1
2
3
4
5
Constrained walk with S = {−1, 1} (Dyck paths)
5
2n
. #N {x −→ N} ∼ √
n
n
10
15
Meanders
20
Exponent
1
2
0
1
2
3
4
5
Constrained walk with S = {−1, 1} (Dyck paths)
5
2n
. #N {x −→ N} ∼ √
n
2n
n
. #N {x −→ y } ∼ 3/2
n
n
10
15
Meanders
Excursions
20
1
2
3
Exponent
2
Exponent
0
1
2
3
4
5
Constrained walk with S = {−1, 1} (Dyck paths)
5
10
15
20
2n
1
. #N {x −→ N} ∼ √
Meanders
Exponent
2
n
2n
3
n
. #N {x −→ y } ∼ 3/2
Excursions
Exponent
2
n
q
n
. Reflection principle cancels the first term π2 √2 n
n
0
1
2
3
4
5
Constrained walk with S = {−1, 1} (Dyck paths)
5
.
.
.
.
10
15
20
2n
1
#N {x −→ N} ∼ √
Meanders
Exponent
2
n
2n
3
n
#N {x −→ y } ∼ 3/2
Excursions
Exponent
2
n
q
n
Reflection principle cancels the first term π2 √2 n
Wiener-Hopf techniques in probability theory
n
0
1
2
3
4
5
Constrained walk with S = {−1, 1} (Dyck paths)
5
.
.
.
.
.
10
15
20
2n
#N {x −→ N} ∼ √
Meanders
Exponent
n
2n
n
#N {x −→ y } ∼ 3/2
Excursions
Exponent
n
q
n
Reflection principle cancels the first term π2 √2 n
Wiener-Hopf techniques in probability theory
See Bousquet-Mélou & Petkovšek ’00; Banderier & Flajolet ’02
n
1
2
3
2
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
The simple walk in two-dimensional wedges
1
2
3
2
&
3
2
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
The simple walk in two-dimensional wedges
. Half-plane:
one-dimensional case
6
6
?
. Dyck paths
-
. Total number of walks:
Exponent 12
. Excursions:
Exponent 2 =
3
2
+
1
2
3
2
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
The simple walk in two-dimensional wedges
. Quarter plane: product of
two one-dimensional cases
6
6
?
. Reflection principle
-
. Total number of walks:
Exponent 1 = 12 + 12
. Excursions:
Exponent 3
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
The simple walk in two-dimensional wedges
. Slit plane:
6
Bousquet-Mélou & Schaeffer ’00
6
?
. Highly non-convex cone
-
. Total number of walks:
Exponent 14
. Excursions:
Exponent
3
2
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
The simple walk in two-dimensional wedges
. 45◦ :
6
6
?
-
Gouyou-Beauchamps ’86
. See
Bousquet-Mélou & Mishna ’10
. Total number of walks:
Exponent 2
. Excursions:
Exponent 5
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
The simple walk in two-dimensional wedges
. 135◦ : Gessel
6
6
?
. See
Kauers, Koutschan &
Zeilberger ’09, etc.
-
. Total number of walks:
Exponent 23
. Excursions:
Exponent
7
3
3
2
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
The simple walk in two-dimensional wedges
. Walks avoiding a quadrant
6
6
?
. See
Bousquet-Mélou ’15;
Mustapha ’15
-
. Total number of walks:
Exponent 13
. Excursions:
Exponent
5
3
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
The simple walk in two-dimensional wedges
. Arbitrary angular sector θ
6
6
?
. See
Varopoulos ’99; Denisov &
Wachtel ’15
-
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
The simple walk in two-dimensional wedges
. Arbitrary angular sector θ
6
6
?
. See
Varopoulos ’99; Denisov &
Wachtel ’15
-
. Total number of walks:
π
Exponent 2θ
. Excursions:
Exponent
π
θ
+1
Beyond the algebraic exponents 0,
1
2
&
Weighted models in dimension 1
P
Drift s∈S s governs the exponents, which are still 0,
1
2
3
2
&
3
2
The simple walk in two-dimensional wedges
. Arbitrary angular sector θ
6
6
?
. See
Varopoulos ’99; Denisov &
Wachtel ’15
-
. Total number of walks:
π
Exponent 2θ
. Excursions:
Exponent
π
θ
+1
Conclusion: 1D case not enough
. Dramatic change of behavior: every exponent is possible!
. Non-D-finite behaviors (first observed by Varopoulos ’99)
Introduction
Dimension 1: examples & limits
Central idea in dimension > 2: approximation by Brownian motion
Application #1: excursions
Application #2: walks with prescribed length
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
100
Donsker’s theorem (functional central limit theorem)
law
-400
-300
-200
-100
0
RW −→ BM
50000
100000
150000
200000
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
100
Donsker’s theorem (functional central limit theorem)
law
-100
0
RW −→ BM
-400
-300
-200
50000
100000
150000
200000
S(bntc)
√
n
law
−→ {B(t)}t
t
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
Denisov & Wachtel ’15 (excursions for RW in cones of ⊂ Zd )
law
. RW −→ BM
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
Denisov & Wachtel ’15 (excursions for RW in cones of ⊂ Zd )
law
. RW −→ BM
. Mapping theorem: many asymptotic results concerning RW
can be deduced from BM
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
Denisov & Wachtel ’15 (excursions for RW in cones of ⊂ Zd )
law
. RW −→ BM
. Mapping theorem: many asymptotic results concerning RW
can be deduced from BM
E [RW] = E [BM] = 0
. For excursions, α{RW} = α{BM} if
V[RW] = V[BM] = id
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
Denisov & Wachtel ’15 (excursions for RW in cones of ⊂ Zd )
law
. RW −→ BM
. Mapping theorem: many asymptotic results concerning RW
can be deduced from BM
E [RW] = E [BM] = 0
. For excursions, α{RW} = α{BM} if
V[RW] = V[BM] = id
. If V[RW] 6= id then V[M · RW] = id for some M ∈ Md (R)
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
Denisov & Wachtel ’15 (excursions for RW in cones of ⊂ Zd )
law
. RW −→ BM
. Mapping theorem: many asymptotic results concerning RW
can be deduced from BM
E [RW] = E [BM] = 0
. For excursions, α{RW} = α{BM} if
V[RW] = V[BM] = id
. If V[RW] 6= id then V[M · RW] = id for some M ∈ Md (R)
. Cone C becomes M · C
Brownian motion on R
Law of large numbers
X (1) + · · · + X (n) a.s.
−→ E[X (1)]
n1
Central limit theorem
1
X (1) + · · · + X (n)
law
n2
− E[X (1)] −→ N (0, V[X (1)])
n1
Denisov & Wachtel ’15 (excursions for RW in cones of ⊂ Zd )
law
. RW −→ BM
. Mapping theorem: many asymptotic results concerning RW
can be deduced from BM
E [RW] = E [BM] = 0
. For excursions, α{RW} = α{BM} if
V[RW] = V[BM] = id
Remainder of this section: computing α{BM} (easier)
Two derivations of the BM persistence probability in R
Reflection principle
x
6
Px [T(0,∞) > t] = P0 [ min B(u) > −x]
06u6t
t
-
= P0 [|B(t)| < x]
Z x
y2
2
=√
e − 2t dy
2πt 0
Two derivations of the BM persistence probability in R
Reflection principle
x
6
Px [T(0,∞) > t] = P0 [ min B(u) > −x]
06u6t
-
t
= P0 [|B(t)| < x]
Z x
y2
2
=√
e − 2t dy
2πt 0
Heat equation
Function g (t; x) = Px [T(0,∞) > t] satisfies



∂
∂t
− 21 ∆ g (t; x) = 0, ∀x ∈ (0, ∞), ∀t ∈ (0, ∞)
g (0; x) = 1, ∀x ∈ (0, ∞)
g (t; 0) = 0, ∀t ∈ (0, ∞)
Dimension d: explicit expression for Px [TC > t]
Heat equation
Doob ’55
For essentially any domain RC in any dimension d, Px [TC > t] &
p C (t; x, y ) (Px [TC > t] = C p C (t; x, y )dy ) satisfy heat equations
Dimension d: explicit expression for Px [TC > t]
Heat equation
Doob ’55
For essentially any domain RC in any dimension d, Px [TC > t] &
p C (t; x, y ) (Px [TC > t] = C p C (t; x, y )dy ) satisfy heat equations
Dirichlet eigenvalues problem
C
t d−1
∩C
S
t
1
-
Chavel ’84
∆Sd−1 m = −λm in Sd−1 ∩ C
m = 0
in ∂(Sd−1 ∩ C )
Dimension d: explicit expression for Px [TC > t]
Heat equation
Doob ’55
For essentially any domain RC in any dimension d, Px [TC > t] &
p C (t; x, y ) (Px [TC > t] = C p C (t; x, y )dy ) satisfy heat equations
Dirichlet eigenvalues problem
C
t d−1
∩C
S
t
Chavel ’84
∆Sd−1 m = −λm in Sd−1 ∩ C
m = 0
in ∂(Sd−1 ∩ C )
-
1
Discrete eigenvalues 0 < λ1 < λ2 6 λ3 6 ... and eigenfunctions
m1 , m2 , m3 , ...
Dimension d: explicit expression for Px [TC > t]
Heat equation
Doob ’55
For essentially any domain RC in any dimension d, Px [TC > t] &
p C (t; x, y ) (Px [TC > t] = C p C (t; x, y )dy ) satisfy heat equations
Chavel ’84
Dirichlet eigenvalues problem
C
t d−1
∩C
S
t
∆Sd−1 m = −λm in Sd−1 ∩ C
m = 0
in ∂(Sd−1 ∩ C )
-
1
Discrete eigenvalues 0 < λ1 < λ2 6 λ3 6 ... and eigenfunctions
m1 , m2 , m3 , ...
Series expansion
DeBlassie ’87; Bañuelos & Smits ’97
∞
X
Px [TC > t] =
Bj (|x|2 /t)mj (x/|x|)
j=1
Asymptotics of the non-exit probability
DeBlassie ’87; Bañuelos & Smits ’97
Series expansion
Px [TC > t] =
∞
X
Bj (|x|2 /t)mj (x/|x|),
j=1
with
. Bj hypergeometric
. series expansion very well suited for asymptotics
Asymptotics of the non-exit probability
DeBlassie ’87; Bañuelos & Smits ’97
Series expansion
Px [TC > t] =
∞
X
Bj (|x|2 /t)mj (x/|x|),
j=1
with
. Bj hypergeometric
. series expansion very well suited for asymptotics
Asymptotic result
DeBlassie ’87; Bañuelos & Smits ’97
Px [TC > t] ∼ κ · u(x) · t −α ,
Asymptotics of the non-exit probability
DeBlassie ’87; Bañuelos & Smits ’97
Series expansion
Px [TC > t] =
∞
X
Bj (|x|2 /t)mj (x/|x|),
j=1
with
. Bj hypergeometric
. series expansion very well suited for asymptotics
Asymptotic result
DeBlassie ’87; Bañuelos & Smits ’97
Px [TC > t] ∼ κ · u(x) · t −α ,
q
with α = 2 λ1 + ( d2 − 1)2 − ( d2 − 1) linked to the first eigenvalue
Asymptotics of the non-exit probability
DeBlassie ’87; Bañuelos & Smits ’97
Series expansion
Px [TC > t] =
∞
X
Bj (|x|2 /t)mj (x/|x|),
j=1
with
. Bj hypergeometric
. series expansion very well suited for asymptotics
Asymptotic result
DeBlassie ’87; Bañuelos & Smits ’97
Px [TC > t] ∼ κ · u(x) · t −α ,
q
with α = 2 λ1 + ( d2 − 1)2 − ( d2 − 1) linked to the first eigenvalue
Exercise
π
of the persistence probability for a simple
Recover the exponent 2θ
random walk in a two-dimensional wedge of opening angle θ
Introduction
Dimension 1: examples & limits
Central idea in dimension > 2: approximation by Brownian motion
Application #1: excursions
Application #2: walks with prescribed length
Example #1: Gouyou-Beauchamps model
In the quarter plane
6
@
I
@ @
R
@
-
Example #1: Gouyou-Beauchamps model
In the quarter plane
Hypotheses on the moments:
6
E[GB] = (1, 0)+(1, −1)+(−1, 0)+(−1, 1)
@
I
@ @
R
@
= (0, 0)
-
Example #1: Gouyou-Beauchamps model
In the quarter plane
Hypotheses on the moments:
6
E[GB] = (1, 0)+(1, −1)+(−1, 0)+(−1, 1)
@
I
@ @
R
@
-
= (0, 0)
4 −2
V[GB] =
6= id
−2
4
Example #1: Gouyou-Beauchamps model
In the quarter plane
Hypotheses on the moments:
6
E[GB] = (1, 0)+(1, −1)+(−1, 0)+(−1, 1)
@
I
@ @
R
@
-
= (0, 0)
4 −2
V[GB] =
6= id
−2
4
Changing the cone
6
6
?
-
Example #1: Gouyou-Beauchamps model
In the quarter plane
Hypotheses on the moments:
6
E[GB] = (1, 0)+(1, −1)+(−1, 0)+(−1, 1)
@
I
@ @
R
@
-
= (0, 0)
4 −2
V[GB] =
6= id
−2
4
Changing the cone
. Wedge of angle θ =
6
6
?
-
π
4
. Total number of walks:
π
Exponent 2θ
=2
. Excursions:
Exponent
π
θ
+1=5
Example #2: quadrant walks
A scarecrow
6
@
I
@6
@
R
@
-
Example #2: quadrant walks
A scarecrow
6
. E = (0, 0) & V =
@
I
@6
@
R
@
-
4 −3
−3
4
6= id
Example #2: quadrant walks
A scarecrow
4 −3
. E = (0, 0) & V =
6= id
−3
4
. θ = arccos − 41 =⇒ α = πθ + 1 ∈
/Q
6
@
I
@6
@
R
@
-
Example #2: quadrant walks
A scarecrow
6
@
I
@6
@
R
@
-
4 −3
. E = (0, 0) & V =
6= id
−3
4
. θ = arccos − 41 =⇒ α = πθ + 1 ∈
/Q
P∞
n
n
.
n=0 #N2 {(0, 0) −→ (0, 0)}t
non-D-finite
Example #2: quadrant walks
A scarecrow
6
@
I
@6
@
R
@
-
4 −3
. E = (0, 0) & V =
6= id
−3
4
. θ = arccos − 41 =⇒ α = πθ + 1 ∈
/Q
P∞
n
n
.
n=0 #N2 {(0, 0) −→ (0, 0)}t
non-D-finite
In dimension 2 (excursions only)
Bostan, R. & Salvy ’14
. Systematic computation of α = arccos{algebraic number}
Example #2: quadrant walks
A scarecrow
6
@
I
@6
@
R
@
-
4 −3
. E = (0, 0) & V =
6= id
−3
4
. θ = arccos − 41 =⇒ α = πθ + 1 ∈
/Q
P∞
n
n
.
n=0 #N2 {(0, 0) −→ (0, 0)}t
non-D-finite
In dimension 2 (excursions only)
Bostan, R. & Salvy ’14
. Systematic computation of α = arccos{algebraic number}
. Walks with small steps in N2 :
. α ∈ Q iff
. generating function of the excursions is D-finite iff
. the group of the model is finite
Example #2: quadrant walks
A scarecrow
6
@
I
@6
@
R
@
-
4 −3
. E = (0, 0) & V =
6= id
−3
4
. θ = arccos − 41 =⇒ α = πθ + 1 ∈
/Q
P∞
n
n
.
n=0 #N2 {(0, 0) −→ (0, 0)}t
non-D-finite
In dimension 2 (excursions only)
Bostan, R. & Salvy ’14
. Systematic computation of α = arccos{algebraic number}
. Walks with small steps in N2 :
. α ∈ Q iff
. generating function of the excursions is D-finite iff
. the group of the model is finite
. If
P
s∈S s
6= 0, first perform a Cramér transform
Three-dimensional models
Example: Kreweras 3D
Model with jumps:
Three-dimensional models
Example: Kreweras 3D
Model with jumps:
q
Exponent α = 2 λ1 +
1
4
−
1
2
Three-dimensional models
Example: Kreweras 3D
Model with jumps:
q
Exponent α = 2 λ1 +
1
4
−
1
2
Three-dimensional models
Example: Kreweras 3D
Model with jumps:
q
Exponent α = 2 λ1 +
Value of λ1 ? λ1 ∈ Q?
1
4
−
1
2
Three-dimensional models
Example: Kreweras 3D
Model with jumps:
q
Exponent α = 2 λ1 +
1
4
−
1
2
Value of λ1 ? λ1 ∈ Q?
General theory (still to be done!)
. Classification & resolution of some finite group models
Bostan, Bousquet-Mélou, Kauers & Melczer ’16
. Asymptotic simulation
Bacher, Kauers & Yatchak ’16
Conjectured Kreweras exponent: 3.3257569
. Equivalence finite group iff D-finite generating functions?
Introduction
Dimension 1: examples & limits
Central idea in dimension > 2: approximation by Brownian motion
Application #1: excursions
Application #2: walks with prescribed length
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
Case #1: interior drift
6
1
-
P
s∈S s
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
P
s∈S s
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
6
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
6
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
6
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
6
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
. Exponent α =
*
-
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
. Exponent α =
*
1
2
Non-universal exponents: six cases
Excursions: formula for α independent of the drift
P
s∈S s
Case #1: interior drift
. Law of large numbers:
P[∀n, S(n) ∈ C ] > 0
6
. Exponent α = 0
1
-
. Cannot be used as a filter to detect
non-D-finiteness
Case #2: boundary drift
. Half-plane case
. Exponent α =
*
1
2
. Cannot be used as a filter to detect
non-D-finiteness
. Exponent α =
boundary
i
2
for non-smooth
Non-universal exponents: six cases
Case #3: directed drift
6
. Half-plane case
. Exponent α =
3
-
3
2
. Cannot be used as a filter to detect
non-D-finiteness
Non-universal exponents: six cases
Case #3: directed drift
6
. Half-plane case
. Exponent α =
3
-
3
2
. Cannot be used as a filter to detect
non-D-finiteness
Case #4: zero drift
. See
6
Varopoulos ’99; Denisov & Wachtel ’15
. Exponent
q
α1 = 2 λ1 + ( d2 − 1)2 − ( d2 − 1)
•
-
. Can be used as a filter to detect
non-D-finiteness
Non-universal exponents: six cases
Case #5: polar interior drift
6
]
JJ
J
J
J
. See
Duraj ’14
. Exponent 2α1 + 1
-
. Can be used as a filter to detect
non-D-finiteness
Non-universal exponents: six cases
Case #5: polar interior drift
6
]
JJ
J
J
J
. See
Duraj ’14
. Exponent 2α1 + 1
-
. Can be used as a filter to detect
non-D-finiteness
Case #6: polar boundary drift
6
BMB
. Exponent α1 + 1
B
B
B
-
. Can be used as a filter to detect
non-D-finiteness
Non-universal exponents: six cases
Case #5: polar interior drift
6
]
JJ
J
J
J
. See
Duraj ’14
. Exponent 2α1 + 1
-
. Can be used as a filter to detect
non-D-finiteness
Case #6: polar boundary drift
6
BMB
. Exponent α1 + 1
B
B
B
-
. Can be used as a filter to detect
non-D-finiteness
Six-exponents-result: joint with R. Garbit & S. Mustapha
A A
@
AA AA@
@
@
-