CDT & Trees
93
In Search for Quantum Gravity: CDT & Friends, Dec. 11-14, 2012, Nijmegen CDT & Trees Timothy Budd Niels Bohr Institute, Copenhagen. [email protected] Collaboration with J. Ambjørn
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Transcript of CDT & Trees
In Search for Quantum Gravity: CDT & Friends,Dec. 11-14, 2012, Nijmegen
CDT & Trees
Timothy BuddNiels Bohr Institute, Copenhagen.
Collaboration with J. Ambjørn
Outline
I Quadrangulations & Trees: ←→
0+
+000
- 0-+
+0 --0
- 00+
--
I CDT & Trees: ←→
I Generalized CDT & Trees: ←→
0+
+000
- 0-+
+0 --0
- 00+
--
I Loop amplitudes & Planar maps: ←→0
0
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
I Quadrangulation
→ mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
I Quadrangulation → mark a point
→ distance labeling → apply rules→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling
→ apply rules→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules
→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection
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[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree
→ add squares → identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection
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6
8
6
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6 6
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57
6
6
4
6
4
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5
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree → add squares
→ identify corners → quadrangulation.
The Cori–Vauquelin–Schaeffer bijection
7
6
6
6
8
6
6
8
6
7
7
7
7
9
7
5
7
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8
6 6
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57
6
6
4
6
4
6
5
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree → add squares → identify corners
→ quadrangulation.
The Cori–Vauquelin–Schaeffer bijection
7
6
6
6
8
6
6
8
6
7
7
7
7
9
7
5
7
7
8
8
6 6
6
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8
8
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7
5
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9
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9
5
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7
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7
7
57
6
6
4
6
4
6
5
[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree → add squares → identify corners
→ quadrangulation.
The Cori–Vauquelin–Schaeffer bijection[Cori, Vauquelin, ’81]
[Schaeffer, ’98]
t t
t + 1
t - 1
t t
t + 1
t + 1
I Quadrangulation → mark a point → distance labeling → apply rules→ labelled tree.
I Labelled tree → add squares → identify corners → quadrangulation.
Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)
I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}×Z2
Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)
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57
6
6
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4
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5
I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}×Z2
Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)
7
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6
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8
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6 6
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7
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7
57
6
6
4
6
4
6
5
I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}×Z2
Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)
+-
-
-
0
-
-
-
0+
-
-
0
-+
-
-
0
-
-
0
-
-
-
0
0
-
-
-
-
-
-
0
0
-
-
--
-
-
-
-
-
-
-
-
0
0
-
-
-
-
0
-
0-
-
-
-
-
-
-
-
-
-
-
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57
6
6
4
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4
6
5
I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}×Z2
Rooting the tree (Miermont, Bouttier, Guitter, Le Gall,...)
+-
-
-
0
-
-
-
0+
-
-
0
-+
-
-
0
-
-
0
-
-
-
0
0
-
-
-
-
-
-
0
0
-
-
--
-
-
-
-
-
-
-
-
0
0
-
-
-
-
0
-
0-
-
-
-
-
-
-
-
-
-
-
I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}×Z2
Causal triangulations/quadrangulations
1
2
3
4
5
6
7
8
9
10
11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
I If we take the origin at t = 0 and the root at final time, all edges ofthe tree are labeled −.
I Quadrangulations ↔ labeled trees. Causal quadr. ↔ trees.I As a direct consequence: With N squares we can build
#
{ }N
= C (N), #
{ }N
= 2C (N)3N , C (N) =1
N + 1
(2NN
)
Causal triangulations/quadrangulations
1
2
3
4
5
6
7
8
9
10
11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
I If we take the origin at t = 0 and the root at final time, all edges ofthe tree are labeled −.
I Quadrangulations ↔ labeled trees. Causal quadr. ↔ trees.I As a direct consequence: With N squares we can build
#
{ }N
= C (N), #
{ }N
= 2C (N)3N , C (N) =1
N + 1
(2NN
)
Causal triangulations/quadrangulations
1
2
3
4
5
6
7
8
9
10
11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
I If we take the origin at t = 0 and the root at final time, all edges ofthe tree are labeled −.
I Quadrangulations ↔ labeled trees. Causal quadr. ↔ trees.I As a direct consequence: With N squares we can build
#
{ }N
= C (N), #
{ }N
= 2C (N)3N , C (N) =1
N + 1
(2NN
)
Causal triangulations/quadrangulations
1
2
3
4
5
6
7
8
9
10
11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
I If we take the origin at t = 0 and the root at final time, all edges ofthe tree are labeled −.
I Quadrangulations ↔ labeled trees. Causal quadr. ↔ trees.
I As a direct consequence: With N squares we can build
#
{ }N
= C (N), #
{ }N
= 2C (N)3N , C (N) =1
N + 1
(2NN
)
Causal triangulations/quadrangulations
1
2
3
4
5
6
7
8
9
10
11
12
13
t
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
(Wednesday’s talks!)
I If we take the origin at t = 0 and the root at final time, all edges ofthe tree are labeled −.
I Quadrangulations ↔ labeled trees. Causal quadr. ↔ trees.I As a direct consequence: With N squares we can build
#
{ }N
= C (N), #
{ }N
= 2C (N)3N , C (N) =1
N + 1
(2NN
)
Including boundaries [Bettinelli ’11]
I A quadrangulations withboundary length 2l
and anorigin.
I Applying the same prescriptionwe obtain a forest rooted at theboundary.
I The labels on the boundaryarise from a (closed) randomwalk.
I A (possibly empty) tree growsat the end of every +-edge.
I There is a bijection [Bettinelli]
56
76
56
54
34
54
32
34
34
56
78
98
98
76
{Quadrangulations with originand boundary length 2l
}↔ {(+,−)-sequences} × {tree}l
Including boundaries [Bettinelli ’11]
I A quadrangulations withboundary length 2l and anorigin.
I Applying the same prescriptionwe obtain a forest rooted at theboundary.
I The labels on the boundaryarise from a (closed) randomwalk.
I A (possibly empty) tree growsat the end of every +-edge.
I There is a bijection [Bettinelli]
56
76
56
54
34
54
32
34
34
56
78
98
98
76
{Quadrangulations with originand boundary length 2l
}↔ {(+,−)-sequences} × {tree}l
Including boundaries [Bettinelli ’11]
I A quadrangulations withboundary length 2l and anorigin.
I Applying the same prescriptionwe obtain a forest rooted at theboundary.
I The labels on the boundaryarise from a (closed) randomwalk.
I A (possibly empty) tree growsat the end of every +-edge.
I There is a bijection [Bettinelli]
5
6
7
6
5
65 4
34
5
4
3
23
4
3
4
5
6
78
9
898
76
56
76
56
54
34
54
32
34
34
56
78
98
98
76
{Quadrangulations with originand boundary length 2l
}↔ {(+,−)-sequences} × {tree}l
Including boundaries [Bettinelli ’11]
I A quadrangulations withboundary length 2l and anorigin.
I Applying the same prescriptionwe obtain a forest rooted at theboundary.
I The labels on the boundaryarise from a (closed) randomwalk.
I A (possibly empty) tree growsat the end of every +-edge.
I There is a bijection [Bettinelli]
5
6
7
6
5
65 4
34
5
4
3
23
4
3
4
5
6
78
9
898
76
56
76
56
54
34
54
32
34
34
56
78
98
98
76
{Quadrangulations with originand boundary length 2l
}↔ {(+,−)-sequences} × {tree}l
Including boundaries [Bettinelli ’11]
I A quadrangulations withboundary length 2l and anorigin.
I Applying the same prescriptionwe obtain a forest rooted at theboundary.
I The labels on the boundaryarise from a (closed) randomwalk.
I A (possibly empty) tree growsat the end of every +-edge.
I There is a bijection [Bettinelli]
5
6
7
6
5
65 4
34
5
4
3
23
4
3
4
5
6
78
9
898
76
-
-+
+-+
+
+-
-+
+
+-
-+-
-
-
-
-
-+-+
+
+
+
{Quadrangulations with originand boundary length 2l
}↔ {(+,−)-sequences} × {tree}l
Including boundaries [Bettinelli ’11]
I A quadrangulations withboundary length 2l and anorigin.
I Applying the same prescriptionwe obtain a forest rooted at theboundary.
I The labels on the boundaryarise from a (closed) randomwalk.
I A (possibly empty) tree growsat the end of every +-edge.
I There is a bijection [Bettinelli]
5
6
7
6
5
65 4
34
5
4
3
23
4
3
4
5
6
78
9
898
76
-
-+
+-+
+
+-
-+
+
+-
-+-
-
-
-
-
-+-+
+
+
+
{Quadrangulations with originand boundary length 2l
}↔ {(+,−)-sequences} × {tree}l
Disk amplitudes
w(g , l) = z(g)l
w(g , x) =∞∑l=0
w(g , l)x l =1
1− z(g)x
w(g , l) =
(2ll
)z(g)l
w(g , x) =1√
1− 4z(g)x
Generating functionfor unlabeled trees:
z(g) =1−√
1− 4g
2g
0+
+000
- 0-+
+0 --0
- 00+
--
Generating functionfor labeled trees:
z(g) =1−√
1− 12g
6g
Disk amplitudes
w(g , l) = z(g)l
w(g , x) =∞∑l=0
w(g , l)x l =1
1− z(g)x
w(g , l) =
(2ll
)z(g)l
w(g , x) =1√
1− 4z(g)x
Generating functionfor unlabeled trees:
z(g) =1−√
1− 4g
2g
0+
+000
- 0-+
+0 --0
- 00+
--
Generating functionfor labeled trees:
z(g) =1−√
1− 12g
6g
Disk amplitudes
w(g , l) = z(g)l
w(g , x) =∞∑l=0
w(g , l)x l =1
1− z(g)x
w(g , l) =
(2ll
)z(g)l
w(g , x) =1√
1− 4z(g)x
Generating functionfor unlabeled trees:
z(g) =1−√
1− 4g
2g
0+
+000
- 0-+
+0 --0
- 00+
--
Generating functionfor labeled trees:
z(g) =1−√
1− 12g
6g
Disk amplitudes
w(g , l) = z(g)l
w(g , x) =∞∑l=0
w(g , l)x l =1
1− z(g)x
w(g , l) =
(2ll
)z(g)l
w(g , x) =1√
1− 4z(g)x
Generating functionfor unlabeled trees:
z(g) =1−√
1− 4g
2g
0+
+000
- 0-+
+0 --0
- 00+
--
Generating functionfor labeled trees:
z(g) =1−√
1− 12g
6g
Continuum limit
w(g , x) =1
1− zx
WΛ(X ) =1
X + Z
w(g , x) =1√
1− 4zx
W ′Λ(X ) =1√
X + Z
z(g) = 1−√
1−4g2g
Z =√
Λ
0+
+000
- 0-+
+0 --0
- 00+
--
z(g) = 1−√
1−12g6g
Z =√
Λ
I Expanding around critical point in terms of “lattice spacing” ε:
g = gc(1− Λε2), z(g) = zc(1− Zε), x = xc(1− X ε)
I CDT disk amplitude: WΛ(X ) = 1X+√
Λ
I DT disk amplitude with marked point: W ′Λ(X ) = 1√X+√
Λ. Integrate
w.r.t. Λ to remove mark: WΛ(X ) = 23 (X − 1
2
√Λ)√
X +√
Λ.
Continuum limit
w(g , x) =1
1− zx
WΛ(X ) =1
X + Z
w(g , x) =1√
1− 4zx
W ′Λ(X ) =1√
X + Z
z(g) = 1−√
1−4g2g
Z =√
Λ
0+
+000
- 0-+
+0 --0
- 00+
--
z(g) = 1−√
1−12g6g
Z =√
Λ
I Expanding around critical point in terms of “lattice spacing” ε:
g = gc(1− Λε2), z(g) = zc(1− Zε), x = xc(1− X ε)
I CDT disk amplitude: WΛ(X ) = 1X+√
Λ
I DT disk amplitude with marked point: W ′Λ(X ) = 1√X+√
Λ. Integrate
w.r.t. Λ to remove mark: WΛ(X ) = 23 (X − 1
2
√Λ)√
X +√
Λ.
Continuum limit
w(g , x) =1
1− zx
WΛ(X ) =1
X + Z
w(g , x) =1√
1− 4zx
W ′Λ(X ) =1√
X + Z
z(g) = 1−√
1−4g2g
Z =√
Λ
0+
+000
- 0-+
+0 --0
- 00+
--
z(g) = 1−√
1−12g6g
Z =√
Λ
I Expanding around critical point in terms of “lattice spacing” ε:
g = gc(1− Λε2), z(g) = zc(1− Zε), x = xc(1− X ε)
I CDT disk amplitude: WΛ(X ) = 1X+√
Λ
I DT disk amplitude with marked point: W ′Λ(X ) = 1√X+√
Λ. Integrate
w.r.t. Λ to remove mark: WΛ(X ) = 23 (X − 1
2
√Λ)√
X +√
Λ.
Continuum limit
w(g , x) =1
1− zx
WΛ(X ) =1
X + Z
w(g , x) =1√
1− 4zx
W ′Λ(X ) =1√
X + Z
z(g) = 1−√
1−4g2g
Z =√
Λ
0+
+000
- 0-+
+0 --0
- 00+
--
z(g) = 1−√
1−12g6g
Z =√
Λ
I Expanding around critical point in terms of “lattice spacing” ε:
g = gc(1− Λε2), z(g) = zc(1− Zε), x = xc(1− X ε)
I CDT disk amplitude: WΛ(X ) = 1X+√
Λ
I DT disk amplitude with marked point: W ′Λ(X ) = 1√X+√
Λ. Integrate
w.r.t. Λ to remove mark: WΛ(X ) = 23 (X − 1
2
√Λ)√
X +√
Λ.
Continuum limit
w(g , x) =1
1− zx
WΛ(X ) =1
X + Z
w(g , x) =1√
1− 4zx
W ′Λ(X ) =1√
X + Z
z(g) = 1−√
1−4g2g
Z =√
Λ
0+
+000
- 0-+
+0 --0
- 00+
--
z(g) = 1−√
1−12g6g
Z =√
Λ
I Expanding around critical point in terms of “lattice spacing” ε:
g = gc(1− Λε2), z(g) = zc(1− Zε), x = xc(1− X ε)
I CDT disk amplitude: WΛ(X ) = 1X+√
Λ
I DT disk amplitude with marked point: W ′Λ(X ) = 1√X+√
Λ. Integrate
w.r.t. Λ to remove mark: WΛ(X ) = 23 (X − 1
2
√Λ)√X +
√Λ.
Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
I Assign coupling a to the local maxima of thedistance function.
I In terms of labeled trees:
I Local maximum if +/0coming in and any numberof 0/−’s going out.
5 4
44
4
4 4
5
3
3
5
5
I We can write down equations for the generating functions
-
= g∞∑k=0
(-
+0
++
)k= g
(1−
-
−0−
+
)−1
+
=0
= g
[ ∞∑k=0
(-
+0
++
)k+ (a− 1)
∞∑k=0
(-
+0
)k]
=-
+ g(a− 1)(
1−-
−0
)−1
Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
I Assign coupling a to the local maxima of thedistance function.
I In terms of labeled trees:
I Local maximum if +/0coming in and any numberof 0/−’s going out.
5 4
44
4
4 4
5
3
3
5
5
I We can write down equations for the generating functions
-
= g∞∑k=0
(-
+0
++
)k= g
(1−
-
−0−
+
)−1
+
=0
= g
[ ∞∑k=0
(-
+0
++
)k+ (a− 1)
∞∑k=0
(-
+0
)k]
=-
+ g(a− 1)(
1−-
−0
)−1
Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
I Assign coupling a to the local maxima of thedistance function.
I In terms of labeled trees:
I Local maximum if +/0coming in and any numberof 0/−’s going out.
5 4
44
4
4 4
5
3
3
5
5
I We can write down equations for the generating functions
-
= g∞∑k=0
(-
+0
++
)k= g
(1−
-
−0−
+
)−1
+
=0
= g
[ ∞∑k=0
(-
+0
++
)k+ (a− 1)
∞∑k=0
(-
+0
)k]
=-
+ g(a− 1)(
1−-
−0
)−1
Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
I Assign coupling a to the local maxima of thedistance function.
I In terms of labeled trees:
I Local maximum if +/0coming in and any numberof 0/−’s going out.
5 4
4
5
5
5
I We can write down equations for the generating functions
-
= g∞∑k=0
(-
+0
++
)k= g
(1−
-
−0−
+
)−1
+
=0
= g
[ ∞∑k=0
(-
+0
++
)k+ (a− 1)
∞∑k=0
(-
+0
)k]
=-
+ g(a− 1)(
1−-
−0
)−1
Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
I Assign coupling a to the local maxima of thedistance function.
I In terms of labeled trees:
I Local maximum if +/0coming in and any numberof 0/−’s going out.
+�0
0�-
0�-
0�-
0�-
I We can write down equations for the generating functions
-
= g∞∑k=0
(-
+0
++
)k= g
(1−
-
−0−
+
)−1
+
=0
= g
[ ∞∑k=0
(-
+0
++
)k+ (a− 1)
∞∑k=0
(-
+0
)k]
=-
+ g(a− 1)(
1−-
−0
)−1
Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
I Assign coupling a to the local maxima of thedistance function.
I In terms of labeled trees:
I Local maximum if +/0coming in and any numberof 0/−’s going out.
+�0
0�-
0�-
0�-
0�-
I We can write down equations for the generating functions
-
= g∞∑k=0
(-
+0
++
)k= g
(1−
-
−0−
+
)−1
+
=0
= g
[ ∞∑k=0
(-
+0
++
)k+ (a− 1)
∞∑k=0
(-
+0
)k]
=-
+ g(a− 1)(
1−-
−0
)−1
Generalized CDT [Ambjørn, Loll, Westra, Zohren ’07]
I Assign coupling a to the local maxima of thedistance function.
I In terms of labeled trees:
I Local maximum if +/0coming in and any numberof 0/−’s going out.
+�0
0�-
0�-
0�-
0�-
I We can write down equations for the generating functions
-
= g∞∑k=0
(-
+0
++
)k= g
(1−
-
−0−
+
)−1
+
=0
= g
[ ∞∑k=0
(-
+0
++
)k+ (a− 1)
∞∑k=0
(-
+0
)k]
=-
+ g(a− 1)(
1−-
−0
)−1
-
= g(
1−-
−0−
+
)−1
+
=0
=-
+ g(a− 1)(
1−-
−0
)−1
I Combine into one equation for z−(g , a) =-
:
3z4− − 4z3
− + (1 + 2g(1− 2a))z2− − g2 = 0
I Phase diagram for weighted labeled trees (constant a):
a = 0
a = 1
CDT
DT
01
12
1
8
1
6
1
4
1
24
5
24
g
1
6
1
2
1
3
z-Hg, aL
-
= g(
1−-
−0−
+
)−1
+
=0
=-
+ g(a− 1)(
1−-
−0
)−1
I Combine into one equation for z−(g , a) =-
:
3z4− − 4z3
− + (1 + 2g(1− 2a))z2− − g2 = 0
I Phase diagram for weighted labeled trees (constant a):
a = 0
a = 1
CDT
DT
01
12
1
8
1
6
1
4
1
24
5
24
g
1
6
1
2
1
3
z-Hg, aL
-
= g(
1−-
−0−
+
)−1
+
=0
=-
+ g(a− 1)(
1−-
−0
)−1
I Combine into one equation for z−(g , a) =-
:
3z4− − 4z3
− + (1 + 2g(1− 2a))z2− − g2 = 0
I Phase diagram for weighted labeled trees (constant a):
2 z3- 3 z4
- g2= 0
a = 0
a = 1
01
12
1
8
1
6
1
4
1
24
5
24
g
1
6
1
2
1
3
z-Hg, aL
-
= g(
1−-
−0−
+
)−1
+
=0
=-
+ g(a− 1)(
1−-
−0
)−1
I Combine into one equation for z−(g , a) =-
:
3z4− − 4z3
− + (1 + 2g(1− 2a))z2− − g2 = 0
I Phase diagram for weighted labeled trees (constant a):
2 z3- 3 z4
- g2= 0
a = 0
a = 1
01
12
1
8
1
6
1
4
1
24
5
24
g
1
6
1
2
1
3
z-Hg, aL
N = 2000, a = 0, Nmax = 0
N = 5000, a = 0.00007, Nmax = 11
N = 7000, a = 0.0002, Nmax = 37
N = 7000, a = 0.004, Nmax = 221
N = 4000, a = 0.02, Nmax = 362
N = 2500, a = 1, Nmax = 1216
I The number of local maxima Nmax(a) scales with N at the criticalpoint,
〈Nmax(a)〉NN
= 2(a
2
)2/3
+O(a),〈Nmax(a = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale a ∝ N−3/2, i.e. a = gsε
3 as observed in [ALWZ ’07].
I This is the only scaling leading to a continuum limit qualitativelydifferent from DT and CDT.
I Continuum limit g = gc(a)(1− Λε2),z− = z−,c(1− Zε), a = gsε
3:
Z 3 −(
Λ + 3(gs
2
)2/3)Z − gs = 0
I “Cup function” WΛ(X ) = 1X+Z . Agrees with
[ALWZ ’07].
I The number of local maxima Nmax(a) scales with N at the criticalpoint,
〈Nmax(a)〉NN
= 2(a
2
)2/3
+O(a),〈Nmax(a = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale a ∝ N−3/2, i.e. a = gsε
3 as observed in [ALWZ ’07].
I This is the only scaling leading to a continuum limit qualitativelydifferent from DT and CDT.
I Continuum limit g = gc(a)(1− Λε2),z− = z−,c(1− Zε), a = gsε
3:
Z 3 −(
Λ + 3(gs
2
)2/3)Z − gs = 0
I “Cup function” WΛ(X ) = 1X+Z . Agrees with
[ALWZ ’07].
I The number of local maxima Nmax(a) scales with N at the criticalpoint,
〈Nmax(a)〉NN
= 2(a
2
)2/3
+O(a),〈Nmax(a = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale a ∝ N−3/2, i.e. a = gsε
3 as observed in [ALWZ ’07].
I This is the only scaling leading to a continuum limit qualitativelydifferent from DT and CDT.
I Continuum limit g = gc(a)(1− Λε2),z− = z−,c(1− Zε), a = gsε
3:
Z 3 −(
Λ + 3(gs
2
)2/3)Z − gs = 0
I “Cup function” WΛ(X ) = 1X+Z . Agrees with
[ALWZ ’07].
I The number of local maxima Nmax(a) scales with N at the criticalpoint,
〈Nmax(a)〉NN
= 2(a
2
)2/3
+O(a),〈Nmax(a = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale a ∝ N−3/2, i.e. a = gsε
3 as observed in [ALWZ ’07].
I This is the only scaling leading to a continuum limit qualitativelydifferent from DT and CDT.
I Continuum limit g = gc(a)(1− Λε2),z− = z−,c(1− Zε), a = gsε
3:
Z 3 −(
Λ + 3(gs
2
)2/3)Z − gs = 0
I “Cup function” WΛ(X ) = 1X+Z . Agrees with
[ALWZ ’07].
I The number of local maxima Nmax(a) scales with N at the criticalpoint,
〈Nmax(a)〉NN
= 2(a
2
)2/3
+O(a),〈Nmax(a = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale a ∝ N−3/2, i.e. a = gsε
3 as observed in [ALWZ ’07].
I This is the only scaling leading to a continuum limit qualitativelydifferent from DT and CDT.
I Continuum limit g = gc(a)(1− Λε2),z− = z−,c(1− Zε), a = gsε
3:
Z 3 −(
Λ + 3(gs
2
)2/3)Z − gs = 0
I “Cup function” WΛ(X ) = 1X+Z . Agrees with
[ALWZ ’07].
Two loop identity in generalized CDT
I Consider surfaces with two boundaries separatedby a geodesic distance D.
I One can assign time T1, T2 to the boundaries(|T1 − T2| ≤ D) and study a “merging” process.
I For a given surface the foliation depends onT1 − T2, hence also Nmax and its weight.
I However, in [ALWZ ’07] it was shown that theamplitude is independent of T1 − T2.
I Refoliation symmetry at the quantum level in thepresence of topology change!
I Can we better understand this symmetry at thediscrete level?
I For simplicity set the boundarylengths to zero.Straightforward generalization to finite boundaries.
Two loop identity in generalized CDT
I Consider surfaces with two boundaries separatedby a geodesic distance D.
I One can assign time T1, T2 to the boundaries(|T1 − T2| ≤ D) and study a “merging” process.
I For a given surface the foliation depends onT1 − T2, hence also Nmax and its weight.
I However, in [ALWZ ’07] it was shown that theamplitude is independent of T1 − T2.
I Refoliation symmetry at the quantum level in thepresence of topology change!
I Can we better understand this symmetry at thediscrete level?
I For simplicity set the boundarylengths to zero.Straightforward generalization to finite boundaries.
Two loop identity in generalized CDT
I Consider surfaces with two boundaries separatedby a geodesic distance D.
I One can assign time T1, T2 to the boundaries(|T1 − T2| ≤ D) and study a “merging” process.
I For a given surface the foliation depends onT1 − T2, hence also Nmax and its weight.
I However, in [ALWZ ’07] it was shown that theamplitude is independent of T1 − T2.
I Refoliation symmetry at the quantum level in thepresence of topology change!
I Can we better understand this symmetry at thediscrete level?
I For simplicity set the boundarylengths to zero.Straightforward generalization to finite boundaries.
Two loop identity in generalized CDT
I Consider surfaces with two boundaries separatedby a geodesic distance D.
I One can assign time T1, T2 to the boundaries(|T1 − T2| ≤ D) and study a “merging” process.
I For a given surface the foliation depends onT1 − T2, hence also Nmax and its weight.
I However, in [ALWZ ’07] it was shown that theamplitude is independent of T1 − T2.
I Refoliation symmetry at the quantum level in thepresence of topology change!
I Can we better understand this symmetry at thediscrete level?
I For simplicity set the boundarylengths to zero.Straightforward generalization to finite boundaries.
Two loop identity in generalized CDT
I Consider surfaces with two boundaries separatedby a geodesic distance D.
I One can assign time T1, T2 to the boundaries(|T1 − T2| ≤ D) and study a “merging” process.
I For a given surface the foliation depends onT1 − T2, hence also Nmax and its weight.
I However, in [ALWZ ’07] it was shown that theamplitude is independent of T1 − T2.
I Refoliation symmetry at the quantum level in thepresence of topology change!
I Can we better understand this symmetry at thediscrete level?
I For simplicity set the boundarylengths to zero.Straightforward generalization to finite boundaries.
Two loop identity in generalized CDT
I Consider surfaces with two boundaries separatedby a geodesic distance D.
I One can assign time T1, T2 to the boundaries(|T1 − T2| ≤ D) and study a “merging” process.
I For a given surface the foliation depends onT1 − T2, hence also Nmax and its weight.
I However, in [ALWZ ’07] it was shown that theamplitude is independent of T1 − T2.
I Refoliation symmetry at the quantum level in thepresence of topology change!
I Can we better understand this symmetry at thediscrete level?
I For simplicity set the boundarylengths to zero.Straightforward generalization to finite boundaries.
Two loop identity in generalized CDT
I Consider surfaces with two boundaries separatedby a geodesic distance D.
I One can assign time T1, T2 to the boundaries(|T1 − T2| ≤ D) and study a “merging” process.
I For a given surface the foliation depends onT1 − T2, hence also Nmax and its weight.
I However, in [ALWZ ’07] it was shown that theamplitude is independent of T1 − T2.
I Refoliation symmetry at the quantum level in thepresence of topology change!
I Can we better understand this symmetry at thediscrete level?
I For simplicity set the boundarylengths to zero.Straightforward generalization to finite boundaries.
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
8
7
7
7
7
9
9
7
6
8
8
68
6
6
6
8
6
8
8
5
5
7
55
7
5
7
7
5
7
5
5
75
7
5
7
6
6
6
4
6
4
44
46
4
4
4
6
4 4
6
4
6
6
6
3
5
3
55
5
5
5
3
3
5
3
5
5
5
3
3
5
3
5
3
3
4
2
6
2
2
4
4
44
22
4
2
4
2
44
44 4
4
2
1
1
33
5
5
3
3
5
3
1
3
3
3
3
5
53
3
3
4
0
2
4
24
2
2
2
4
4
4
2
2
4
2
4
2
3
3
3
1
3
3
1
1
3
1
3
3
3
3
0
2
2
2
22
2
2
1
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
8
7
7
7
7
7
6
8
68
6
6
6
6
8
5
5
7
55
7
5
7
7
5
7
5
5
75
7
5
7
6
6
6
4
6
4
44
46
4
4
4
6
4 4
6
4
6
6
6
3
5
3
55
5
5
5
3
3
5
3
5
5
5
3
3
5
3
5
3
3
4
2
2
2
44
22
4
2
4
2
44
44 4
4
2
1
1
33
5
5
3
3
3
1
3
3
3
3
5
53
3
3
4
0
2
24
2
2
2
4
4
4
2
2
2
2
3
3
3
1
3
1
1
3
1
3
3
0
2
2
2
2
2
2
1
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
0
0
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 0D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
0
2
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 2D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
8
7
7
7
7
7
6
9
69
6
6
6
6
8
5
5
7
55
8
5
8
9
5
7
5
5
75
8
5
9
7
8
8
4
6
4
44
48
4
4
4
8
4 4
7
4
8
8
8
3
7
3
77
7
7
7
3
3
7
3
5
6
7
3
3
7
3
7
3
3
6
2
2
2
66
22
4
2
6
2
66
66 6
6
2
1
1
55
7
5
5
3
4
1
5
5
5
4
6
75
5
3
4
0
2
46
4
2
3
5
6
6
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3
t t
t + 1
t - 1
t t
t + 1
t + 1
T1 = 0T2 = 2D = 4
I No local minima ⇒ the labeling is canonical!
I There exists a bijection preserving the number of local maxima:{ }T1,T2
←→
{0
0
}T1,T2
←→
{0
2
}T ′
1 ,T′2
←→
{ }T ′
1 ,T′2
Conclusions & Outlook
I ConlusionsI The Cori–Vauquelin–Schaeffer bijection encodes 2d geometry in
trees, which are simple objects from an analytical point of view.I The bijection is ideal for studying “proper-time foliations” of random
surfaces.I In the setting of generalized CDT, the bijection exposes symmetries
in certain amplitudes with prescibed time on the boundaries.
I What I’ve not shown (see our forthcoming paper)I Similar bijections exist for triangulations, but more involved.I Explicit expressions can be derived for transition amplitudes
G(L1, L2;T ) in generalized CDT.
I OutlookI What is the exact structure of these symmetries? Related to
conformal symmetry (Virasoro algebra)?I Straightforward extension to higher genus. Sum over topologies in
non-critical string theory?
Conclusions & Outlook
I ConlusionsI The Cori–Vauquelin–Schaeffer bijection encodes 2d geometry in
trees, which are simple objects from an analytical point of view.I The bijection is ideal for studying “proper-time foliations” of random
surfaces.I In the setting of generalized CDT, the bijection exposes symmetries
in certain amplitudes with prescibed time on the boundaries.
I What I’ve not shown (see our forthcoming paper)I Similar bijections exist for triangulations, but more involved.I Explicit expressions can be derived for transition amplitudes
G(L1, L2;T ) in generalized CDT.
I OutlookI What is the exact structure of these symmetries? Related to
conformal symmetry (Virasoro algebra)?I Straightforward extension to higher genus. Sum over topologies in
non-critical string theory?
Conclusions & Outlook
I ConlusionsI The Cori–Vauquelin–Schaeffer bijection encodes 2d geometry in
trees, which are simple objects from an analytical point of view.I The bijection is ideal for studying “proper-time foliations” of random
surfaces.I In the setting of generalized CDT, the bijection exposes symmetries
in certain amplitudes with prescibed time on the boundaries.
I What I’ve not shown (see our forthcoming paper)I Similar bijections exist for triangulations, but more involved.I Explicit expressions can be derived for transition amplitudes
G(L1, L2;T ) in generalized CDT.
I OutlookI What is the exact structure of these symmetries? Related to
conformal symmetry (Virasoro algebra)?I Straightforward extension to higher genus. Sum over topologies in
non-critical string theory?
Merry Christmas!
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