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Quantum Gravity in Paris,Mar. 20, 2013
Generalized CDT as ascaling limit of planar maps
Timothy Budd
(collaboration with J. Ambjørn)
Niels Bohr Institute, [email protected]
http://www.nbi.dk/~budd/
Outline
I Introduction to (generalized) CDT in 2d
I Enumeration using labeled trees
I Continuum limit and two-point function
I Scaling limit of planar maps
I Loop identities
Causal Dynamical Triangulations in 2d
I CDT in 2d is a statistical system with partition function
ZCDT =∑T
1
CTgN(T )
I ZCDT (g) is a generating function for the number of causaltriangulations T of S2 with N triangles.
I The triangulations have a foliatedstructure
I May as well view them as causalquadrangulations with a uniquelocal maximum of the distancefunction from the origin.
I What if we allow more than onelocal maximum?
t
Causal Dynamical Triangulations in 2d
I CDT in 2d is a statistical system with partition function
ZCDT =∑T
1
CTgN(T )
I ZCDT (g) is a generating function for the number of causaltriangulations T of S2 with N triangles.
I The triangulations have a foliatedstructure
I May as well view them as causalquadrangulations with a uniquelocal maximum of the distancefunction from the origin.
I What if we allow more than onelocal maximum?
t
Causal Dynamical Triangulations in 2d
I CDT in 2d is a statistical system with partition function
ZCDT =∑T
1
CTgN(T )
I ZCDT (g) is a generating function for the number of causaltriangulations T of S2 with N triangles.
I The triangulations have a foliatedstructure
I May as well view them as causalquadrangulations with a uniquelocal maximum of the distancefunction from the origin.
I What if we allow more than onelocal maximum?
t
Causal Dynamical Triangulations in 2d
I CDT in 2d is a statistical system with partition function
ZCDT =∑T
1
CTgN(T )
I ZCDT (g) is a generating function for the number of causaltriangulations T of S2 with N triangles.
I The triangulations have a foliatedstructure
I May as well view them as causalquadrangulations with a uniquelocal maximum of the distancefunction from the origin.
I What if we allow more than onelocal maximum?
t
Generalized CDTI Allow spatial topology to change in time.
Assign a coupling gs to each baby universe.
I The model was solved in the continuum bygluing together chunks of CDT. [Ambjørn,
Loll, Westra, Zohren ’07]
I Can we understand the geometry in moredetail by obtaining generalized CDT as ascaling limit of a discrete model?
I Generalized CDT partition function
Z (g , g) =∑Q
1
CQgNgNmax ,
sum over quadrangulations Q withN faces, a marked origin, and Nmax
local maxima of the distance to theorigin.
−→
Generalized CDTI Allow spatial topology to change in time.
Assign a coupling gs to each baby universe.
I The model was solved in the continuum bygluing together chunks of CDT. [Ambjørn,
Loll, Westra, Zohren ’07]
I Can we understand the geometry in moredetail by obtaining generalized CDT as ascaling limit of a discrete model?
I Generalized CDT partition function
Z (g , g) =∑Q
1
CQgNgNmax ,
sum over quadrangulations Q withN faces, a marked origin, and Nmax
local maxima of the distance to theorigin.
−→
Generalized CDTI Allow spatial topology to change in time.
Assign a coupling gs to each baby universe.
I The model was solved in the continuum bygluing together chunks of CDT. [Ambjørn,
Loll, Westra, Zohren ’07]
I Can we understand the geometry in moredetail by obtaining generalized CDT as ascaling limit of a discrete model?
I Generalized CDT partition function
Z (g , g) =∑Q
1
CQgNgNmax ,
sum over quadrangulations Q withN faces, a marked origin, and Nmax
local maxima of the distance to theorigin.
−→
Generalized CDTI Allow spatial topology to change in time.
Assign a coupling gs to each baby universe.
I The model was solved in the continuum bygluing together chunks of CDT. [Ambjørn,
Loll, Westra, Zohren ’07]
I Can we understand the geometry in moredetail by obtaining generalized CDT as ascaling limit of a discrete model?
I Generalized CDT partition function
Z (g , g) =∑Q
1
CQgNgNmax ,
sum over quadrangulations Q withN faces, a marked origin, and Nmax
local maxima of the distance to theorigin.
−→
Generalized CDTI Allow spatial topology to change in time.
Assign a coupling gs to each baby universe.
I The model was solved in the continuum bygluing together chunks of CDT. [Ambjørn,
Loll, Westra, Zohren ’07]
I Can we understand the geometry in moredetail by obtaining generalized CDT as ascaling limit of a discrete model?
I Generalized CDT partition function
Z (g , g) =∑Q
1
CQgNgNmax ,
sum over quadrangulations Q withN faces, a marked origin, and Nmax
local maxima of the distance to theorigin.
−→
N = 2000, g = 0, Nmax = 1
N = 5000, g = 0.00007, Nmax = 12
N = 7000, g = 0.0002, Nmax = 38
N = 7000, g = 0.004, Nmax = 221
N = 4000, g = 0.02, Nmax = 362
N = 2500, g = 1, Nmax = 1216
Causal triangulations and trees
I Union of all left-most geodesicsrunning away from the origin.
I Simple enumeration of planar trees:
#
{ }N
= C (N), C (N) =1
N + 1
(2NN
)[Malyshev, Yambartsev, Zamyatin ’01]
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
I Union of all left-most geodesicsrunning towards the origin.
I Both generalize to generalized CDTleading to different representations.
Causal triangulations and treesI Union of all left-most geodesics
running away from the origin.
I Simple enumeration of planar trees:
#
{ }N
= C (N), C (N) =1
N + 1
(2NN
)[Malyshev, Yambartsev, Zamyatin ’01]
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
I Union of all left-most geodesicsrunning towards the origin.
I Both generalize to generalized CDTleading to different representations.
Causal triangulations and treesI Union of all left-most geodesics
running away from the origin.
I Simple enumeration of planar trees:
#
{ }N
= C (N), C (N) =1
N + 1
(2NN
)[Malyshev, Yambartsev, Zamyatin ’01]
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
I Union of all left-most geodesicsrunning towards the origin.
I Both generalize to generalized CDTleading to different representations.
Causal triangulations and treesI Union of all left-most geodesics
running away from the origin.
I Simple enumeration of planar trees:
#
{ }N
= C (N), C (N) =1
N + 1
(2NN
)[Malyshev, Yambartsev, Zamyatin ’01]
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
I Union of all left-most geodesicsrunning towards the origin.
I Both generalize to generalized CDTleading to different representations.
Causal triangulations and treesI Union of all left-most geodesics
running away from the origin.
I Simple enumeration of planar trees:
#
{ }N
= C (N), C (N) =1
N + 1
(2NN
)[Malyshev, Yambartsev, Zamyatin ’01]
[Krikun, Yambartsev ’08]
[Durhuus, Jonsson, Wheater ’09]
I Union of all left-most geodesicsrunning towards the origin.
I Both generalize to generalized CDTleading to different representations.
Causal triangulations and trees
I Labeled planar trees: Schaeffer’s bijection.
I Unlabeled planar maps (one face per localmaximum).
Causal triangulations and trees
I Labeled planar trees: Schaeffer’s bijection.
I Unlabeled planar maps (one face per localmaximum).
Causal triangulations and trees
I Labeled planar trees: Schaeffer’s bijection.
I Unlabeled planar maps (one face per localmaximum).
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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[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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[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 [e.g. Chassaing, Schaeffer ’04]
I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}
Rooting the tree [e.g. Chassaing, Schaeffer ’04]
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I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}
Rooting the tree [e.g. Chassaing, Schaeffer ’04]
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I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}
Rooting the tree [e.g. Chassaing, Schaeffer ’04]
+-
-
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--
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0-
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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,−
}
Rooting the tree [e.g. Chassaing, Schaeffer ’04]
+-
-
-
0
-
-
-
0+
-
-
0
-+
-
-
0
-
-
0
-
-
-
0
0
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0
0
-
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--
-
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0
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0
-
0-
-
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-
-
-
-
-
-
-
-
I We will be using the bijection:{Quadrangulations with originand marked edge
}↔{
Rooted planar treeslabelled by +, 0,−
}
Assigning couplings to local maxima
I Assign coupling g to the local maxima of thedistance function.
I In terms of labeled trees:
5 4
44
4
4 4
5
3
3
5
5
I Generating function z0(g , g) for number of rooted labeled trees withN edges and Nmax local maxima.
I Similarly z1(g , g) but local maximum at the root not counted.
I Satisfy recursion relations:
z1 =∞∑k=0
(z1 + z0 + z0)k gk = (1− gz1 − 2gz0)−1
z0 =∞∑k=0
(z1 + z0 + z0)k gk + (g− 1)∞∑k=0
(z1 + z0)k gk
= z1 + (g− 1) (1− gz1 − gz0)−1
Assigning couplings to local maxima
I Assign coupling g to the local maxima of thedistance function.
I In terms of labeled trees:
5 4
44
4
4 4
5
3
3
5
5
I Generating function z0(g , g) for number of rooted labeled trees withN edges and Nmax local maxima.
I Similarly z1(g , g) but local maximum at the root not counted.
I Satisfy recursion relations:
z1 =∞∑k=0
(z1 + z0 + z0)k gk = (1− gz1 − 2gz0)−1
z0 =∞∑k=0
(z1 + z0 + z0)k gk + (g− 1)∞∑k=0
(z1 + z0)k gk
= z1 + (g− 1) (1− gz1 − gz0)−1
Assigning couplings to local maxima
I Assign coupling g to the local maxima of thedistance function.
I In terms of labeled trees:
5 4
44
4
4 4
5
3
3
5
5
I Generating function z0(g , g) for number of rooted labeled trees withN edges and Nmax local maxima.
I Similarly z1(g , g) but local maximum at the root not counted.
I Satisfy recursion relations:
z1 =∞∑k=0
(z1 + z0 + z0)k gk = (1− gz1 − 2gz0)−1
z0 =∞∑k=0
(z1 + z0 + z0)k gk + (g− 1)∞∑k=0
(z1 + z0)k gk
= z1 + (g− 1) (1− gz1 − gz0)−1
Assigning couplings to local maxima
I Assign coupling g to the local maxima of thedistance function.
I In terms of labeled trees:
5 4
4
5
5
5
I Generating function z0(g , g) for number of rooted labeled trees withN edges and Nmax local maxima.
I Similarly z1(g , g) but local maximum at the root not counted.
I Satisfy recursion relations:
z1 =∞∑k=0
(z1 + z0 + z0)k gk = (1− gz1 − 2gz0)−1
z0 =∞∑k=0
(z1 + z0 + z0)k gk + (g− 1)∞∑k=0
(z1 + z0)k gk
= z1 + (g− 1) (1− gz1 − gz0)−1
Assigning couplings to local maxima
I Assign coupling g to the local maxima of thedistance function.
I In terms of labeled trees:
+�0
0�-
0�-
0�-
0�-
I Generating function z0(g , g) for number of rooted labeled trees withN edges and Nmax local maxima.
I Similarly z1(g , g) but local maximum at the root not counted.
I Satisfy recursion relations:
z1 =∞∑k=0
(z1 + z0 + z0)k gk = (1− gz1 − 2gz0)−1
z0 =∞∑k=0
(z1 + z0 + z0)k gk + (g− 1)∞∑k=0
(z1 + z0)k gk
= z1 + (g− 1) (1− gz1 − gz0)−1
Assigning couplings to local maxima
I Assign coupling g to the local maxima of thedistance function.
I In terms of labeled trees:
+�0
0�-
0�-
0�-
0�-
I Generating function z0(g , g) for number of rooted labeled trees withN edges and Nmax local maxima.
I Similarly z1(g , g) but local maximum at the root not counted.
I Satisfy recursion relations:
z1 =∞∑k=0
(z1 + z0 + z0)k gk = (1− gz1 − 2gz0)−1
z0 =∞∑k=0
(z1 + z0 + z0)k gk + (g− 1)∞∑k=0
(z1 + z0)k gk
= z1 + (g− 1) (1− gz1 − gz0)−1
Assigning couplings to local maxima
I Assign coupling g to the local maxima of thedistance function.
I In terms of labeled trees:
+�0
0�-
0�-
0�-
0�-
I Generating function z0(g , g) for number of rooted labeled trees withN edges and Nmax local maxima.
I Similarly z1(g , g) but local maximum at the root not counted.
I Satisfy recursion relations:
z1 =∞∑k=0
(z1 + z0 + z0)k gk = (1− gz1 − 2gz0)−1
z0 =∞∑k=0
(z1 + z0 + z0)k gk + (g− 1)∞∑k=0
(z1 + z0)k gk
= z1 + (g− 1) (1− gz1 − gz0)−1
z1 = (1− gz1 − 2gz0)−1
z0 = z1 + (g− 1) (1− gz1 − gz0)−1
I Combine into one equation for z1(g , g):
3g2z41 − 4gz3
1 + (1 + 2g(1− 2g))z21 − 1 = 0
I Phase diagram for weighted labeled trees (constant g):
g = 0g = 1
z1 = (1− gz1 − 2gz0)−1
z0 = z1 + (g− 1) (1− gz1 − gz0)−1
I Combine into one equation for z1(g , g):
3g2z41 − 4gz3
1 + (1 + 2g(1− 2g))z21 − 1 = 0
I Phase diagram for weighted labeled trees (constant g):
g = 0g = 1
z1 = (1− gz1 − 2gz0)−1
z0 = z1 + (g− 1) (1− gz1 − gz0)−1
I Combine into one equation for z1(g , g):
3g2z41 − 4gz3
1 + (1 + 2g(1− 2g))z21 − 1 = 0
I Phase diagram for weighted labeled trees (constant g):
g = 0g = 1
z1 = (1− gz1 − 2gz0)−1
z0 = z1 + (g− 1) (1− gz1 − gz0)−1
I Combine into one equation for z1(g , g):
3g2z41 − 4gz3
1 + (1 + 2g(1− 2g))z21 − 1 = 0
I Phase diagram for weighted labeled trees (constant g):
g = 0g = 1
I The number of local maxima Nmax(g) scales with N at the criticalpoint,
〈Nmax(g)〉NN
= 2(g
2
)2/3
+O(g),〈Nmax(g = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale g ∝ N−3/2, i.e. g = 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(g)(1− Λε2),z1 = z1,c(1− Z1ε), g = gsε
3:
Z 31 −
(Λ + 3
(gs2
)2/3)Z1 − gs = 0
I Can compute:
I The number of local maxima Nmax(g) scales with N at the criticalpoint,
〈Nmax(g)〉NN
= 2(g
2
)2/3
+O(g),〈Nmax(g = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale g ∝ N−3/2, i.e. g = 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(g)(1− Λε2),z1 = z1,c(1− Z1ε), g = gsε
3:
Z 31 −
(Λ + 3
(gs2
)2/3)Z1 − gs = 0
I Can compute:
I The number of local maxima Nmax(g) scales with N at the criticalpoint,
〈Nmax(g)〉NN
= 2(g
2
)2/3
+O(g),〈Nmax(g = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale g ∝ N−3/2, i.e. g = 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(g)(1− Λε2),z1 = z1,c(1− Z1ε), g = gsε
3:
Z 31 −
(Λ + 3
(gs2
)2/3)Z1 − gs = 0
I Can compute:
I The number of local maxima Nmax(g) scales with N at the criticalpoint,
〈Nmax(g)〉NN
= 2(g
2
)2/3
+O(g),〈Nmax(g = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale g ∝ N−3/2, i.e. g = 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(g)(1− Λε2),z1 = z1,c(1− Z1ε), g = gsε
3:
Z 31 −
(Λ + 3
(gs2
)2/3)Z1 − gs = 0
I Can compute:
I The number of local maxima Nmax(g) scales with N at the criticalpoint,
〈Nmax(g)〉NN
= 2(g
2
)2/3
+O(g),〈Nmax(g = 1)〉N
N= 1/2
I Therefore, to obtain a finite continuum density of critical points oneshould scale g ∝ N−3/2, i.e. g = 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(g)(1− Λε2),z1 = z1,c(1− Z1ε), g = gsε
3:
Z 31 −
(Λ + 3
(gs2
)2/3)Z1 − gs = 0
I Can compute:
Two-point function
I Amplitude for having root atdistance T from origin.
I Given by T -derivative of Z0(T )and Z1(T ), which are thescaling limits of the generatingfunctions z0(t) and z1(t) oflabeled trees with label t on theroot.
I They satisfy
z1(t) =1
1− gz1(t − 1)− gz0(t)− gz0(t + 1)
z0(t) = z1(t) +g− 1
1− gz1(t − 1)− gz0(t)
Two-point function
I Amplitude for having root atdistance T from origin.
I Given by T -derivative of Z0(T )and Z1(T ), which are thescaling limits of the generatingfunctions z0(t) and z1(t) oflabeled trees with label t on theroot.
I They satisfy
z1(t) =1
1− gz1(t − 1)− gz0(t)− gz0(t + 1)
z0(t) = z1(t) +g− 1
1− gz1(t − 1)− gz0(t)
Two-point function
I Amplitude for having root atdistance T from origin.
I Given by T -derivative of Z0(T )and Z1(T ), which are thescaling limits of the generatingfunctions z0(t) and z1(t) oflabeled trees with label t on theroot.
I They satisfy
z1(t) =1
1− gz1(t − 1)− gz0(t)− gz0(t + 1)
z0(t) = z1(t) +g− 1
1− gz1(t − 1)− gz0(t)
I Solution is (using methods of [Bouttier, Di Francesco, Guitter, ’03]):
z1(t) = z11− σt
1− σt+1
1− (1− β)σ − βσt+3
1− (1− β)σ − βσt+2,
z0(t) = z01− σt
1− (1− β)σ − βσt+1
(1− (1− β)σ)2 − β2σt+3
1− (1− β)σ − βσt+2,
with β = β(g , g) and σ = σ(g , g) fixed by
g(1 + σ)(1 + βσ)z1 − σ(1− 2g z0) = 0,
(1− β)σ − g(1 + σ)z1 + g(1− σ + 2βσ)z0 = 0.
I Continuum limit, g = gsε3, g = gc(1− Λε2), t = T/ε, gives
dZ0(T )
dT= Σ3 gs
α
Σ sinh ΣT + α cosh ΣT(Σ cosh ΣT + α sinh ΣT
)3
gs→∞−→ Λ3/4 cosh(Λ1/4T ′)
sinh3(Λ1/4T ′)T ′ = g1/6
s T
I DT two-point function appears as gs →∞! [Ambjørn, Watabiki, ’95]
I Solution is (using methods of [Bouttier, Di Francesco, Guitter, ’03]):
z1(t) = z11− σt
1− σt+1
1− (1− β)σ − βσt+3
1− (1− β)σ − βσt+2,
z0(t) = z01− σt
1− (1− β)σ − βσt+1
(1− (1− β)σ)2 − β2σt+3
1− (1− β)σ − βσt+2,
with β = β(g , g) and σ = σ(g , g) fixed by
g(1 + σ)(1 + βσ)z1 − σ(1− 2g z0) = 0,
(1− β)σ − g(1 + σ)z1 + g(1− σ + 2βσ)z0 = 0.
I Continuum limit, g = gsε3, g = gc(1− Λε2), t = T/ε, gives
dZ0(T )
dT= Σ3 gs
α
Σ sinh ΣT + α cosh ΣT(Σ cosh ΣT + α sinh ΣT
)3
gs→∞−→ Λ3/4 cosh(Λ1/4T ′)
sinh3(Λ1/4T ′)T ′ = g1/6
s T
I DT two-point function appears as gs →∞! [Ambjørn, Watabiki, ’95]
I Solution is (using methods of [Bouttier, Di Francesco, Guitter, ’03]):
z1(t) = z11− σt
1− σt+1
1− (1− β)σ − βσt+3
1− (1− β)σ − βσt+2,
z0(t) = z01− σt
1− (1− β)σ − βσt+1
(1− (1− β)σ)2 − β2σt+3
1− (1− β)σ − βσt+2,
with β = β(g , g) and σ = σ(g , g) fixed by
g(1 + σ)(1 + βσ)z1 − σ(1− 2g z0) = 0,
(1− β)σ − g(1 + σ)z1 + g(1− σ + 2βσ)z0 = 0.
I Continuum limit, g = gsε3, g = gc(1− Λε2), t = T/ε, gives
dZ0(T )
dT= Σ3 gs
α
Σ sinh ΣT + α cosh ΣT(Σ cosh ΣT + α sinh ΣT
)3
gs→∞−→ Λ3/4 cosh(Λ1/4T ′)
sinh3(Λ1/4T ′)T ′ = g1/6
s T
I DT two-point function appears as gs →∞! [Ambjørn, Watabiki, ’95]
Planar maps
2
1
1
1
3
3
3
4
4
0
2
2
2
2
2
1
3
1
3
3
2
t t
t +1
t - 1
t t
t +1
t +1
I Bijection between quadrangulations with Nmax local maxima andplanar maps with Nmax faces!
Planar maps
2
1
1
1
3
3
3
4
4
2
2
2
2
2
1
3
1
3
3
2
t t
t +1
t - 1
t t
t +1
t +1
I Bijection between quadrangulations with Nmax local maxima andplanar maps with Nmax faces!
Planar maps
2
1
1
1
3
3
3
4
4
2
2
2
2
2
1
3
1
3
3
2
t t
t +1
t - 1
t t
t +1
t +1
2
1
1
1
3 3
3
4
4
0
2
2
2
2
2
1
3
1
332
t t
t +1
t - 1
t t
t +1
t +1
I Bijection between quadrangulations with Nmax local maxima andplanar maps with Nmax faces!
Planar maps
2
1
1
1
3
3
3
4
4
2
2
2
2
2
1
3
1
3
3
2
t t
t +1
t - 1
t t
t +1
t +1
2
1
1
1
3 3
3
4
4
0
2
2
2
2
2
1
3
1
332
t t
t +1
t - 1
t t
t +1
t +1
I Bijection between quadrangulations with Nmax local maxima andplanar maps with Nmax faces!
Planar maps
2
1
1
1
3
3
3
4
4
2
2
2
2
2
1
3
1
3
3
2
t t
t +1
t - 1
t t
t +1
t +1
t t
t +1
t - 1
t t
t +1
t +1
I Bijection between quadrangulations with Nmax local maxima andplanar maps with Nmax faces!
Planar maps
t t
t +1
t - 1
t t
t +1
t +1
I Bijection between quadrangulations with Nmax local maxima andplanar maps with Nmax faces!
Two-point function for planar maps
2
1
1
1
3 3
3
4
4
2
2
2
2
2
1
3
1
3
32
I We know generating functions for trees and therefore obtain anexplicit generating function
z0(t + 1)− z0(t) =∞∑
N=0
∞∑n=0
Nt(N, n)gNgn
for the number Nt(N, n) of planar maps with N edges, n faces, anda marked point at distance t from the root.
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 Only need to keep the labels of the marked points!
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 is ideal for studying
“proper-time foliations” of random surfaces.I Generalized CDT appears naturally as the scaling limit of random
planar maps with a fixed finite number of faces.I Continuum DT (Brownian map?) seems to be recovered by taking
gs → ∞.I The relation to planar maps explains the mysterious loop-loop
identities in the continuum.
I OutlookI Is there a convergence towards a random measure on metric spaces,
i.e. analogue of the Brownian map? Should first try to understand2d geometry of random causal triangulations.
I What is the structure of the symmetries mentioned above?I Various stochastic processes involved in generalized CDT. How are
they related?
Further reading: arXiv:1302.1763
These slides and more: http://www.nbi.dk/~budd/
Questions?
Conclusions & Outlook
I ConlusionsI The Cori–Vauquelin–Schaeffer bijection is ideal for studying
“proper-time foliations” of random surfaces.I Generalized CDT appears naturally as the scaling limit of random
planar maps with a fixed finite number of faces.I Continuum DT (Brownian map?) seems to be recovered by taking
gs → ∞.I The relation to planar maps explains the mysterious loop-loop
identities in the continuum.
I OutlookI Is there a convergence towards a random measure on metric spaces,
i.e. analogue of the Brownian map? Should first try to understand2d geometry of random causal triangulations.
I What is the structure of the symmetries mentioned above?I Various stochastic processes involved in generalized CDT. How are
they related?
Further reading: arXiv:1302.1763
These slides and more: http://www.nbi.dk/~budd/
Questions?
Conclusions & Outlook
I ConlusionsI The Cori–Vauquelin–Schaeffer bijection is ideal for studying
“proper-time foliations” of random surfaces.I Generalized CDT appears naturally as the scaling limit of random
planar maps with a fixed finite number of faces.I Continuum DT (Brownian map?) seems to be recovered by taking
gs → ∞.I The relation to planar maps explains the mysterious loop-loop
identities in the continuum.
I OutlookI Is there a convergence towards a random measure on metric spaces,
i.e. analogue of the Brownian map? Should first try to understand2d geometry of random causal triangulations.
I What is the structure of the symmetries mentioned above?I Various stochastic processes involved in generalized CDT. How are
they related?
Further reading: arXiv:1302.1763
These slides and more: http://www.nbi.dk/~budd/
Questions?
Appendix: canonical labeling
0
2
Appendix: canonical labeling
1
11
0
1
2
Appendix: canonical labeling
2
2
2
22
2
2
2
1
11
0
2
2
2
21
2
2
Appendix: canonical labeling
3
3
3
3
3
3
3
3
3
3
2
2
2
22
2
2
2
1
1
3
1
3
0
2
2
3
2
23
3
3
3
3
1
2
2
3
Appendix: canonical labeling
4
4
44
4
4
4
4
4 4
4
3
3
3
3
3
3
3
3
3
3
2
2
2
22
4
2
2
2
1
1
3
4
1
43
4
0
2
4
4
2
3
4
2
4
23
3
3
3
3
1 4
2
4
4
2
4
4
4
3
Appendix: canonical labeling
5
5
55
5
5
5
5
5
5
4
4
44
4
4
4
4
4 4
4
3
3
3
3
3
5
3
3
3
3
3
2
2
2
22
4
2
2
2
1
1
55
5
5
3
4
1
5
5
5
4
5
5
3
4
0
2
4
4
2
3
5
4
2
4
2
5
5
3
3
3
3
3
5
1
5
4
2
4
4
2
4
4
4
3
Appendix: canonical labeling
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Appendix: canonical labeling
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Appendix: canonical labeling
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Appendix: canonical labeling
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Including boundaries [Bouttier, Guitter ’09, Bettinelli ’11, Curien, Miermont ’12]
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 [Bouttier, Guitter ’09, Bettinelli ’11, Curien, Miermont ’12]
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 [Bouttier, Guitter ’09, Bettinelli ’11, Curien, Miermont ’12]
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 [Bouttier, Guitter ’09, Bettinelli ’11, Curien, Miermont ’12]
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 [Bouttier, Guitter ’09, Bettinelli ’11, Curien, Miermont ’12]
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 [Bouttier, Guitter ’09, Bettinelli ’11, Curien, Miermont ’12]
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.
