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Hopf algebras and factorial divergent powerseries: Algebraic tools for graphical enumeration
Michael Borinsky1
Humboldt-University BerlinDepartments of Physics and Mathematics
Workshop on Enumerative Combinatorics, ESI, October 2017-
Renormalized asymptotic enumeration of Feynman diagrams,Annals of Physics, October 2017 arXiv:1703.00840
[email protected]. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 1
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Overview
1 Motivation
2 Asymptotics of multigraph enumeration
3 Ring of factorially divergent power series
4 Counting subgraph-restricted graphs
5 Conclusions
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 2
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Motivation
Perturbation expansions in quantum field theory may beorganized as sums of integrals.
Each integral maybe represented as a graph.
Usually these expansions have vanishing radius of convergence.The coefficients behave as fn ≈ CAnΓ(n + β) for large n.
Divergence is believed to be caused by proliferation of graphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 3
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Motivation
Perturbation expansions in quantum field theory may beorganized as sums of integrals.
Each integral maybe represented as a graph.
Usually these expansions have vanishing radius of convergence.The coefficients behave as fn ≈ CAnΓ(n + β) for large n.
Divergence is believed to be caused by proliferation of graphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 3
![Page 5: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/5.jpg)
Motivation
Perturbation expansions in quantum field theory may beorganized as sums of integrals.
Each integral maybe represented as a graph.
Usually these expansions have vanishing radius of convergence.The coefficients behave as fn ≈ CAnΓ(n + β) for large n.
Divergence is believed to be caused by proliferation of graphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 3
![Page 6: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/6.jpg)
Motivation
Perturbation expansions in quantum field theory may beorganized as sums of integrals.
Each integral maybe represented as a graph.
Usually these expansions have vanishing radius of convergence.The coefficients behave as fn ≈ CAnΓ(n + β) for large n.
Divergence is believed to be caused by proliferation of graphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 3
![Page 7: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/7.jpg)
In zero dimensions the integration becomes trivial.
Observables are generating functions of certain classes ofgraphs [Hurst, 1952, Cvitanovic et al., 1978, Argyres et al.,2001, Molinari and Manini, 2006].
For instance, the partition function in ϕ3 theory is formally,
Zϕ3(~) =
∫R
dx√2π~
e1~
(− x2
2+ x3
3!
).
‘Formal’ ⇒ expand under the integral sign and integrate overGaussian term by term,
Zϕ3(~) :=
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 4
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In zero dimensions the integration becomes trivial.
Observables are generating functions of certain classes ofgraphs [Hurst, 1952, Cvitanovic et al., 1978, Argyres et al.,2001, Molinari and Manini, 2006].
For instance, the partition function in ϕ3 theory is formally,
Zϕ3(~) =
∫R
dx√2π~
e1~
(− x2
2+ x3
3!
).
‘Formal’ ⇒ expand under the integral sign and integrate overGaussian term by term,
Zϕ3(~) :=
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 4
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In zero dimensions the integration becomes trivial.
Observables are generating functions of certain classes ofgraphs [Hurst, 1952, Cvitanovic et al., 1978, Argyres et al.,2001, Molinari and Manini, 2006].
For instance, the partition function in ϕ3 theory is formally,
Zϕ3(~) =
∫R
dx√2π~
e1~
(− x2
2+ x3
3!
).
‘Formal’ ⇒ expand under the integral sign and integrate overGaussian term by term,
Zϕ3(~) :=
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 4
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In zero dimensions the integration becomes trivial.
Observables are generating functions of certain classes ofgraphs [Hurst, 1952, Cvitanovic et al., 1978, Argyres et al.,2001, Molinari and Manini, 2006].
For instance, the partition function in ϕ3 theory is formally,
Zϕ3(~) =
∫R
dx√2π~
e1~
(− x2
2+ x3
3!
).
‘Formal’ ⇒ expand under the integral sign and integrate overGaussian term by term,
Zϕ3(~) :=
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 4
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Zϕ3
is the exponential generating function of 3-valentmultigraphs,
Zϕ3(~) =
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~ =∑
cubic graphs Γ
~|E(Γ)|−|V (Γ)|
|Aut Γ|
The parameter ~ counts the excess of the graph.
Zϕ3(~) = φ
(1 +
1
8+
1
12+
1
128+
1
288+
1
96
+1
48+
1
16+
1
16+
1
8+
1
24+ . . .
)= 1 +
(1
8+
1
12
)~ +
385
1152~2 + . . .
where φ(Γ) = ~|E(Γ)|−|V (Γ)| maps a graph with excess n to ~n.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 5
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Zϕ3
is the exponential generating function of 3-valentmultigraphs,
Zϕ3(~) =
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~ =∑
cubic graphs Γ
~|E(Γ)|−|V (Γ)|
|Aut Γ|
The parameter ~ counts the excess of the graph.
Zϕ3(~) = φ
(1 +
1
8+
1
12+
1
128+
1
288+
1
96
+1
48+
1
16+
1
16+
1
8+
1
24+ . . .
)= 1 +
(1
8+
1
12
)~ +
385
1152~2 + . . .
where φ(Γ) = ~|E(Γ)|−|V (Γ)| maps a graph with excess n to ~n.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 5
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Zϕ3
is the exponential generating function of 3-valentmultigraphs,
Zϕ3(~) =
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~ =∑
cubic graphs Γ
~|E(Γ)|−|V (Γ)|
|Aut Γ|
The parameter ~ counts the excess of the graph.
Zϕ3(~) = φ
(1 +
1
8+
1
12+
1
128+
1
288+
1
96
+1
48+
1
16+
1
16+
1
8+
1
24+ . . .
)= 1 +
(1
8+
1
12
)~ +
385
1152~2 + . . .
where φ(Γ) = ~|E(Γ)|−|V (Γ)| maps a graph with excess n to ~n.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 5
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Zϕ3
is the exponential generating function of 3-valentmultigraphs,
Zϕ3(~) =
∞∑n=0
~n(2n − 1)!![x2n]ex3
3!~ =∑
cubic graphs Γ
~|E(Γ)|−|V (Γ)|
|Aut Γ|
The parameter ~ counts the excess of the graph.
Zϕ3(~) = φ
(1 +
1
8+
1
12+
1
128+
1
288+
1
96
+1
48+
1
16+
1
16+
1
8+
1
24+ . . .
)= 1 +
(1
8+
1
12
)~ +
385
1152~2 + . . .
where φ(Γ) = ~|E(Γ)|−|V (Γ)| maps a graph with excess n to ~n.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 5
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Arbitrary degree distribution
More general with arbitrary degree distribution,
Z (~) =∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~
=∑
graphs Γ
~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
|Aut Γ|
where the sum is over all multigraphs Γ and |v | is the degreeof vertex v .
This corresponds to the pairing model of multigraphs [Benderand Canfield, 1978].
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 6
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Arbitrary degree distribution
More general with arbitrary degree distribution,
Z (~) =∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~
=∑
graphs Γ
~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
|Aut Γ|
where the sum is over all multigraphs Γ and |v | is the degreeof vertex v .
This corresponds to the pairing model of multigraphs [Benderand Canfield, 1978].
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 6
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Arbitrary degree distribution
More general with arbitrary degree distribution,
Z (~) =∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~
=∑
graphs Γ
~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
|Aut Γ|
where the sum is over all multigraphs Γ and |v | is the degreeof vertex v .
This corresponds to the pairing model of multigraphs [Benderand Canfield, 1978].
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 6
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Arbitrary degree distribution
More general with arbitrary degree distribution,
Z (~) =∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~
=∑
graphs Γ
~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
|Aut Γ|
where the sum is over all multigraphs Γ and |v | is the degreeof vertex v .
This corresponds to the pairing model of multigraphs [Benderand Canfield, 1978].
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 6
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We obtain a series of polynomials
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~ = φ(
1 +1
8+
1
12+
1
8
+1
128+
1
288+
1
96+
1
48
+1
16+
1
16+
1
8+
1
24+ · · ·
)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
= 1 +
((1
8+
1
12
)λ2
3 +1
8λ4
)~
+
(385
1152λ4
3 +35
64λ2
3λ4 +35
384λ2
4 +7
48λ3λ5 +
1
48λ6
)~2 + · · ·
=∞∑n=0
Pn(λ3, λ4, . . .)~n
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 7
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We obtain a series of polynomials
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~ = φ(
1 +1
8+
1
12+
1
8
+1
128+
1
288+
1
96+
1
48
+1
16+
1
16+
1
8+
1
24+ · · ·
)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
= 1 +
((1
8+
1
12
)λ2
3 +1
8λ4
)~
+
(385
1152λ4
3 +35
64λ2
3λ4 +35
384λ2
4 +7
48λ3λ5 +
1
48λ6
)~2 + · · ·
=∞∑n=0
Pn(λ3, λ4, . . .)~n
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 7
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We obtain a series of polynomials
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~ = φ(
1 +1
8+
1
12+
1
8
+1
128+
1
288+
1
96+
1
48
+1
16+
1
16+
1
8+
1
24+ · · ·
)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
= 1 +
((1
8+
1
12
)λ2
3 +1
8λ4
)~
+
(385
1152λ4
3 +35
64λ2
3λ4 +35
384λ2
4 +7
48λ3λ5 +
1
48λ6
)~2 + · · ·
=∞∑n=0
Pn(λ3, λ4, . . .)~n
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 7
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We obtain a series of polynomials
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
λkk!
xk
~ = φ(
1 +1
8+
1
12+
1
8
+1
128+
1
288+
1
96+
1
48
+1
16+
1
16+
1
8+
1
24+ · · ·
)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏
v∈V (Γ) λ|v |
= 1 +
((1
8+
1
12
)λ2
3 +1
8λ4
)~
+
(385
1152λ4
3 +35
64λ2
3λ4 +35
384λ2
4 +7
48λ3λ5 +
1
48λ6
)~2 + · · ·
=∞∑n=0
Pn(λ3, λ4, . . .)~n
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 7
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Example
Take the degree distribution λk = −1 for all k ≥ 3.
Z stir(~) :=∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
−1k!
xk
~
= φ(
1 +1
8+
1
12+
1
8+ . . .
)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|
= 1 +
(1
8(−1)2 +
1
12(−1)2 +
1
8(−1)1
)~ + . . .
= 1 +1
12~ +
1
288~2 − 139
51840~3 + . . . = e
∑∞n=1
Bn+1n(n+1)
~n
⇒ Stirling series as a signed sum over multigraphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 8
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Example
Take the degree distribution λk = −1 for all k ≥ 3.
Z stir(~) :=∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
−1k!
xk
~
= φ(
1 +1
8+
1
12+
1
8+ . . .
)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|
= 1 +
(1
8(−1)2 +
1
12(−1)2 +
1
8(−1)1
)~ + . . .
= 1 +1
12~ +
1
288~2 − 139
51840~3 + . . . = e
∑∞n=1
Bn+1n(n+1)
~n
⇒ Stirling series as a signed sum over multigraphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 8
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Example
Take the degree distribution λk = −1 for all k ≥ 3.
Z stir(~) :=∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
−1k!
xk
~
= φ(
1 +1
8+
1
12+
1
8+ . . .
)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|
= 1 +
(1
8(−1)2 +
1
12(−1)2 +
1
8(−1)1
)~ + . . .
= 1 +1
12~ +
1
288~2 − 139
51840~3 + . . . = e
∑∞n=1
Bn+1n(n+1)
~n
⇒ Stirling series as a signed sum over multigraphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 8
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Example
Take the degree distribution λk = −1 for all k ≥ 3.
Z stir(~) :=∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
−1k!
xk
~
= φ(
1 +1
8+
1
12+
1
8+ . . .
)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|
= 1 +
(1
8(−1)2 +
1
12(−1)2 +
1
8(−1)1
)~ + . . .
= 1 +1
12~ +
1
288~2 − 139
51840~3 + . . .
= e∑∞
n=1Bn+1n(n+1)
~n
⇒ Stirling series as a signed sum over multigraphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 8
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Example
Take the degree distribution λk = −1 for all k ≥ 3.
Z stir(~) :=∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
−1k!
xk
~
= φ(
1 +1
8+
1
12+
1
8+ . . .
)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|
= 1 +
(1
8(−1)2 +
1
12(−1)2 +
1
8(−1)1
)~ + . . .
= 1 +1
12~ +
1
288~2 − 139
51840~3 + . . . = e
∑∞n=1
Bn+1n(n+1)
~n
⇒ Stirling series as a signed sum over multigraphs.
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Example
Take the degree distribution λk = −1 for all k ≥ 3.
Z stir(~) :=∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥3
−1k!
xk
~
= φ(
1 +1
8+
1
12+
1
8+ . . .
)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|
= 1 +
(1
8(−1)2 +
1
12(−1)2 +
1
8(−1)1
)~ + . . .
= 1 +1
12~ +
1
288~2 − 139
51840~3 + . . . = e
∑∞n=1
Bn+1n(n+1)
~n
⇒ Stirling series as a signed sum over multigraphs.
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Define a functional F : R[[x ]]→ R[[~]]
F : S(x) 7→∞∑n=0
~n(2n − 1)!![x2n]ex2
2 +S(x)
~
where S(x) = − x2
2 +∑
k≥3λkk! x
k .
F maps a degree sequence to the corresponding generatingfunction of multigraphs.
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Define a functional F : R[[x ]]→ R[[~]]
F : S(x) 7→∞∑n=0
~n(2n − 1)!![x2n]ex2
2 +S(x)
~
where S(x) = − x2
2 +∑
k≥3λkk! x
k .
F maps a degree sequence to the corresponding generatingfunction of multigraphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 9
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Define a functional F : R[[x ]]→ R[[~]]
F : S(x) 7→∞∑n=0
~n(2n − 1)!![x2n]ex2
2 +S(x)
~
where S(x) = − x2
2 +∑
k≥3λkk! x
k .
F maps a degree sequence to the corresponding generatingfunction of multigraphs.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 9
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Expressions as
F [S(x)] =∞∑n=0
~n(2n − 1)!![x2n]ex2
2 +S(x)
~
are inconvenient to expand, because of the ~ and the 1~ in the
exponent ⇒ we have to sum over a diagonal.
Theorem
This can be expressed without a diagonal summation [MB, 2017]:
F [S(x)] =∞∑n=0
~n(2n + 1)!![y2n+1]x(y),
where x(y) is the (power series) solution of
y2
2= −S(x(y)).
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Expressions as
F [S(x)] =∞∑n=0
~n(2n − 1)!![x2n]ex2
2 +S(x)
~
are inconvenient to expand, because of the ~ and the 1~ in the
exponent ⇒ we have to sum over a diagonal.
Theorem
This can be expressed without a diagonal summation [MB, 2017]:
F [S(x)] =∞∑n=0
~n(2n + 1)!![y2n+1]x(y),
where x(y) is the (power series) solution of
y2
2= −S(x(y)).
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 10
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Expressions as
F [S(x)] =∞∑n=0
~n(2n − 1)!![x2n]ex2
2 +S(x)
~
are inconvenient to expand, because of the ~ and the 1~ in the
exponent ⇒ we have to sum over a diagonal.
Theorem
This can be expressed without a diagonal summation [MB, 2017]:
F [S(x)] =∞∑n=0
~n(2n + 1)!![y2n+1]x(y),
where x(y) is the (power series) solution of
y2
2= −S(x(y)).
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 10
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Asymptotics
Calculate asymptotics of F [S(x)] by singu-larity analysis of x(y):
⇒ Locate the dominant singularity of x(y) byanalysis of the (generalized) hyperellipticcurve,
y2
2= −S(x).
The dominant singularity of x(y) coincideswith a branch-cut of the local parametriza-tion of the curve.
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Asymptotics
Calculate asymptotics of F [S(x)] by singu-larity analysis of x(y):
⇒ Locate the dominant singularity of x(y) byanalysis of the (generalized) hyperellipticcurve,
y2
2= −S(x).
The dominant singularity of x(y) coincideswith a branch-cut of the local parametriza-tion of the curve.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 11
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Asymptotics
Calculate asymptotics of F [S(x)] by singu-larity analysis of x(y):
⇒ Locate the dominant singularity of x(y) byanalysis of the (generalized) hyperellipticcurve,
y2
2= −S(x).
The dominant singularity of x(y) coincideswith a branch-cut of the local parametriza-tion of the curve.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 11
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Asymptotics
Calculate asymptotics of F [S(x)] by singu-larity analysis of x(y):
⇒ Locate the dominant singularity of x(y) byanalysis of the (generalized) hyperellipticcurve,
y2
2= −S(x).
The dominant singularity of x(y) coincideswith a branch-cut of the local parametriza-tion of the curve.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 11
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Asymptotics
Calculate asymptotics of F [S(x)] by singu-larity analysis of x(y):
⇒ Locate the dominant singularity of x(y) byanalysis of the (generalized) hyperellipticcurve,
y2
2= −S(x).
The dominant singularity of x(y) coincideswith a branch-cut of the local parametriza-tion of the curve.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 11
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−4 −2 0 2 4
−4
−2
0
2
4
x
y
Figure: Example: The curve y2
2 = x2
2 −x3
3! associated to Zϕ3
.
⇒ x(y) has a (dominant) branch-cut singularity at y = ρ = 2√3
,
where x(ρ) = τ = 2.
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−4 −2 0 2 4
−4
−2
0
2
4
x
y
Figure: Example: The curve y2
2 = x2
2 −x3
3! associated to Zϕ3
.
⇒ x(y) has a (dominant) branch-cut singularity at y = ρ = 2√3
,
where x(ρ) = τ = 2.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 12
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⇒ Near the dominant singularity:
x(y)− τ = −d1
√1− y
ρ+∞∑j=2
dj
(1− y
ρ
) j2
[yn]x(y) ∼ Cρ−nn−32
(1 +
∞∑k=1
eknk
)
Therefore,
[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)
=2nΓ(n + 3
2 )√π
[y2n+1]x(y)
∼ C ′2nρ−2nΓ(n)
(1 +
∞∑k=1
e ′knk
)
What is the value of C ′ and the e ′k?
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
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⇒ Near the dominant singularity:
x(y)− τ = −d1
√1− y
ρ+∞∑j=2
dj
(1− y
ρ
) j2
[yn]x(y) ∼ Cρ−nn−32
(1 +
∞∑k=1
eknk
)
Therefore,
[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)
=2nΓ(n + 3
2 )√π
[y2n+1]x(y)
∼ C ′2nρ−2nΓ(n)
(1 +
∞∑k=1
e ′knk
)
What is the value of C ′ and the e ′k?
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
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⇒ Near the dominant singularity:
x(y)− τ = −d1
√1− y
ρ+∞∑j=2
dj
(1− y
ρ
) j2
[yn]x(y) ∼ Cρ−nn−32
(1 +
∞∑k=1
eknk
)
Therefore,
[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)
=2nΓ(n + 3
2 )√π
[y2n+1]x(y)
∼ C ′2nρ−2nΓ(n)
(1 +
∞∑k=1
e ′knk
)
What is the value of C ′ and the e ′k?
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
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⇒ Near the dominant singularity:
x(y)− τ = −d1
√1− y
ρ+∞∑j=2
dj
(1− y
ρ
) j2
[yn]x(y) ∼ Cρ−nn−32
(1 +
∞∑k=1
eknk
)
Therefore,
[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)
=2nΓ(n + 3
2 )√π
[y2n+1]x(y)
∼ C ′2nρ−2nΓ(n)
(1 +
∞∑k=1
e ′knk
)
What is the value of C ′ and the e ′k?
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
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⇒ Near the dominant singularity:
x(y)− τ = −d1
√1− y
ρ+∞∑j=2
dj
(1− y
ρ
) j2
[yn]x(y) ∼ Cρ−nn−32
(1 +
∞∑k=1
eknk
)
Therefore,
[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)
=2nΓ(n + 3
2 )√π
[y2n+1]x(y)
∼ C ′2nρ−2nΓ(n)
(1 +
∞∑k=1
e ′knk
)
What is the value of C ′ and the e ′k?
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
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⇒ Near the dominant singularity:
x(y)− τ = −d1
√1− y
ρ+∞∑j=2
dj
(1− y
ρ
) j2
[yn]x(y) ∼ Cρ−nn−32
(1 +
∞∑k=1
eknk
)
Therefore,
[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)
=2nΓ(n + 3
2 )√π
[y2n+1]x(y)
∼ C ′2nρ−2nΓ(n)
(1 +
∞∑k=1
e ′knk
)
What is the value of C ′ and the e ′k?
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
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⇒ Near the dominant singularity:
x(y)− τ = −d1
√1− y
ρ+∞∑j=2
dj
(1− y
ρ
) j2
[yn]x(y) ∼ Cρ−nn−32
(1 +
∞∑k=1
eknk
)
Therefore,
[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)
=2nΓ(n + 3
2 )√π
[y2n+1]x(y)
∼ C ′2nρ−2nΓ(n)
(1 +
∞∑k=1
e ′knk
)
What is the value of C ′ and the e ′k?
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 13
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Theorem ([MB, 2017])
Let (x , y) = (τ, ρ) be the location of the dominant branch-cut
singularity of y2
2 = −S(x). Then
[~n]F [S(x)](~) =R−1∑k=0
ckA−(n−k)Γ(n − k) +O
(A−nΓ(n − R)
),
where A = −S(τ) and
ck =1
2π[~k ]F [S(τ)− S(x + τ)](−~).
⇒ The asymptotic expansion can be expressed as a generatingfunction of graphs.
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Theorem ([MB, 2017])
Let (x , y) = (τ, ρ) be the location of the dominant branch-cut
singularity of y2
2 = −S(x). Then
[~n]F [S(x)](~) =R−1∑k=0
ckA−(n−k)Γ(n − k) +O
(A−nΓ(n − R)
),
where A = −S(τ) and
ck =1
2π[~k ]F [S(τ)− S(x + τ)](−~).
⇒ The asymptotic expansion can be expressed as a generatingfunction of graphs.
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Theorem ([MB, 2017])
Let (x , y) = (τ, ρ) be the location of the dominant branch-cut
singularity of y2
2 = −S(x). Then
[~n]F [S(x)](~) =R−1∑k=0
ckA−(n−k)Γ(n − k) +O
(A−nΓ(n − R)
),
where A = −S(τ) and
ck =1
2π[~k ]F [S(τ)− S(x + τ)](−~).
⇒ The asymptotic expansion can be expressed as a generatingfunction of graphs.
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Example
For cubic graphs or equivalently ϕ3 theory, we are interestedin S(x) = − x2
2 + x3
3! ,
F [S(x)] (~) = φ(1 +
1
8+
1
12+
1
128+ . . .
)1 +
5
24~ +
385
1152~2 +
85085
82944~3 + · · ·
We find τ = 2, A = 23 and the coefficients of the asymptotic
expansion
∞∑k=0
ck~k =1
2πF [S(τ)− S(τ + x)](−~) =
1
2πF [−x2
2+
x3
3!](−~)
=1
2π
(1− 5
24~ +
385
1152~2 − 85085
82944~3 + . . .
)⇒ The asymptotic expansion is [~n]F [S(x)] (~) =∑R−1
k=0 ckA−n+kΓ(n − k) +O(A−n+RΓ(n − R)).
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Example
For cubic graphs or equivalently ϕ3 theory, we are interestedin S(x) = − x2
2 + x3
3! ,
F [S(x)] (~) = φ(1 +
1
8+
1
12+
1
128+ . . .
)1 +
5
24~ +
385
1152~2 +
85085
82944~3 + · · ·
We find τ = 2, A = 23 and the coefficients of the asymptotic
expansion
∞∑k=0
ck~k =1
2πF [S(τ)− S(τ + x)](−~) =
1
2πF [−x2
2+
x3
3!](−~)
=1
2π
(1− 5
24~ +
385
1152~2 − 85085
82944~3 + . . .
)⇒ The asymptotic expansion is [~n]F [S(x)] (~) =∑R−1
k=0 ckA−n+kΓ(n − k) +O(A−n+RΓ(n − R)).
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Example
For cubic graphs or equivalently ϕ3 theory, we are interestedin S(x) = − x2
2 + x3
3! ,
F [S(x)] (~) = φ(1 +
1
8+
1
12+
1
128+ . . .
)1 +
5
24~ +
385
1152~2 +
85085
82944~3 + · · ·
We find τ = 2, A = 23 and the coefficients of the asymptotic
expansion
∞∑k=0
ck~k =1
2πF [S(τ)− S(τ + x)](−~) =
1
2πF [−x2
2+
x3
3!](−~)
=1
2π
(1− 5
24~ +
385
1152~2 − 85085
82944~3 + . . .
)
⇒ The asymptotic expansion is [~n]F [S(x)] (~) =∑R−1k=0 ckA
−n+kΓ(n − k) +O(A−n+RΓ(n − R)).
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Example
For cubic graphs or equivalently ϕ3 theory, we are interestedin S(x) = − x2
2 + x3
3! ,
F [S(x)] (~) = φ(1 +
1
8+
1
12+
1
128+ . . .
)1 +
5
24~ +
385
1152~2 +
85085
82944~3 + · · ·
We find τ = 2, A = 23 and the coefficients of the asymptotic
expansion
∞∑k=0
ck~k =1
2πF [S(τ)− S(τ + x)](−~) =
1
2πF [−x2
2+
x3
3!](−~)
=1
2π
(1− 5
24~ +
385
1152~2 − 85085
82944~3 + . . .
)⇒ The asymptotic expansion is [~n]F [S(x)] (~) =∑R−1
k=0 ckA−n+kΓ(n − k) +O(A−n+RΓ(n − R)).
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Ring of factorially divergent power series
Power series which have Poincare asymptotic expansion of theform
fn =R−1∑k=0
ckA−n+kΓ(n − k) +O(A−nΓ(n − R)) ∀R ≥ 0,
form a subring of R[[x ]] which is closed under compositionand inversion of power series [MB, 2016].
For instance, this allows us to calculate the completeasymptotic expansion of constructions such as
logF [S(x)] (~)
in closed form.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 16
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Ring of factorially divergent power series
Power series which have Poincare asymptotic expansion of theform
fn =R−1∑k=0
ckA−n+kΓ(n − k) +O(A−nΓ(n − R)) ∀R ≥ 0,
form a subring of R[[x ]] which is closed under compositionand inversion of power series [MB, 2016].
For instance, this allows us to calculate the completeasymptotic expansion of constructions such as
logF [S(x)] (~)
in closed form.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 16
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Ring of factorially divergent power series
Power series which have Poincare asymptotic expansion of theform
fn =R−1∑k=0
ckA−n+kΓ(n − k) +O(A−nΓ(n − R)) ∀R ≥ 0,
form a subring of R[[x ]] which is closed under compositionand inversion of power series [MB, 2016].
For instance, this allows us to calculate the completeasymptotic expansion of constructions such as
logF [S(x)] (~)
in closed form.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 16
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Renormalization
Renormalization → Restriction on the allowed bridgelesssubgraphs.
P is a set of forbidden subgraphs.
We wish to have a map
φP(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ has no subgraph in P
0 else
which is compatible with our generating function andasymptotic techniques.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 17
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Renormalization
Renormalization → Restriction on the allowed bridgelesssubgraphs.
P is a set of forbidden subgraphs.
We wish to have a map
φP(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ has no subgraph in P
0 else
which is compatible with our generating function andasymptotic techniques.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 17
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Renormalization
Renormalization → Restriction on the allowed bridgelesssubgraphs.
P is a set of forbidden subgraphs.
We wish to have a map
φP(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ has no subgraph in P
0 else
which is compatible with our generating function andasymptotic techniques.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 17
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Renormalization
Renormalization → Restriction on the allowed bridgelesssubgraphs.
P is a set of forbidden subgraphs.
We wish to have a map
φP(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ has no subgraph in P
0 else
which is compatible with our generating function andasymptotic techniques.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 17
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Renormalization
Renormalization → Restriction on the allowed bridgelesssubgraphs.
P is a set of forbidden subgraphs.
We wish to have a map
φP(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ has no subgraph in P
0 else
which is compatible with our generating function andasymptotic techniques.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 17
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G is the Q-algebra generated by multigraphs.
Split G = G− ⊕ G+ such that
G+ is the set of graphs without subgraphs in P.
G− is the set of graphs with subgraphs in P.
We know a map φ : G → R[[~]].
Construct φP such that φP |G+ = φ and φP |G− = 0.
This is a Riemann-Hilbert problem.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
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G is the Q-algebra generated by multigraphs.
Split G = G− ⊕ G+ such that
G+ is the set of graphs without subgraphs in P.
G− is the set of graphs with subgraphs in P.
We know a map φ : G → R[[~]].
Construct φP such that φP |G+ = φ and φP |G− = 0.
This is a Riemann-Hilbert problem.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
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G is the Q-algebra generated by multigraphs.
Split G = G− ⊕ G+ such that
G+ is the set of graphs without subgraphs in P.
G− is the set of graphs with subgraphs in P.
We know a map φ : G → R[[~]].
Construct φP such that φP |G+ = φ and φP |G− = 0.
This is a Riemann-Hilbert problem.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
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G is the Q-algebra generated by multigraphs.
Split G = G− ⊕ G+ such that
G+ is the set of graphs without subgraphs in P.
G− is the set of graphs with subgraphs in P.
We know a map φ : G → R[[~]].
Construct φP such that φP |G+ = φ and φP |G− = 0.
This is a Riemann-Hilbert problem.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
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G is the Q-algebra generated by multigraphs.
Split G = G− ⊕ G+ such that
G+ is the set of graphs without subgraphs in P.
G− is the set of graphs with subgraphs in P.
We know a map φ : G → R[[~]].
Construct φP such that φP |G+ = φ and φP |G− = 0.
This is a Riemann-Hilbert problem.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
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G is the Q-algebra generated by multigraphs.
Split G = G− ⊕ G+ such that
G+ is the set of graphs without subgraphs in P.
G− is the set of graphs with subgraphs in P.
We know a map φ : G → R[[~]].
Construct φP such that φP |G+ = φ and φP |G− = 0.
This is a Riemann-Hilbert problem.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
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G is the Q-algebra generated by multigraphs.
Split G = G− ⊕ G+ such that
G+ is the set of graphs without subgraphs in P.
G− is the set of graphs with subgraphs in P.
We know a map φ : G → R[[~]].
Construct φP such that φP |G+ = φ and φP |G− = 0.
This is a Riemann-Hilbert problem.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 18
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Hopf algebra of graphs
Pick a set of bridgeless graphs P, such that
if γ1 ⊂ γ2 then γ1, γ2 ∈ P iff γ1, γ2/γ1 ∈ P (1)
if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)
∅ ∈ P (3)
H is the Q-algebra of graphs in P.
Define a coaction on G:
∆ : G → H⊗ G
Γ 7→∑γ⊂Γ
s.t.γ∈P
γ ⊗ Γ/γ
H is a left-comodule over G.
∆ can be set up on H. ∆ : H → H⊗H.
H is a Hopf algebra Connes and Kreimer [2001].
(1) implies ∆ is coassociative (id⊗∆) ◦∆ = (∆⊗ id) ◦∆.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19
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Hopf algebra of graphs
Pick a set of bridgeless graphs P, such that
if γ1 ⊂ γ2 then γ1, γ2 ∈ P iff γ1, γ2/γ1 ∈ P (1)
if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)
∅ ∈ P (3)
H is the Q-algebra of graphs in P.
Define a coaction on G:
∆ : G → H⊗ G
Γ 7→∑γ⊂Γ
s.t.γ∈P
γ ⊗ Γ/γ
H is a left-comodule over G.
∆ can be set up on H. ∆ : H → H⊗H.
H is a Hopf algebra Connes and Kreimer [2001].
(1) implies ∆ is coassociative (id⊗∆) ◦∆ = (∆⊗ id) ◦∆.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19
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Hopf algebra of graphs
Pick a set of bridgeless graphs P, such that
if γ1 ⊂ γ2 then γ1, γ2 ∈ P iff γ1, γ2/γ1 ∈ P (1)
if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)
∅ ∈ P (3)
H is the Q-algebra of graphs in P.
Define a coaction on G:
∆ : G → H⊗ G
Γ 7→∑γ⊂Γ
s.t.γ∈P
γ ⊗ Γ/γ
H is a left-comodule over G.
∆ can be set up on H. ∆ : H → H⊗H.
H is a Hopf algebra Connes and Kreimer [2001].
(1) implies ∆ is coassociative (id⊗∆) ◦∆ = (∆⊗ id) ◦∆.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19
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Hopf algebra of graphs
Pick a set of bridgeless graphs P, such that
if γ1 ⊂ γ2 then γ1, γ2 ∈ P iff γ1, γ2/γ1 ∈ P (1)
if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)
∅ ∈ P (3)
H is the Q-algebra of graphs in P.
Define a coaction on G:
∆ : G → H⊗ G
Γ 7→∑γ⊂Γ
s.t.γ∈P
γ ⊗ Γ/γ
H is a left-comodule over G.
∆ can be set up on H. ∆ : H → H⊗H.
H is a Hopf algebra Connes and Kreimer [2001].
(1) implies ∆ is coassociative (id⊗∆) ◦∆ = (∆⊗ id) ◦∆.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19
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Hopf algebra of graphs
Pick a set of bridgeless graphs P, such that
if γ1 ⊂ γ2 then γ1, γ2 ∈ P iff γ1, γ2/γ1 ∈ P (1)
if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)
∅ ∈ P (3)
H is the Q-algebra of graphs in P.
Define a coaction on G:
∆ : G → H⊗ G
Γ 7→∑γ⊂Γ
s.t.γ∈P
γ ⊗ Γ/γ
H is a left-comodule over G.
∆ can be set up on H. ∆ : H → H⊗H.
H is a Hopf algebra Connes and Kreimer [2001].
(1) implies ∆ is coassociative (id⊗∆) ◦∆ = (∆⊗ id) ◦∆.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19
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Hopf algebra of graphs
Pick a set of bridgeless graphs P, such that
if γ1 ⊂ γ2 then γ1, γ2 ∈ P iff γ1, γ2/γ1 ∈ P (1)
if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)
∅ ∈ P (3)
H is the Q-algebra of graphs in P.
Define a coaction on G:
∆ : G → H⊗ G
Γ 7→∑γ⊂Γ
s.t.γ∈P
γ ⊗ Γ/γ
H is a left-comodule over G.
∆ can be set up on H. ∆ : H → H⊗H.
H is a Hopf algebra Connes and Kreimer [2001].
(1) implies ∆ is coassociative (id⊗∆) ◦∆ = (∆⊗ id) ◦∆.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19
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Hopf algebra of graphs
Pick a set of bridgeless graphs P, such that
if γ1 ⊂ γ2 then γ1, γ2 ∈ P iff γ1, γ2/γ1 ∈ P (1)
if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)
∅ ∈ P (3)
H is the Q-algebra of graphs in P.
Define a coaction on G:
∆ : G → H⊗ G
Γ 7→∑γ⊂Γ
s.t.γ∈P
γ ⊗ Γ/γ
H is a left-comodule over G.
∆ can be set up on H. ∆ : H → H⊗H.
H is a Hopf algebra Connes and Kreimer [2001].
(1) implies ∆ is coassociative (id⊗∆) ◦∆ = (∆⊗ id) ◦∆.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 19
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The coaction shall keep information about the number ofedges it was connected in the original graph.
⇒ The graphs in P have ‘legs’ or ‘hairs’.
Example: Suppose ∅, , ∈ P, then
∆ =∑γ⊂s.t.γ∈P
γ ⊗ /γ = 1⊗ + 3 ⊗ + ⊗
where we had to consider the subgraphs
, , ,
the complete and the empty subgraph.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 20
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The coaction shall keep information about the number ofedges it was connected in the original graph.
⇒ The graphs in P have ‘legs’ or ‘hairs’.
Example: Suppose ∅, , ∈ P, then
∆ =∑γ⊂s.t.γ∈P
γ ⊗ /γ = 1⊗ + 3 ⊗ + ⊗
where we had to consider the subgraphs
, , ,
the complete and the empty subgraph.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 20
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The coaction shall keep information about the number ofedges it was connected in the original graph.
⇒ The graphs in P have ‘legs’ or ‘hairs’.
Example: Suppose ∅, , ∈ P, then
∆ =∑γ⊂s.t.γ∈P
γ ⊗ /γ = 1⊗ + 3 ⊗ + ⊗
where we had to consider the subgraphs
, , ,
the complete and the empty subgraph.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 20
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The coaction shall keep information about the number ofedges it was connected in the original graph.
⇒ The graphs in P have ‘legs’ or ‘hairs’.
Example: Suppose ∅, , ∈ P, then
∆ =∑γ⊂s.t.γ∈P
γ ⊗ /γ = 1⊗ + 3 ⊗ + ⊗
where we had to consider the subgraphs
, , ,
the complete and the empty subgraph.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 20
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Gives us an action on algebra morphisms G → R[[~]]. Forψ : H → R[[~]] and φ : G → R[[~]],
ψ ? φ : G → R[[~]] ψ ? φ = m ◦ (ψ ⊗ φ) ◦∆
and a product of algebra morphisms H → R[[~]]. Forξ : H → R[[~]] and ψ : H → R[[~]],
ξ ? ψ : H → R[[~]] ξ ? ψ = m ◦ (ξ ⊗ ψ) ◦∆
Coassociativity of ∆ implies associativity of ?:
(ξ ? ψ) ? φ = ξ ? (ψ ? φ)
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Gives us an action on algebra morphisms G → R[[~]]. Forψ : H → R[[~]] and φ : G → R[[~]],
ψ ? φ : G → R[[~]] ψ ? φ = m ◦ (ψ ⊗ φ) ◦∆
and a product of algebra morphisms H → R[[~]]. Forξ : H → R[[~]] and ψ : H → R[[~]],
ξ ? ψ : H → R[[~]] ξ ? ψ = m ◦ (ξ ⊗ ψ) ◦∆
Coassociativity of ∆ implies associativity of ?:
(ξ ? ψ) ? φ = ξ ? (ψ ? φ)
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 21
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Gives us an action on algebra morphisms G → R[[~]]. Forψ : H → R[[~]] and φ : G → R[[~]],
ψ ? φ : G → R[[~]] ψ ? φ = m ◦ (ψ ⊗ φ) ◦∆
and a product of algebra morphisms H → R[[~]]. Forξ : H → R[[~]] and ψ : H → R[[~]],
ξ ? ψ : H → R[[~]] ξ ? ψ = m ◦ (ξ ⊗ ψ) ◦∆
Coassociativity of ∆ implies associativity of ?:
(ξ ? ψ) ? φ = ξ ? (ψ ? φ)
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 21
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The set of all algebra morphisms H → R[[~]] with the?-product forms a group.
The identity ε : H → R[[~]] maps the empty graph to 1 andall other graphs to 0.
The inverse of ψ : H → R[[~]] maybe calculated recursively:
ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ
s.t.γ∈P
ψ?−1(γ)ψ(Γ/γ)
Corresponds to Moebius-Inversion on the subgraph poset.
Simplifies on many (physical) cases to a (functional) inversionproblem on power series.
Solves the Riemann-Hilbert problem:Invert φ : Γ 7→ ~|E(Γ)|−|V (Γ)| restricted to H. Then
(φ|?−1H ? φ)(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ ∈ G+
0 else
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 22
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The set of all algebra morphisms H → R[[~]] with the?-product forms a group.
The identity ε : H → R[[~]] maps the empty graph to 1 andall other graphs to 0.
The inverse of ψ : H → R[[~]] maybe calculated recursively:
ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ
s.t.γ∈P
ψ?−1(γ)ψ(Γ/γ)
Corresponds to Moebius-Inversion on the subgraph poset.
Simplifies on many (physical) cases to a (functional) inversionproblem on power series.
Solves the Riemann-Hilbert problem:Invert φ : Γ 7→ ~|E(Γ)|−|V (Γ)| restricted to H. Then
(φ|?−1H ? φ)(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ ∈ G+
0 else
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 22
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The set of all algebra morphisms H → R[[~]] with the?-product forms a group.
The identity ε : H → R[[~]] maps the empty graph to 1 andall other graphs to 0.
The inverse of ψ : H → R[[~]] maybe calculated recursively:
ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ
s.t.γ∈P
ψ?−1(γ)ψ(Γ/γ)
Corresponds to Moebius-Inversion on the subgraph poset.
Simplifies on many (physical) cases to a (functional) inversionproblem on power series.
Solves the Riemann-Hilbert problem:Invert φ : Γ 7→ ~|E(Γ)|−|V (Γ)| restricted to H. Then
(φ|?−1H ? φ)(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ ∈ G+
0 else
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 22
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The set of all algebra morphisms H → R[[~]] with the?-product forms a group.
The identity ε : H → R[[~]] maps the empty graph to 1 andall other graphs to 0.
The inverse of ψ : H → R[[~]] maybe calculated recursively:
ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ
s.t.γ∈P
ψ?−1(γ)ψ(Γ/γ)
Corresponds to Moebius-Inversion on the subgraph poset.
Simplifies on many (physical) cases to a (functional) inversionproblem on power series.
Solves the Riemann-Hilbert problem:Invert φ : Γ 7→ ~|E(Γ)|−|V (Γ)| restricted to H. Then
(φ|?−1H ? φ)(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ ∈ G+
0 else
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 22
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The set of all algebra morphisms H → R[[~]] with the?-product forms a group.
The identity ε : H → R[[~]] maps the empty graph to 1 andall other graphs to 0.
The inverse of ψ : H → R[[~]] maybe calculated recursively:
ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ
s.t.γ∈P
ψ?−1(γ)ψ(Γ/γ)
Corresponds to Moebius-Inversion on the subgraph poset.
Simplifies on many (physical) cases to a (functional) inversionproblem on power series.
Solves the Riemann-Hilbert problem:Invert φ : Γ 7→ ~|E(Γ)|−|V (Γ)| restricted to H. Then
(φ|?−1H ? φ)(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ ∈ G+
0 else
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 22
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The set of all algebra morphisms H → R[[~]] with the?-product forms a group.
The identity ε : H → R[[~]] maps the empty graph to 1 andall other graphs to 0.
The inverse of ψ : H → R[[~]] maybe calculated recursively:
ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ
s.t.γ∈P
ψ?−1(γ)ψ(Γ/γ)
Corresponds to Moebius-Inversion on the subgraph poset.
Simplifies on many (physical) cases to a (functional) inversionproblem on power series.
Solves the Riemann-Hilbert problem:Invert φ : Γ 7→ ~|E(Γ)|−|V (Γ)| restricted to H. Then
(φ|?−1H ? φ)(Γ) =
{~|E(Γ)|−|V (Γ)| if Γ ∈ G+
0 else
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 22
![Page 91: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/91.jpg)
The identity van Suijlekom [2007], Yeats [2008],
∆X =∑
graphs Γ
∏v∈V (Γ)
X(|v |)P
⊗ Γ
|Aut Γ|,
where X =∑
graphs ΓΓ
|Aut Γ| and
X(k)P =
∑Γ
s.t.Γ∈P andΓ has k legs
Γ
|Aut Γ|.
can be used to make this accessible for asymptotic analysis:
φ|?−1H ? φ (X ) = m ◦
(φ|?−1H ⊗ φ
)◦∆X
=∑
graphs Γ
∏v∈V (Γ)
φ|?−1H
(X
(|v |)P
) φ(Γ)
|Aut Γ|
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 23
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The identity van Suijlekom [2007], Yeats [2008],
∆X =∑
graphs Γ
∏v∈V (Γ)
X(|v |)P
⊗ Γ
|Aut Γ|,
where X =∑
graphs ΓΓ
|Aut Γ| and
X(k)P =
∑Γ
s.t.Γ∈P andΓ has k legs
Γ
|Aut Γ|.
can be used to make this accessible for asymptotic analysis:
φ|?−1H ? φ (X ) = m ◦
(φ|?−1H ⊗ φ
)◦∆X
=∑
graphs Γ
∏v∈V (Γ)
φ|?−1H
(X
(|v |)P
) φ(Γ)
|Aut Γ|
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 23
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φ|?−1H ? φ (X ) =
∑graphs Γ
∏v∈V (Γ)
φ|?−1H
(X
(|v |)P
) φ(Γ)
|Aut Γ|
The generating function of all graphs with arbitrary weight foreach vertex degree is
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
λkk!
xk
~
=∑
graphs Γ
∏v∈V (Γ)
λ|v |
~|E(Γ)|−|V (Γ)|
|Aut Γ|
Therefore,
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 24
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φ|?−1H ? φ (X ) =
∑graphs Γ
∏v∈V (Γ)
φ|?−1H
(X
(|v |)P
) φ(Γ)
|Aut Γ|
The generating function of all graphs with arbitrary weight foreach vertex degree is
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
λkk!
xk
~
=∑
graphs Γ
∏v∈V (Γ)
λ|v |
~|E(Γ)|−|V (Γ)|
|Aut Γ|
Therefore,
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 24
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φ|?−1H ? φ (X ) =
∑graphs Γ
∏v∈V (Γ)
φ|?−1H
(X
(|v |)P
) φ(Γ)
|Aut Γ|
The generating function of all graphs with arbitrary weight foreach vertex degree is
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
λkk!
xk
~
=∑
graphs Γ
∏v∈V (Γ)
λ|v |
~|E(Γ)|−|V (Γ)|
|Aut Γ|
Therefore,
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 24
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φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
In quantum field theories we want the set P to be all graphswith a bounded number of legs.
This corresponds to restrictions on the edge-connectivity ofthe graphs.
In renormalizable quantum field theories the expressions
φ|?−1H
(X
(k)P
)are relatively easy to expand.
The generating functions φ|?−1H
(X
(k)P
)are called
counterterms in this context.
They are (almost) the generating functions of the number ofprimitive elements of H:
Γ is primitive iff ∆Γ = Γ⊗ 1 + 1⊗ Γ.
We can obtain the full asymptotic expansions in these cases.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25
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φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
In quantum field theories we want the set P to be all graphswith a bounded number of legs.
This corresponds to restrictions on the edge-connectivity ofthe graphs.
In renormalizable quantum field theories the expressions
φ|?−1H
(X
(k)P
)are relatively easy to expand.
The generating functions φ|?−1H
(X
(k)P
)are called
counterterms in this context.
They are (almost) the generating functions of the number ofprimitive elements of H:
Γ is primitive iff ∆Γ = Γ⊗ 1 + 1⊗ Γ.
We can obtain the full asymptotic expansions in these cases.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25
![Page 98: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/98.jpg)
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
In quantum field theories we want the set P to be all graphswith a bounded number of legs.
This corresponds to restrictions on the edge-connectivity ofthe graphs.
In renormalizable quantum field theories the expressions
φ|?−1H
(X
(k)P
)are relatively easy to expand.
The generating functions φ|?−1H
(X
(k)P
)are called
counterterms in this context.
They are (almost) the generating functions of the number ofprimitive elements of H:
Γ is primitive iff ∆Γ = Γ⊗ 1 + 1⊗ Γ.
We can obtain the full asymptotic expansions in these cases.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25
![Page 99: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/99.jpg)
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
In quantum field theories we want the set P to be all graphswith a bounded number of legs.
This corresponds to restrictions on the edge-connectivity ofthe graphs.
In renormalizable quantum field theories the expressions
φ|?−1H
(X
(k)P
)are relatively easy to expand.
The generating functions φ|?−1H
(X
(k)P
)are called
counterterms in this context.
They are (almost) the generating functions of the number ofprimitive elements of H:
Γ is primitive iff ∆Γ = Γ⊗ 1 + 1⊗ Γ.
We can obtain the full asymptotic expansions in these cases.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25
![Page 100: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/100.jpg)
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
In quantum field theories we want the set P to be all graphswith a bounded number of legs.
This corresponds to restrictions on the edge-connectivity ofthe graphs.
In renormalizable quantum field theories the expressions
φ|?−1H
(X
(k)P
)are relatively easy to expand.
The generating functions φ|?−1H
(X
(k)P
)are called
counterterms in this context.
They are (almost) the generating functions of the number ofprimitive elements of H:
Γ is primitive iff ∆Γ = Γ⊗ 1 + 1⊗ Γ.
We can obtain the full asymptotic expansions in these cases.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25
![Page 101: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/101.jpg)
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
In quantum field theories we want the set P to be all graphswith a bounded number of legs.
This corresponds to restrictions on the edge-connectivity ofthe graphs.
In renormalizable quantum field theories the expressions
φ|?−1H
(X
(k)P
)are relatively easy to expand.
The generating functions φ|?−1H
(X
(k)P
)are called
counterterms in this context.
They are (almost) the generating functions of the number ofprimitive elements of H:
Γ is primitive iff ∆Γ = Γ⊗ 1 + 1⊗ Γ.
We can obtain the full asymptotic expansions in these cases.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25
![Page 102: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/102.jpg)
φ|?−1H ? φ (X ) =
∞∑n=0
~n(2n − 1)!![x2n]e
∑k≥0
xk
k!φ|?−1H
(X
(k)P
)~
In quantum field theories we want the set P to be all graphswith a bounded number of legs.
This corresponds to restrictions on the edge-connectivity ofthe graphs.
In renormalizable quantum field theories the expressions
φ|?−1H
(X
(k)P
)are relatively easy to expand.
The generating functions φ|?−1H
(X
(k)P
)are called
counterterms in this context.
They are (almost) the generating functions of the number ofprimitive elements of H:
Γ is primitive iff ∆Γ = Γ⊗ 1 + 1⊗ Γ.
We can obtain the full asymptotic expansions in these cases.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 25
![Page 103: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/103.jpg)
Examples
Expressed as densities over all multigraphs:
In ϕ3-theory (i.e. three-valent multigraphs):
P(Γ is 2-edge-connected) = e−1(1− 23
211n + . . .
).
P(Γ is cyclically 4-edge-connected) = e−103
(1− 133
31n + · · ·
)In ϕ4-theory (i.e. four-valent multigraphs):
P(Γ is cyclically 6-edge-connected) = e−154
(1− 126 1
n + · · ·).
Arbitrary high order terms can be obtained by iterativelysolving implicit equations.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 26
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Examples
Expressed as densities over all multigraphs:
In ϕ3-theory (i.e. three-valent multigraphs):
P(Γ is 2-edge-connected) = e−1(1− 23
211n + . . .
).
P(Γ is cyclically 4-edge-connected) = e−103
(1− 133
31n + · · ·
)In ϕ4-theory (i.e. four-valent multigraphs):
P(Γ is cyclically 6-edge-connected) = e−154
(1− 126 1
n + · · ·).
Arbitrary high order terms can be obtained by iterativelysolving implicit equations.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 26
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Examples
Expressed as densities over all multigraphs:
In ϕ3-theory (i.e. three-valent multigraphs):
P(Γ is 2-edge-connected) = e−1(1− 23
211n + . . .
).
P(Γ is cyclically 4-edge-connected) = e−103
(1− 133
31n + · · ·
)In ϕ4-theory (i.e. four-valent multigraphs):
P(Γ is cyclically 6-edge-connected) = e−154
(1− 126 1
n + · · ·).
Arbitrary high order terms can be obtained by iterativelysolving implicit equations.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 26
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Examples
Expressed as densities over all multigraphs:
In ϕ3-theory (i.e. three-valent multigraphs):
P(Γ is 2-edge-connected) = e−1(1− 23
211n + . . .
).
P(Γ is cyclically 4-edge-connected) = e−103
(1− 133
31n + · · ·
)
In ϕ4-theory (i.e. four-valent multigraphs):
P(Γ is cyclically 6-edge-connected) = e−154
(1− 126 1
n + · · ·).
Arbitrary high order terms can be obtained by iterativelysolving implicit equations.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 26
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Examples
Expressed as densities over all multigraphs:
In ϕ3-theory (i.e. three-valent multigraphs):
P(Γ is 2-edge-connected) = e−1(1− 23
211n + . . .
).
P(Γ is cyclically 4-edge-connected) = e−103
(1− 133
31n + · · ·
)In ϕ4-theory (i.e. four-valent multigraphs):
P(Γ is cyclically 6-edge-connected) = e−154
(1− 126 1
n + · · ·).
Arbitrary high order terms can be obtained by iterativelysolving implicit equations.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 26
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Examples
Expressed as densities over all multigraphs:
In ϕ3-theory (i.e. three-valent multigraphs):
P(Γ is 2-edge-connected) = e−1(1− 23
211n + . . .
).
P(Γ is cyclically 4-edge-connected) = e−103
(1− 133
31n + · · ·
)In ϕ4-theory (i.e. four-valent multigraphs):
P(Γ is cyclically 6-edge-connected) = e−154
(1− 126 1
n + · · ·).
Arbitrary high order terms can be obtained by iterativelysolving implicit equations.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 26
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Conclusions
The ‘zero-dimensional path integral’ is a convenient tool toenumerate multigraphs by excess.
Asymptotics are easily accessible: The asymptotic expansionalso enumerates graphs.
With Hopf algebra techniques restrictions on the set ofenumerated graphs can be imposed.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27
![Page 110: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/110.jpg)
Conclusions
The ‘zero-dimensional path integral’ is a convenient tool toenumerate multigraphs by excess.
Asymptotics are easily accessible: The asymptotic expansionalso enumerates graphs.
With Hopf algebra techniques restrictions on the set ofenumerated graphs can be imposed.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27
![Page 111: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/111.jpg)
Conclusions
The ‘zero-dimensional path integral’ is a convenient tool toenumerate multigraphs by excess.
Asymptotics are easily accessible: The asymptotic expansionalso enumerates graphs.
With Hopf algebra techniques restrictions on the set ofenumerated graphs can be imposed.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27
![Page 112: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/112.jpg)
Conclusions
The ‘zero-dimensional path integral’ is a convenient tool toenumerate multigraphs by excess.
Asymptotics are easily accessible: The asymptotic expansionalso enumerates graphs.
With Hopf algebra techniques restrictions on the set ofenumerated graphs can be imposed.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27
![Page 113: Hopf algebras and factorial divergent power series: … › ~kratt › esi4 › borinsky.pdfRenormalized asymptotic enumeration of Feynman diagrams, Annals of Physics, October 2017](https://reader033.vdocuments.net/reader033/viewer/2022060507/5f1fa9b7d395af03317e8163/html5/thumbnails/113.jpg)
EN Argyres, AFW van Hameren, RHP Kleiss, andCG Papadopoulos. Zero-dimensional field theory. The EuropeanPhysical Journal C-Particles and Fields, 19(3):567–582, 2001.
Edward A Bender and E Rodney Canfield. The asymptotic numberof labeled graphs with given degree sequences. Journal ofCombinatorial Theory, Series A, 24(3):296–307, 1978.
A Connes and D Kreimer. Renormalization in quantum field theoryand the Riemann–Hilbert Problem ii: The β-function,diffeomorphisms and the renormalization group.Communications in Mathematical Physics, 216(1):215–241,2001.
P Cvitanovic, B Lautrup, and RB Pearson. Number and weights ofFeynman diagrams. Physical Review D, 18(6):1939, 1978.
CA Hurst. The enumeration of graphs in the Feynman-Dysontechnique. In Proceedings of the Royal Society of London A:Mathematical, Physical and Engineering Sciences, volume 214,pages 44–61. The Royal Society, 1952.
MB. Generating asymptotics for factorially divergent sequences.arXiv preprint arXiv:1603.01236, 2016.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27
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MB. Renormalized asymptotic enumeration of feynman diagrams.arXiv preprint arXiv:1703.00840, 2017.
LG Molinari and N Manini. Enumeration of many-body skeletondiagrams. The European Physical Journal B-Condensed Matterand Complex Systems, 51(3):331–336, 2006.
Walter D van Suijlekom. Renormalization of gauge fields: A hopfalgebra approach. Communications in Mathematical Physics,276(3):773–798, 2007.
Karen Yeats. Growth estimates for dyson-schwinger equations.arXiv preprint arXiv:0810.2249, 2008.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27