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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
−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
−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
⇒ 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
⇒ 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
⇒ 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
⇒ 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
⇒ 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
⇒ 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
⇒ 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
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.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 14
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.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 14
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.
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 14
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)).
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 15
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)).
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 15
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)).
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 15
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)).
M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 15
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27
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M. Borinsky (HU Berlin) Algebraic tools for graphical enumeration 27