Hopf algebras and factorial divergent power series: … › ~kratt › esi4 ›...

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

1borinsky@physik.hu-berlin.deM. 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.

Motivation

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(~) =

dx√2π~

(− x2

‘Formal’ ⇒ expand under the integral sign and integrate overGaussian term by term,

Zϕ3(~) :=

∞∑n=0

~n(2n − 1)!![x2n]ex3

Zϕ3(~) =

dx√2π~

(− x2

Zϕ3(~) :=

∞∑n=0

~n(2n − 1)!![x2n]ex3

Zϕ3(~) =

dx√2π~

(− x2

Zϕ3(~) :=

∞∑n=0

~n(2n − 1)!![x2n]ex3

Zϕ3(~) =

dx√2π~

(− x2

Zϕ3(~) :=

∞∑n=0

~n(2n − 1)!![x2n]ex3

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

24+ . . .

)= 1 +

1152~2 + . . .

where φ(Γ) = ~|E(Γ)|−|V (Γ)| maps a graph with excess n to ~n.

Zϕ3(~) =

∞∑n=0

~n(2n − 1)!![x2n]ex3

3!~ =∑

cubic graphs Γ

~|E(Γ)|−|V (Γ)|

|Aut Γ|

Zϕ3(~) = φ

24+ . . .

)= 1 +

1152~2 + . . .

Zϕ3(~) =

∞∑n=0

~n(2n − 1)!![x2n]ex3

3!~ =∑

cubic graphs Γ

~|E(Γ)|−|V (Γ)|

|Aut Γ|

Zϕ3(~) = φ

24+ . . .

)= 1 +

1152~2 + . . .

Zϕ3(~) =

∞∑n=0

~n(2n − 1)!![x2n]ex3

3!~ =∑

cubic graphs Γ

~|E(Γ)|−|V (Γ)|

|Aut Γ|

Zϕ3(~) = φ

24+ . . .

)= 1 +

1152~2 + . . .

Arbitrary degree distribution

More general with arbitrary degree distribution,

Z (~) =∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

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].

Z (~) =∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

graphs Γ

~|E(Γ)|−|V (Γ)|∏

v∈V (Γ) λ|v |

|Aut Γ|

Z (~) =∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

graphs Γ

~|E(Γ)|−|V (Γ)|∏

v∈V (Γ) λ|v |

|Aut Γ|

Z (~) =∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

graphs Γ

~|E(Γ)|−|V (Γ)|∏

v∈V (Γ) λ|v |

|Aut Γ|

We obtain a series of polynomials

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

~ = φ(

24+ · · ·

)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏

v∈V (Γ) λ|v |

1152λ4

3λ4 +35

384λ2

48λ3λ5 +

)~2 + · · ·

=∞∑n=0

Pn(λ3, λ4, . . .)~n

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

~ = φ(

24+ · · ·

)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏

v∈V (Γ) λ|v |

1152λ4

3λ4 +35

384λ2

48λ3λ5 +

)~2 + · · ·

=∞∑n=0

Pn(λ3, λ4, . . .)~n

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

~ = φ(

24+ · · ·

)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏

v∈V (Γ) λ|v |

1152λ4

3λ4 +35

384λ2

48λ3λ5 +

)~2 + · · ·

=∞∑n=0

Pn(λ3, λ4, . . .)~n

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

~ = φ(

24+ · · ·

)where φ(Γ) = ~|E(Γ)|−|V (Γ)|∏

v∈V (Γ) λ|v |

1152λ4

3λ4 +35

384λ2

48λ3λ5 +

)~2 + · · ·

=∞∑n=0

Pn(λ3, λ4, . . .)~n

Example

Take the degree distribution λk = −1 for all k ≥ 3.

Z stir(~) :=∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

−1k!

8+ . . .

)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|

8(−1)2 +

12(−1)2 +

8(−1)1

)~ + . . .

= 1 +1

288~2 − 139

51840~3 + . . . = e

∑∞n=1

Bn+1n(n+1)

⇒ Stirling series as a signed sum over multigraphs.

Example

Z stir(~) :=∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

−1k!

8+ . . .

)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|

8(−1)2 +

12(−1)2 +

8(−1)1

)~ + . . .

= 1 +1

288~2 − 139

51840~3 + . . . = e

∑∞n=1

Bn+1n(n+1)

Example

Z stir(~) :=∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

−1k!

8+ . . .

)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|

8(−1)2 +

12(−1)2 +

8(−1)1

)~ + . . .

= 1 +1

288~2 − 139

51840~3 + . . . = e

∑∞n=1

Bn+1n(n+1)

Example

Z stir(~) :=∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

−1k!

8+ . . .

)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|

8(−1)2 +

12(−1)2 +

8(−1)1

)~ + . . .

= 1 +1

288~2 − 139

51840~3 + . . .

= e∑∞

n=1Bn+1n(n+1)

Example

Z stir(~) :=∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

−1k!

8+ . . .

)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|

8(−1)2 +

12(−1)2 +

8(−1)1

)~ + . . .

= 1 +1

288~2 − 139

51840~3 + . . . = e

∑∞n=1

Bn+1n(n+1)

Example

Z stir(~) :=∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥3

−1k!

8+ . . .

)where φ(Γ) = (−1)|V (Γ)|~|E(Γ)|−|V (Γ)|

8(−1)2 +

12(−1)2 +

8(−1)1

)~ + . . .

= 1 +1

288~2 − 139

51840~3 + . . . = e

∑∞n=1

Bn+1n(n+1)

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

F maps a degree sequence to the corresponding generatingfunction of multigraphs.

F : S(x) 7→∞∑n=0

~n(2n − 1)!![x2n]ex2

2 +S(x)

where S(x) = − x2

2 +∑

k≥3λkk! x

F : S(x) 7→∞∑n=0

~n(2n − 1)!![x2n]ex2

2 +S(x)

where S(x) = − x2

2 +∑

k≥3λkk! x

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

2= −S(x(y)).

Expressions as

F [S(x)] =∞∑n=0

~n(2n − 1)!![x2n]ex2

2 +S(x)

Theorem

F [S(x)] =∞∑n=0

~n(2n + 1)!![y2n+1]x(y),

2= −S(x(y)).

Expressions as

F [S(x)] =∞∑n=0

~n(2n − 1)!![x2n]ex2

2 +S(x)

Theorem

F [S(x)] =∞∑n=0

~n(2n + 1)!![y2n+1]x(y),

2= −S(x(y)).

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,

2= −S(x).

The dominant singularity of x(y) coincideswith a branch-cut of the local parametriza-tion of the curve.

Asymptotics

2= −S(x).

Asymptotics

2= −S(x).

Asymptotics

2= −S(x).

Asymptotics

2= −S(x).

−4 −2 0 2 4

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.

−4 −2 0 2 4

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.

⇒ Near the dominant singularity:

x(y)− τ = −d1

√1− y

ρ+∞∑j=2

(1− y

[yn]x(y) ∼ Cρ−nn−32

∞∑k=1

Therefore,

[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)

=2nΓ(n + 3

2 )√π

[y2n+1]x(y)

∼ C ′2nρ−2nΓ(n)

∞∑k=1

e ′knk

What is the value of C ′ and the e ′k?

x(y)− τ = −d1

√1− y

ρ+∞∑j=2

(1− y

∞∑k=1

Therefore,

[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)

=2nΓ(n + 3

2 )√π

[y2n+1]x(y)

∞∑k=1

e ′knk

x(y)− τ = −d1

√1− y

ρ+∞∑j=2

(1− y

∞∑k=1

Therefore,

[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)

=2nΓ(n + 3

2 )√π

[y2n+1]x(y)

∞∑k=1

e ′knk

x(y)− τ = −d1

√1− y

ρ+∞∑j=2

(1− y

∞∑k=1

Therefore,

[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)

=2nΓ(n + 3

2 )√π

[y2n+1]x(y)

∞∑k=1

e ′knk

x(y)− τ = −d1

√1− y

ρ+∞∑j=2

(1− y

∞∑k=1

Therefore,

[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)

=2nΓ(n + 3

2 )√π

[y2n+1]x(y)

∞∑k=1

e ′knk

x(y)− τ = −d1

√1− y

ρ+∞∑j=2

(1− y

∞∑k=1

Therefore,

[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)

=2nΓ(n + 3

2 )√π

[y2n+1]x(y)

∞∑k=1

e ′knk

x(y)− τ = −d1

√1− y

ρ+∞∑j=2

(1− y

∞∑k=1

Therefore,

[~n]F [S(x)] = (2n + 1)!![y2n+1]x(y)

=2nΓ(n + 3

2 )√π

[y2n+1]x(y)

∞∑k=1

e ′knk

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

2π[~k ]F [S(τ)− S(x + τ)](−~).

⇒ The asymptotic expansion can be expressed as a generatingfunction of graphs.

singularity of y2

2 = −S(x). Then

[~n]F [S(x)](~) =R−1∑k=0

(A−nΓ(n − R)

2π[~k ]F [S(τ)− S(x + τ)](−~).

singularity of y2

2 = −S(x). Then

[~n]F [S(x)](~) =R−1∑k=0

(A−nΓ(n − R)

2π[~k ]F [S(τ)− S(x + τ)](−~).

Example

For cubic graphs or equivalently ϕ3 theory, we are interestedin S(x) = − x2

2 + x3

F [S(x)] (~) = φ(1 +

128+ . . .

1152~2 +

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)](−~) =

2πF [−x2

3!](−~)

(1− 5

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)).

Example

2 + x3

F [S(x)] (~) = φ(1 +

128+ . . .

1152~2 +

82944~3 + · · ·

expansion

∞∑k=0

ck~k =1

2πF [S(τ)− S(τ + x)](−~) =

2πF [−x2

3!](−~)

(1− 5

1152~2 − 85085

82944~3 + . . .

Example

2 + x3

F [S(x)] (~) = φ(1 +

128+ . . .

1152~2 +

82944~3 + · · ·

expansion

∞∑k=0

ck~k =1

2πF [S(τ)− S(τ + x)](−~) =

2πF [−x2

3!](−~)

(1− 5

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)).

Example

2 + x3

F [S(x)] (~) = φ(1 +

128+ . . .

1152~2 +

82944~3 + · · ·

expansion

∞∑k=0

ck~k =1

2πF [S(τ)− S(τ + x)](−~) =

2πF [−x2

3!](−~)

(1− 5

1152~2 − 85085

82944~3 + . . .

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.

fn =R−1∑k=0

logF [S(x)] (~)

in closed form.

fn =R−1∑k=0

logF [S(x)] (~)

in closed form.

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.

Renormalization

φP(Γ) =

0 else

Renormalization

φP(Γ) =

0 else

Renormalization

φP(Γ) =

0 else

Renormalization

φP(Γ) =

0 else

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.

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) ◦∆.

if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)

∅ ∈ P (3)

∆ : G → H⊗ G

Γ 7→∑γ⊂Γ

s.t.γ∈P

γ ⊗ Γ/γ

if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)

∅ ∈ P (3)

∆ : G → H⊗ G

Γ 7→∑γ⊂Γ

s.t.γ∈P

γ ⊗ Γ/γ

if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)

∅ ∈ P (3)

∆ : G → H⊗ G

Γ 7→∑γ⊂Γ

s.t.γ∈P

γ ⊗ Γ/γ

if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)

∅ ∈ P (3)

∆ : G → H⊗ G

Γ 7→∑γ⊂Γ

s.t.γ∈P

γ ⊗ Γ/γ

if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)

∅ ∈ P (3)

∆ : G → H⊗ G

Γ 7→∑γ⊂Γ

s.t.γ∈P

γ ⊗ Γ/γ

if γ1, γ2 ∈ P then γ1 ∪ γ2 ∈ P (2)

∅ ∈ P (3)

∆ : G → H⊗ G

Γ 7→∑γ⊂Γ

s.t.γ∈P

γ ⊗ Γ/γ

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.

∆ =∑γ⊂s.t.γ∈P

γ ⊗ /γ = 1⊗ + 3 ⊗ + ⊗

∆ =∑γ⊂s.t.γ∈P

γ ⊗ /γ = 1⊗ + 3 ⊗ + ⊗

∆ =∑γ⊂s.t.γ∈P

γ ⊗ /γ = 1⊗ + 3 ⊗ + ⊗

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)

ψ ? φ : G → R[[~]] ψ ? φ = m ◦ (ψ ⊗ φ) ◦∆

ξ ? ψ : H → R[[~]] ξ ? ψ = m ◦ (ξ ⊗ ψ) ◦∆

(ξ ? ψ) ? φ = ξ ? (ψ ? φ)

ψ ? φ : G → R[[~]] ψ ? φ = m ◦ (ψ ⊗ φ) ◦∆

ξ ? ψ : H → R[[~]] ξ ? ψ = m ◦ (ξ ⊗ ψ) ◦∆

(ξ ? ψ) ? φ = ξ ? (ψ ? φ)

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

ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ

s.t.γ∈P

ψ?−1(γ)ψ(Γ/γ)

(φ|?−1H ? φ)(Γ) =

{~|E(Γ)|−|V (Γ)| if Γ ∈ G+

0 else

ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ

s.t.γ∈P

ψ?−1(γ)ψ(Γ/γ)

(φ|?−1H ? φ)(Γ) =

{~|E(Γ)|−|V (Γ)| if Γ ∈ G+

0 else

ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ

s.t.γ∈P

ψ?−1(γ)ψ(Γ/γ)

(φ|?−1H ? φ)(Γ) =

{~|E(Γ)|−|V (Γ)| if Γ ∈ G+

0 else

ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ

s.t.γ∈P

ψ?−1(γ)ψ(Γ/γ)

(φ|?−1H ? φ)(Γ) =

{~|E(Γ)|−|V (Γ)| if Γ ∈ G+

0 else

ψ?−1(Γ) = −ψ(Γ)−∑γ(Γ

s.t.γ∈P

ψ?−1(γ)ψ(Γ/γ)

(φ|?−1H ? φ)(Γ) =

{~|E(Γ)|−|V (Γ)| if Γ ∈ G+

0 else

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 =

s.t.Γ∈P andΓ has k legs

|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

(|v |)P

) φ(Γ)

|Aut Γ|

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 =

s.t.Γ∈P andΓ has k legs

|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

(|v |)P

) φ(Γ)

|Aut Γ|

φ|?−1H ? φ (X ) =

∑graphs Γ

∏v∈V (Γ)

φ|?−1H

(|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

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

k!φ|?−1H

φ|?−1H ? φ (X ) =

∑graphs Γ

∏v∈V (Γ)

φ|?−1H

(|v |)P

) φ(Γ)

|Aut Γ|

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

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

k!φ|?−1H

φ|?−1H ? φ (X ) =

∑graphs Γ

∏v∈V (Γ)

φ|?−1H

(|v |)P

) φ(Γ)

|Aut Γ|

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

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

k!φ|?−1H

φ|?−1H ? φ (X ) =

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

k!φ|?−1H

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

)are relatively easy to expand.

The generating functions φ|?−1H

)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.

φ|?−1H ? φ (X ) =

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

k!φ|?−1H

φ|?−1H

)are called

φ|?−1H ? φ (X ) =

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

k!φ|?−1H

φ|?−1H

)are called

φ|?−1H ? φ (X ) =

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

k!φ|?−1H

φ|?−1H

)are called

φ|?−1H ? φ (X ) =

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

k!φ|?−1H

φ|?−1H

)are called

φ|?−1H ? φ (X ) =

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

k!φ|?−1H

φ|?−1H

)are called

φ|?−1H ? φ (X ) =

∞∑n=0

~n(2n − 1)!![x2n]e

∑k≥0

k!φ|?−1H

φ|?−1H

)are called

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):

(1− 126 1

n + · · ·).

Arbitrary high order terms can be obtained by iterativelysolving implicit equations.

Examples

211n + . . .

(1− 133

31n + · · ·

(1− 126 1

n + · · ·).

Examples

211n + . . .

(1− 133

31n + · · ·

(1− 126 1

n + · · ·).

Examples

211n + . . .

(1− 133

31n + · · ·

In ϕ4-theory (i.e. four-valent multigraphs):

(1− 126 1

n + · · ·).

Examples

211n + . . .

(1− 133

31n + · · ·

(1− 126 1

n + · · ·).

Examples

211n + . . .

(1− 133

31n + · · ·

(1− 126 1

n + · · ·).

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.

Conclusions

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.

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.

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