Edgecolored graphs as higher-dimensional maps Valentin Bonzom LIPN, - - PowerPoint PPT Presentation

Edge–colored graphs as higher-dimensional maps

Valentin Bonzom

LIPN, Paris University 13 October 17, 2016 RGP2016

Discretization of manifolds

◮ 2D discrete surfaces: triangulations, p–angulations and

combinatorial maps

◮ 3D triangulations: gluings of tetrahedra

Discretization of manifolds

◮ 2D discrete surfaces: triangulations, p–angulations and

combinatorial maps

◮ 3D triangulations: gluings of tetrahedra ◮ How to represent them in a suitable fashion for combinatorics? ◮ Equivalent of p–angulations? ◮ Enumeration? ◮ Topological recursion, integrable hierarchies?

Combinatorial maps

Graph with cyclic ordering of edges incident to each vertex =

Combinatorial maps

Graph with cyclic ordering of edges incident to each vertex = Cyclic ordering defines faces: follow the corners

2p–angulation

◮ Faces of degree 2p ◮ Duality: vertices of degree 2p ◮ Euler’s relation with E(M) = pV (M)

F(M) − E(M) + V (M) = F(M) − (p − 1)V (M) = 2 − 2g(M)

Two properties of the genus to generalize

◮ g(M) ≥ 0

⇒ bound on F(M) linear in V (M)

◮ Maximizing F(M) at fixed V (M) equivalent to g(M) = 0

Main challenge: generalize that!

2p–angulation

◮ Faces of degree 2p ◮ Duality: vertices of degree 2p ◮ Euler’s relation with E(M) = pV (M)

F(M) − E(M) + V (M) = F(M) − (p − 1)V (M) = 2 − 2g(M)

Two properties of the genus to generalize

◮ g(M) ≥ 0

⇒ bound on F(M) linear in V (M)

◮ Maximizing F(M) at fixed V (M) equivalent to g(M) = 0

Main challenge: generalize that!

2p–angulation

◮ Faces of degree 2p ◮ Duality: vertices of degree 2p ◮ Euler’s relation with E(M) = pV (M)

F(M) − E(M) + V (M) = F(M) − (p − 1)V (M) = 2 − 2g(M)

Two properties of the genus to generalize

◮ g(M) ≥ 0

⇒ bound on F(M) linear in V (M)

◮ Maximizing F(M) at fixed V (M) equivalent to g(M) = 0

Main challenge: generalize that!

What do we know? Maps: from Tutte to today

Enumeration

◮ Count maps with possible decorations (Ising, Potts, loops) ◮ Exact generating functions or their properties ◮ Random matrix model techniques ◮ Bijections [Cori-Vauquelin-Schaeffer, Bouttier-Di Francesco-Guitter] ◮ Tutte’s equations and topological recursion [Borot, Eynard, Orantin]

Physics motivations and applications

◮ Two-dimensional quantum gravity coupled to matter ◮ Liouville theory coupled to conformal field theories ◮ Statistical mechanics ◮ Celebrated KPZ relations

What do we know? II

Geometric applications of bijections

◮ Two-point, three-point functions ◮ Local limit ◮ Continuum limit [Brownian sphere]

Enumerative geometry

◮ Intersection numbers [Kontsevich-Witten] ◮ Hurwitz numbers ◮ Integrable hierarchies, etc.

In higher dimensions

First approach based on generalizing random matrices

ln

• HN×N

dM e−N(tr M2+t tr M4) =

• quadrangulations Q

N2−2g(Q)tn(Q)

◮ From matrices to tensors

Ma1a2 − → Ta1a2...ad

[Ambjorn-Durhuus-Jonsson, Gross, Sasakura] all in 91

◮ Tensor integrals generate

stranded graphs Local rules

Tabc Tcde Tebf Tfda Tabc Tdef

Problems with tensors and stranded graphs

3 origins of difficulty

◮ Random tensor techniques: non-existent

Contrast with random matrices: eigenvalues and orthogonal polynomials e.g.

◮ So look at the stranded graphs directly?

Problems with tensors and stranded graphs

3 origins of difficulty

◮ Random tensor techniques: non-existent

Contrast with random matrices: eigenvalues and orthogonal polynomials e.g.

◮ So look at the stranded graphs directly? ◮ Topology: One can associate topology to graph, very singular! ◮ Combinatorics

◮ Closed strands represent simplices of codimension 2

ex: vertices in 2D, edges in 3D, triangles in 4D, etc.

◮ One needs to count/classify graphs w.r.t. number of closed strands

Nearly impossible!

◮ As a result, essential missing piece is generalization of genus!

No analytical progress for 20 years!

Objectives

Objectives: Family of triangulations s.t.

◮ No topological restriction ◮ Extension of genus: degree ω(T) of T

∆d−2(T) − α∆d(T) = d − ω(T), ω(T) ≥ 0

◮ Infinite number of triangulations at fixed ω ◮ Balance on α

Possible with colored triangulations!

How to represent triangulations?

Triangulations

◮ Gluing of simplices (tetrahedra, pentachora, etc.) ◮ Defined by attaching maps ◮ Ensemble of triangulations defined by constraints on attaching maps

Various ensembles

◮ Various ensembles in topology (simplicial, CW, ∆–complexes, etc.) ◮ Not suitable for combinatorics (too wild) ◮ Digging through old work, found colored triangulations

Crystallization, graph–encoded manifold: Cristofori’s talk!

◮ Nice for combinatorics: represented by edge–colored graphs

Colored graphs

(d + 1)-colored graphs

◮ Bipartite graphs

black and white vertices

◮ Edges colored with d + 1

possible colors

◮ Vertices of degree d + 1 ◮ All colors incident exactly

• nce at each vertex

1 2 1 3 1 2 3 2 1 3 3 3 2 2 1 1 1 2 2 3 3 2 2 1 1 1 2 3 3 1 2 2 3 1

Colored graphs

(d + 1)-colored graphs

◮ Bipartite graphs

black and white vertices

◮ Edges colored with d + 1

possible colors

◮ Vertices of degree d + 1 ◮ All colors incident exactly

• nce at each vertex

1 1 1 2 2 2 3 3 3

Faces are closed cycles with only two colors.

Colored graphs

(d + 1)-colored graphs

◮ Bipartite graphs

black and white vertices

◮ Edges colored with d + 1

possible colors

◮ Vertices of degree d + 1 ◮ All colors incident exactly

• nce at each vertex

1 1 1 2 2 2 3 3 3

Faces are closed cycles with only two colors.

Colored graphs

(d + 1)-colored graphs

◮ Bipartite graphs

black and white vertices

◮ Edges colored with d + 1

possible colors

◮ Vertices of degree d + 1 ◮ All colors incident exactly

• nce at each vertex

1 1 1 2 2 2 3 3 3

Faces are closed cycles with only two colors.

Triangulations from colored graphs

duality              vertex → d–simplex edge → (d − 1)–simplex face → (d − 2)–simplex k-bubble → (d − k)–simplex

◮ Boundary triangles labeled

by a color c = 0, . . . , d

1 2 3

Colors identify all sub-simplices

Triangulations from colored graphs

duality              vertex → d–simplex edge → (d − 1)–simplex face → (d − 2)–simplex k-bubble → (d − k)–simplex

◮ Boundary triangles labeled

by a color c = 0, . . . , d

◮ Induced colorings ◮ Edges labeled by pair of

colors

1 2 3

12 01 23 03 02 13 Colors identify all sub-simplices

Triangulations from colored graphs

duality              vertex → d–simplex edge → (d − 1)–simplex face → (d − 2)–simplex k-bubble → (d − k)–simplex

◮ Boundary triangles labeled

by a color c = 0, . . . , d

◮ Induced colorings ◮ Edges labeled by pair of

colors

◮ Nodes labeled by three colors

1 2 3

12 01 23 03 02 13

123 013 012 023

Colors identify all sub-simplices

Colored attaching maps

Gluing respecting all induced colorings

012 013 013

1 2 1 2

02

3 3

02 03 03

023 023

01 01

012

Theory of crystallization and GEMs (graph–encoded manifolds): (d + 1)-colored graphs are dual to triangulations of pseudo-manifolds of dimension d

[Pezzana, Ferri, Gagliardi, Cristofori, Casali, Lins].

The 2D case

1 1 1 2 2 2 1 2 1 1 2 2

The 2D case

1 1 1 2 2 2 1 2 1 1 2 2

The 2D case

1 1 1 2 2 2 1 2 1 1 2 2

The 2D case

1 1 1 2 2 2 1 2 1 1 2 2

2p–angle

◮ Gluing of 2p triangles with boundary of color 0 ◮ Dually: Components with all colors but 0

The 3D case

3 3 3 3 1 1 1 1 2 2 2 2

The 3D case

3 3 3 3 1 1 1 1 2 2 2 2

The 3D case

3 3 3 3 1 1 1 1 2 2 2 2

The 3D case

3 3 3 3 1 1 1 1 2 2 2 2

The 3D case

3 3 3 3 1 1 1 1 2 2 2 2

Bubbles as building blocks

◮ Colored graph with colors 0, 1, . . . , d

(triangulation in dim d)

◮ Bubble: connected piece with colors 1, . . . , d

Obtained by removing the color 0

◮ All graphs obtained by gluing bubbles along edges of color 0 ◮ G(B) set of (d + 1)-colored graphs where all bubbles are B

1 3 4 2 1 3 4 2 1 3 4 2

Bubbles II

◮ 2D: only bubbles with 2p vertices

Cycles of colors (1, 2)

1 2

Bubbles II

◮ 2D: only bubbles with 2p vertices

Cycles of colors (1, 2)

1 2 ◮ Many more in higher dimensions ◮ Vast world to explore 1 2 1 3 1 2 3 2 1 3 3 3 2 2 1 1 1 2 2 3 3 2 2 1 1 1 2 3 3 1 2 2 3 1

1 2 1 1 1 1 2 2 2 2 1 3 4 3 3 3 3 4 4 4 4

Faces

1 1 1 2 2 2 1 2 1 1 2 2 1

Vertices: two types

◮ cycle with colors (0, 1) ◮ cycle with colors (0, 2)

The problem

◮ Set B a bubble, G ∈ G(B) ◮ Enumerate w.r.t.

◮ # bubbles b(G) ◮ # subsimplices of codimension 2 which belong to bubble boundary

◮ Face of colors (0, c): cycle with colors (0, c)

Number of faces F(G) =

d

• c=1

F0c(G)

◮ Classify graphs according to F(G) at fixed b(G)

Gb(B) =

• F

G(F)

b

(B)

◮ Focus on G(F) b

(B) How to maximize F(G) at fixed number of bubbles b(G)?

Gurau’s degree theorem

Bound on F(G)

There exists ω(G) ≥ 0 F(G) − (d − 1)(p(B) − 1)b(G) = d − ω(G) ≤ d

◮ d = 2

F(G) − (p(B) − 1)b(G) = 2 − ω(G) ⇒ ω(G) = 2g(G)

◮ For d ≥ 3, bound can be saturated only for certain type of bubbles ◮ Maximizing graphs (melonic) are series–parallel

◮ Bijection with trees ◮ Expected from numerics

◮ Gurau–Schaeffer classification according to the degree

Towards other behaviors

◮ Colored graphs built from non-melonic bubbles grow fewer faces ◮ Colored triangulations built from non-melonic bubbles grow fewer

simplices of codimension 2

◮ Need a bubble-dependent degree

F(G) − αBb(G) = d − ωB(G) ≤ d with αB > (d − 1)(p(B) − 1)

◮ Finding αB is challenging!

Results in a nutshell

◮ Set of graphs G(B) ◮ Count number of B with parameter t ◮ Denote B′ the generating function with marked bubble B′

Using Gurau’s degree

◮ Gurau’s degree theorem and Gurau-Schaeffer classification w.r.t.

Gurau’s degree

◮ Gurau’s universality theorem

B′ = G(t)p(B′)

◮ Coupled with Schwinger-Dyson/loop/Tutte-like equations, get

algebraic equation on G(t)

Results in a nutshell

1-bubble graphs

◮ Take a bubble B ◮ Look at all ways to glue boundary triangles 2 by 2 ◮ Sum over its perfect matchings

B =

• perfect matchings

NF(G)

◮ Equivalent of unicellular maps in 2D

n−cycle = Harer-Zagier polynomial(N) = Nn+1 Catalan(n)+O(1/N)

• ◮ Various behaviors
• 4

1 3 2 3 4 3 4 2 1 2 1

• = Catalan(n)

Results in a nutshell

1-bubble graphs

◮ Take a bubble B ◮ Look at all ways to glue boundary triangles 2 by 2 ◮ Sum over its perfect matchings

B =

• perfect matchings

NF(G)

◮ Equivalent of unicellular maps in 2D

n−cycle = Harer-Zagier polynomial(N) = Nn+1 Catalan(n)+O(1/N)

• ◮ Various behaviors
• 3

4 4 2 2 1 2 1 1 4

• = Motzkin(n)

Results in a nutshell

Multiple Non-melonic bubbles

◮ Bubble-dependent degree for a few families and some specific case ◮ Almost-melonic bubbles in arbitrary d ◮ Even dimensions

◮ 4 1 3 2 3 4 3 4 2 1 2 1

reproduces all behaviors of combinatorial maps

◮ Bubble-dependent degree is a genus!

◮ Some specific case in d = 3 [Octahedron in Lionni’s talk] ◮ Maximizing number of faces: universality class of random trees

Methods

Identify graphs of vanishing bubble-dependent degree

◮ Bijection with stuffed hypermaps [VB-Lionni-Rivasseau 2015] ◮ Extends Tutte’s bijection between bipartite quadrangulations and

generic maps

◮ Use bijection to find graphs of vanishing bubble-dependent degree

Counting

◮ Schwinger-Dyson/loop equations exist!

Beyond graphs of vanishing bubble-dependent degree

◮ Topological recursion [VB-Dartois, to appear]

Quartic case, d = 4

i

→

i i 1

→

1i

Universal part

Cycles of color 0 and pairs of vertices → counter–clockwise star–map

j k1 k2 i1 i2 i3 e

ρi ρj ρk

→

i1i2i3 k1k2 j e ρi ρk ρj

Quartic case, d = 4

◮ Maps of arbitrary degree ◮ Monocolored edges, colors 1, . . . , d ◮ Bicolored edges, colors 1c for c = 2, 3, 4

Maximizing faces d

c=1 F0c

◮ Monocolored edges are bridges ◮ Bicolored form planar components ◮ Bicolored types 1c and 1c′ touch on cut–vertices

(similar to O(n) model on planar maps)

The quartic case

◮ Generating function of (rooted) maps for k types of bicolored edges

fk(t, λ) =

• M

t#edges λ#monocol. edges

◮ Algebraicity

fk(t, λ) = 1 − k + tλfk(t, λ)2 + kP(tfk(t, λ)2) implies

• tf 2 = u(1 − u)2

f = k(1 − u)(1 + 3u) − k + 1 + λu(1 − u)2

◮ Generic planar maps for λ = 0 and k = 1

27t2A(t)2 + (1 − 18t)A(t) + 16t − 1 = 0

Explicit singularity analysis for k = 1

◮ Quartic eq on f (t, λ) ◮ For λ < 3, singularity at t1(λ) = 27 4(λ+9)2

f (t, λ) = 4 27(λ + 9) + 16(λ + 3)(λ + 9)3 729(λ − 3) (t1(λ) − t) + 64(λ + 9)11/2 6561(3 − λ)5/2 (t1(λ) − t)3/2 + o

• (t1(λ) − t)3/2

◮ For λ > 3, singularity at t2(λ) = λ 4(1+λ)2

f (t, λ) = 2λ2 − 1 λ2 −4(1 + λ)2 λ5/2

• λ2 − 2λ − 3 (t2(λ)−t)1/2+o
• (t2(λ)−t)1/2

◮ λ = 3, proliferation of baby universes

f (t, λ = 3) = 16 9 − 128 35/3 3 64 − t 2/3 + o 3 64 − t 2/3

Same results with respect to k

◮ k small enough: universality class of maps ◮ k large enough: branching process and square–root singularity ◮ k critical: singularity exponent 2/3 0.02 0.04 0.06 0.08 t 1.2 1.4 1.6 1.8 2.0 2.2 fk k 9

5

k 1 k 4

tk 1,fk 1 tk 2,fk 2

Conclusion

◮ Colored triangulations are genuine generalization of maps ◮ Admit generalization of genus, but bubble-dependent ◮ Universality classes depend on bubbles too, unlike 2D ◮ (At least some) Enumeration is feasible in dim d > 2! ◮ To appear with L. Lionni: enumeration of gluings of octahedra

which maximize the number of edges

◮ Beyond maximizing number of faces in quartic case

→ Topological recursion! [to appear w/ S. Dartois]

◮ More to be studied ◮ Harer–Zagier formula equivalent for unicellular maps?