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Vorono¨ ı Tessellations in the CRT and Continuum Random Maps of Finite Excess

SODA 2018, New Orleans.

Guillaume Chapuy (CNRS – IRIF Paris Diderot) ´ Eric Fusy (CNRS – LIX ´

Ecole Polytechnique)

Christina Goldschmidt (Oxford) Omer Angel (UBC Vancouver) Louigi Addario-Berry (McGill Montr´

eal) Work supported by the grant ERC – Stg 716083 – “CombiTop”

Vorono¨ ı Tessellations in the CRT and Continuum Random Maps of Finite Excess

SODA 2018, New Orleans.

Guillaume Chapuy (CNRS – IRIF Paris Diderot) ´ Eric Fusy (CNRS – LIX ´

Ecole Polytechnique)

Christina Goldschmidt (Oxford) Omer Angel (UBC Vancouver) Louigi Addario-Berry (McGill Montr´

eal) Work supported by the grant ERC – Stg 716083 – “CombiTop”

The Vorono¨ ı vector – main definition of the talk!

• Let Gn be your favorite random graph with n vertices (n → ∞)

Pick k points v1, v2, . . . , vk uniformly at random (k fixed) and call Vi = {x ∈ V (G), d(x, vi) = minj d(x, vj)}

(in case of equality, assign to a random Vi among possible choices)

the i − th Vorono¨ ı cell

The Vorono¨ ı vector – main definition of the talk!

• Let Gn be your favorite random graph with n vertices (n → ∞)

Pick k points v1, v2, . . . , vk uniformly at random (k fixed) and call Vi = {x ∈ V (G), d(x, vi) = minj d(x, vj)}

• Question: what is the limit law of the “Vorono¨

ı vector” ( |V1|

n , |V2| n , . . . , |Vk| n ) ?

player 1 player 2 player 1 player 2

√n

Cycle: deterministic (1

2, 1 2)

Examples with k = 2

“√n × √n-star“: winner takes (almost) all

(in case of equality, assign to a random Vi among possible choices)

the i − th Vorono¨ ı cell

δ1

2,1 2

1 2δ0,1 + 1 2δ1,0

Conjecture and results

• Conjecture [C., published in 2017]

For a random embedded graph of genus g ≥ 0 and any k ≥ 2, the limit law is uniform on the k-simplex. OPEN EVEN FOR PLANAR GRAPHS. In particular for k = 2 points, each of them gets a U[0, 1] fraction of the mass.

Conjecture and results

• Conjecture [C., published in 2017]

For a random embedded graph of genus g ≥ 0 and any k ≥ 2, the limit law is uniform on the k-simplex. OPEN EVEN FOR PLANAR GRAPHS.

• Theorem [Guitter 2017]

True for (g, k) = (0, 2) – two points on planar graph In particular for k = 2 points, each of them gets a U[0, 1] fraction of the mass.

(proof uses sharp tools from planar map enumeration and computer assisted calculations)

Conjecture and results

• Conjecture [C., published in 2017]

For a random embedded graph of genus g ≥ 0 and any k ≥ 2, the limit law is uniform on the k-simplex. OPEN EVEN FOR PLANAR GRAPHS.

• Theorem [Guitter 2017]

True for (g, k) = (0, 2) – two points on planar graph In particular for k = 2 points, each of them gets a U[0, 1] fraction of the mass.

• Theorem [C 2017]

(proof uses sharp tools from planar map enumeration and computer assisted calculations) (proof uses connection to math-φ and the double scaling limit of the 1-matrix model...)

For k = 2 and any g ≥ 0, the second moment matches that of a uniform.

Conjecture and results

• Theorem (main result) [Addario-Berry, Angel, C., Fusy, Goldschmidt, SODA’18]

The uniform Vorono¨ ı property is true for random trees.

• Conjecture [C., published in 2017]

For a random embedded graph of genus g ≥ 0 and any k ≥ 2, the limit law is uniform on the k-simplex. OPEN EVEN FOR PLANAR GRAPHS.

• Theorem [Guitter 2017]

True for (g, k) = (0, 2) – two points on planar graph In fact, true for random one-face maps of genus g ≥ 0 for fixed g. In particular for k = 2 points, each of them gets a U[0, 1] fraction of the mass.

• Theorem [C 2017]

(proof uses sharp tools from planar map enumeration and computer assisted calculations) (proof uses connection to math-φ and the double scaling limit of the 1-matrix model...)

For each g ≥ 0, f ≥ 1, we also have an analogue for random graphs of genus g with f faces (f,g fixed) For k = 2 and any g ≥ 0, the second moment matches that of a uniform.

Conjecture and results

• Theorem (main result) [Addario-Berry, Angel, C., Fusy, Goldschmidt, SODA’18]

The uniform Vorono¨ ı property is true for random trees.

• Conjecture [C., published in 2017]

For a random embedded graph of genus g ≥ 0 and any k ≥ 2, the limit law is uniform on the k-simplex. OPEN EVEN FOR PLANAR GRAPHS.

• Theorem [Guitter 2017]

True for (g, k) = (0, 2) – two points on planar graph In fact, true for random one-face maps of genus g ≥ 0 for fixed g. In particular for k = 2 points, each of them gets a U[0, 1] fraction of the mass.

• Theorem [C 2017]

(proof uses sharp tools from planar map enumeration and computer assisted calculations) (proof uses connection to math-φ and the double scaling limit of the 1-matrix model...)

For each g ≥ 0, f ≥ 1, we also have an analogue for random graphs of genus g with f faces (f,g fixed) For k = 2 and any g ≥ 0, the second moment matches that of a uniform.

Random maps of finite excess

Fix (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. Consider a uniform random map (=embedded graph) M with n edges (n → ∞) such that:

• M has genus g
• M has ℓ faces
• inside the i’th face, M has ni marked vertices numbered from i(1) to ini

clockwise.

Random maps of finite excess

Fix (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. Consider a uniform random map (=embedded graph) M with n edges (n → ∞) such that:

• M has genus g
• M has ℓ faces
• inside the i’th face, M has ni marked vertices numbered from i(1) to ini

clockwise. W.h.p. such a map is formed by a cubic skeleton, with edges subdivided in paths of length O(√n), and trees attached:

Example: (0;3;1,2,1)

11 31 21 22

Random maps of finite excess

Fix (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. Consider a uniform random map (=embedded graph) M with n edges (n → ∞) such that:

• M has genus g
• M has ℓ faces
• inside the i’th face, M has ni marked vertices numbered from i(1) to ini

clockwise. W.h.p. such a map is formed by a cubic skeleton, with edges subdivided in paths of length O(√n), and trees attached:

Example: (0;3;1,2,1)

11 31 21 22

. . . . . .

Random maps of finite excess

Fix (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. Consider a uniform random map (=embedded graph) M with n edges (n → ∞) such that:

• M has genus g
• M has ℓ faces
• inside the i’th face, M has ni marked vertices numbered from i(1) to ini

clockwise. W.h.p. such a map is formed by a cubic skeleton, with edges subdivided in paths of length O(√n), and trees attached:

Example: (0;3;1,2,1)

11 31 21 22 11 31 21 22

. . . . . .

Random maps of finite excess

Fix (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. Consider a uniform random map (=embedded graph) M with n edges (n → ∞) such that:

• M has genus g
• M has ℓ faces
• inside the i’th face, M has ni marked vertices numbered from i(1) to ini

clockwise. W.h.p. such a map is formed by a cubic skeleton, with edges subdivided in paths of length O(√n), and trees attached:

Example: (0;3;1,2,1)

11 31 21 22 11 31 21 22

Note: (0; 1; k)= uniform plane tree with k marked points!

. . . . . .

The number of skeletons is finite and all are equaly likely.

Our most general result: Vorono¨ ı vs. Interval vectors

M ∼ (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. In the map M look at the two vectors of length k =

i ni

• v :=
• |V 1

1 |

n , . . . , |V n1

1

| n

, . . . , |V 1

k |

n , . . . , |V

nℓ ℓ

| n

• Vorono¨

ı vector

• i :=
• |I1

1|

2n , . . . , |In1

1

| 2n , . . . , |I1

k|

2n , . . . , |I

nℓ ℓ

| 2n

• Interval vector

11 31 21 22 where Ii

j is the set of edges sitting along the contour interval

starting at point ij.

Our most general result: Vorono¨ ı vs. Interval vectors

M ∼ (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. In the map M look at the two vectors of length k =

i ni

• v :=
• |V 1

1 |

n , . . . , |V n1

1

| n

, . . . , |V 1

k |

n , . . . , |V

nℓ ℓ

| n

• Vorono¨

ı vector

• i :=
• |I1

1|

2n , . . . , |In1

1

| 2n , . . . , |I1

k|

2n , . . . , |I

nℓ ℓ

| 2n

• Interval vector

11 31 21 22 where Ii

j is the set of edges sitting along the contour interval

starting at point ij.

Our most general result: Vorono¨ ı vs. Interval vectors

M ∼ (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. In the map M look at the two vectors of length k =

i ni

• v :=
• |V 1

1 |

n , . . . , |V n1

1

| n

, . . . , |V 1

k |

n , . . . , |V

nℓ ℓ

| n

• Vorono¨

ı vector

• i :=
• |I1

1|

2n , . . . , |In1

1

| 2n , . . . , |I1

k|

2n , . . . , |I

nℓ ℓ

| 2n

• Interval vector

11 31 21 22 Theorem [AB-A-C-F-G, SODA’18] In the limit, the vectors v and i have the same law! Corollary Random trees have uniform Vorono¨ ı tessellations! where Ii

j is the set of edges sitting along the contour interval

starting at point ij.

Our most general result: Vorono¨ ı vs. Interval vectors

M ∼ (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. In the map M look at the two vectors of length k =

i ni

• v :=
• |V 1

1 |

n , . . . , |V n1

1

| n

, . . . , |V 1

k |

n , . . . , |V

nℓ ℓ

| n

• Vorono¨

ı vector

• i :=
• |I1

1|

2n , . . . , |In1

1

| 2n , . . . , |I1

k|

2n , . . . , |I

nℓ ℓ

| 2n

• Interval vector

11 31 21 22 Theorem [AB-A-C-F-G, SODA’18] In the limit, the vectors v and i have the same law! Corollary Random trees have uniform Vorono¨ ı tessellations! where Ii

j is the set of edges sitting along the contour interval

starting at point ij.

Our most general result: Vorono¨ ı vs. Interval vectors

M ∼ (g; ℓ; n1, . . . , nℓ) with g ≥ 0, ℓ ≥ 1, and with ni ≥ 1. In the map M look at the two vectors of length k =

i ni

• v :=
• |V 1

1 |

n , . . . , |V n1

1

| n

, . . . , |V 1

k |

n , . . . , |V

nℓ ℓ

| n

• Vorono¨

ı vector

• i :=
• |I1

1|

2n , . . . , |In1

1

| 2n , . . . , |I1

k|

2n , . . . , |I

nℓ ℓ

| 2n

• Interval vector

11 31 21 22 Theorem [AB-A-C-F-G, SODA’18] In the limit, the vectors v and i have the same law! Corollary Random trees have uniform Vorono¨ ı tessellations! We DO NOT know how to prove uniformity even for trees without the trick of introducing interval vectors! The proof is by induction on Euler characteristic where Ii

j is the set of edges sitting along the contour interval

starting at point ij. Comments:

Note

finite excess random maps of genus g = general random maps of genus g

11 31 21 22

. . . . . .

n vertices, ∼ n edges, excess O(1) diameter Θ(√n) continuum limit object is “tree-like” n vertices, ∼ n faces, excess Θ(n) diameter Θ(n1/4) continuum limit object is “surface-like”

Note

finite excess random maps of genus g = general random maps of genus g

11 31 21 22

. . . . . .

n vertices, ∼ n edges, excess O(1) diameter Θ(√n) continuum limit object is “tree-like” − → Why would their Voronoi vectors behave similarly ??? n vertices, ∼ n faces, excess Θ(n) diameter Θ(n1/4) continuum limit object is “surface-like”

The proof for trees

Start with k = 2 (two marked points).

Vorono¨ ı game d/2 d/2

The proof for trees

Start with k = 2 (two marked points).

Vorono¨ ı game d/2 d/2 Interval game

The proof for trees

Start with k = 2 (two marked points).

Vorono¨ ı game d/2 d/2 Interval game 90 deg. rotation!! SAME DISTRIBUTION !

... It took us YEARS to find this trick

The proof for trees, continued k ≥ 2

Take k players (here k = 4) and look at the Voronoi and Interval Games. Voronoi Game Interval Game

The proof for trees, continued k ≥ 2

Take k players (here k = 4) and look at the Voronoi and Interval Games. Voronoi Game Interval Game 1 2 3 4

The proof for trees, continued k ≥ 2

Take k players (here k = 4) and look at the Voronoi and Interval Games. Voronoi Game Interval Game 1 2 3 4

δ stop exploration at first time δ when some player reaches a branch point.

The proof for trees, continued k ≥ 2

Take k players (here k = 4) and look at the Voronoi and Interval Games. Voronoi Game Interval Game 1 2 3 4

δ stop exploration at first time δ when some player reaches a branch point. problem splits in two

• subproblems. One

player (here 3) gets to play twice!

1 2 3 4 3

The proof for trees, continued k ≥ 2

Take k players (here k = 4) and look at the Voronoi and Interval Games. Voronoi Game Interval Game 1 2 3 4

δ stop exploration at first time δ when some player reaches a branch point. problem splits in two

• subproblems. One

player (here 3) gets to play twice!

1 2 3 4 3 1 2 3 4

The proof for trees, continued k ≥ 2

Take k players (here k = 4) and look at the Voronoi and Interval Games. Voronoi Game Interval Game 1 2 3 4

δ stop exploration at first time δ when some player reaches a branch point. problem splits in two

• subproblems. One

player (here 3) gets to play twice!

1 2 3 4 3 1 2 4 3

problem AGAIN splits in two subproblems, and AGAIN one player plays twice! (here 4)

1 2 3 4

remove same pieces of length δ as before

4

− → proof complete, by induction!

Conclusion

We only have a “proof from the book” that doesn’t explain anything... WHY would a model of random graphs or random geometry would have uniform Vorono¨ ı tessellations? But the similarity with the main conjecture is puzzling.

THANK YOU