Shifing the thresold of phase transition in 2-SAT and random graphs - - PowerPoint PPT Presentation

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT

Shifing the thresold of phase transition in 2-SAT and random graphs

Sergey Dovgal1,2 Vlady Ravelomanana2

1Université Paris-13, 2Université Paris-Diderot

Acknowledgements: Élie de Panafieu, Fedor Petrov, ipython+sympy+cpp

June 13, 2017

D., Ravelomanana Shifing the phase transition 1 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT

Outline

1

Problem and Motivation

2

Saddle-point method and analytic lemma

3

Distribution of random parameters

4

Lower bound for 2-SAT

D., Ravelomanana Shifing the phase transition 2 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Outline

1

Problem and Motivation

2

Saddle-point method and analytic lemma

3

Distribution of random parameters

4

Lower bound for 2-SAT

D., Ravelomanana Shifing the phase transition 3 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Phase transition in Erdős–Rényi random graphs

n vertices, m edges, m = 1 2n(1 + µn−1/3)

1

“gas” µ → −∞ : planar graph, trees and unicycles, max component size O(log n).

2

“liquid” |µ| = O(1) : complex components appear, max component size O(n2/3).

3

“crystal” µ → +∞ : non-planar, complex compontnes, max component size linear O(n).

D., Ravelomanana Shifing the phase transition 4 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Phase transition in Erdős–Rényi random graphs

n vertices, m edges, m = 1 2n(1 + µn−1/3)

1

“gas” µ → −∞ : planar graph, trees and unicycles, max component size O(log n).

2

“liquid” |µ| = O(1) : complex components appear, max component size O(n2/3).

3

“crystal” µ → +∞ : non-planar, complex compontnes, max component size linear O(n).

D., Ravelomanana Shifing the phase transition 4 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Phase transition in Erdős–Rényi random graphs

n vertices, m edges, m = 1 2n(1 + µn−1/3)

1

“gas” µ → −∞ : planar graph, trees and unicycles, max component size O(log n).

2

“liquid” |µ| = O(1) : complex components appear, max component size O(n2/3).

3

“crystal” µ → +∞ : non-planar, complex compontnes, max component size linear O(n).

D., Ravelomanana Shifing the phase transition 4 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Phase transition :: largest component, n = 1000

D., Ravelomanana Shifing the phase transition 5 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Phase transition :: planarity, n = 1000

D., Ravelomanana Shifing the phase transition 6 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Phase transition :: diameter, n = 1000

D., Ravelomanana Shifing the phase transition 7 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Shifing the phase transition

m = 1 2n(1 + µn−1/3) ⇒ m = αn(1 + µn−1/3)

1 Achlioptas percolation process 2 Degree sequence models 3 Degree set constraint :: current talk

D., Ravelomanana Shifing the phase transition 8 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Shifing the phase transition

m = 1 2n(1 + µn−1/3) ⇒ m = αn(1 + µn−1/3)

1 Achlioptas percolation process 2 Degree sequence models 3 Degree set constraint :: current talk

D., Ravelomanana Shifing the phase transition 8 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Shifing the phase transition

m = 1 2n(1 + µn−1/3) ⇒ m = αn(1 + µn−1/3)

1 Achlioptas percolation process 2 Degree sequence models 3 Degree set constraint :: current talk

D., Ravelomanana Shifing the phase transition 8 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Example of graph with degree constraints

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Figure: Random labeled graph from G26,30,∆ with the set of degree constraints ∆ = {1, 2, 3, 5, 7}.

D., Ravelomanana Shifing the phase transition 9 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Condition for the set Delta and random graph

n – number of vertices m – number of edges

1 1 ∈ ∆

← − Other cases 1 / ∈ ∆ remain open question

2 Period of ∆: p def

= gcd(d1 − d2 : d1, d2 ∈ ∆), p | 2m − n · min(∆) ← − necessary, each degree ∈ ∆

3 2m/n ∈ fixed compact interval of ] min(∆), max(∆)[

D., Ravelomanana Shifing the phase transition 10 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Condition for the set Delta and random graph

n – number of vertices m – number of edges

1 1 ∈ ∆

← − Other cases 1 / ∈ ∆ remain open question

2 Period of ∆: p def

= gcd(d1 − d2 : d1, d2 ∈ ∆), p | 2m − n · min(∆) ← − necessary, each degree ∈ ∆

3 2m/n ∈ fixed compact interval of ] min(∆), max(∆)[

D., Ravelomanana Shifing the phase transition 10 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Condition for the set Delta and random graph

n – number of vertices m – number of edges

1 1 ∈ ∆

← − Other cases 1 / ∈ ∆ remain open question

2 Period of ∆: p def

= gcd(d1 − d2 : d1, d2 ∈ ∆), p | 2m − n · min(∆) ← − necessary, each degree ∈ ∆

3 2m/n ∈ fixed compact interval of ] min(∆), max(∆)[

D., Ravelomanana Shifing the phase transition 10 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Trees with degree constraints

Rooted case

T0               T1

• ∆

∆−1 ∆−1 ∆−1 ∆ − k

def

= {d : d + k ∈ ∆} ω(z) =

• d∈∆

zd d! = zd1 d1! + zd2 d2! + . . . ,      T0(z) = zω(T1(z)), T1(z) = zω′(T1(z)), T2(z) = zω′′(T1(z)).

D., Ravelomanana Shifing the phase transition 11 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Trees with degree constraints

Unrooted case Excercise

A variant of dissymmetry theorem: (∆) =

• +
• 1
• 1

+

• 1

Excercise: what is the EGF for unrooted trees?

D., Ravelomanana Shifing the phase transition 12 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Trees with degree constraints

Unrooted case Excercise

A variant of dissymmetry theorem: (∆) =

• +
• 1
• 1

+

• 1

T0(z) = T1(z)2 2 + U(z) ⇔ U(z) = T0(z) − T1(z)2 2

D., Ravelomanana Shifing the phase transition 12 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Unicycles with degree constraints

Excercise ≥ 3 Excercise: what is the EGF for unicycles?

D., Ravelomanana Shifing the phase transition 13 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Unicycles with degree constraints

≥ 3 V(z) = 1 2

• log

1 1 − T2(z) − T2(z) − T2(z)2 2

• D., Ravelomanana

Shifing the phase transition 13 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

2-core of a graph

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

2-core (the core) and 3-core (the kernel)

3-core of a graph

D., Ravelomanana Shifing the phase transition 14 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Notion of excess

2 1 3 4 −1 4 1 3 2 1 3 4 6 5 2 1 1 2 4 3 2 Excess

def

= # edges - # vertices

D., Ravelomanana Shifing the phase transition 15 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Kernel of a graph

Example: graphs with excess 1

1

1 4

1 2

1 4

1 2

1 6

All possible 3-core multigraphs and their compensation factors. EGF for all connected bicyclic graphs (∆ = Z≥0): W(z) = 1 4 T(z)5 (1 − T(z))2 + 1 4 T(z)6 (1 − T(z))3 + 1 6 T(z)2[3T(z)2 − 2T 3(z)] (1 − T(z))3

• inclusion-exclusion

W(z) ∼ 5 24 · 1 (1 − T(z))3 near z = e−1

D., Ravelomanana Shifing the phase transition 16 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Kernel of a graph

Example: graphs with excess 1

1

1 4

1 2

1 4

1 2

1 6

All possible 3-core multigraphs and their compensation factors. EGF for all connected bicyclic graphs (arbitrary ∆): W∆(z) = 1 4 T4(z)T2(z)4 (1 − T2(z))2 +1 4 T3(z)2T2(z)4 (1 − T2(z))3 +1 6 T3(z)2[3T2(z)2 − 2T2(z)3] (1 − T2(z))3

• inclusion-exclusion

W∆(z) ∼ (???) · T3(z)2(???) (1 − T2(z))3

D., Ravelomanana Shifing the phase transition 16 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

General symbolic lemma

EGF for connected graphs which reduce to given M is: W∆,M(z) = κ(M)

• v∈V

Tdeg(v)(z) n! · P(M, T2(z)) (1 − T2(z))µ P(M, z) =

n

• x=1
• z2µxx

n

• y=x+1

zµxy−1(µxy − (µxy − 1)z)

• ,

           Tk(z) = zω(k)(T1(z)), µxy = # edges between nodes x and y κ(M) = compensation factor µ = # edges

D., Ravelomanana Shifing the phase transition 17 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Role of cubic graphs

Demonstration on the blackboard

Let z be the positive solution of T2( z) = 1. Then EGF for all (not necessary connected) complex multigraphs with excess r, has asymptotics near z, which comes from cubic graphs (degree of each vertex is equal to 3): W∆,r(z) ∼ er0 T3(z)2r (1 − T2(z))3r , er0 = (6r)! 25r32r(3r)!(2r)! .

D., Ravelomanana Shifing the phase transition 18 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Local summary

1 EGF for unrooted trees with degree constraints 2 EGF for unicycles with degree constraints 3 EGF for graphs of fixed excess (main asymptotics)

D., Ravelomanana Shifing the phase transition 19 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Local summary

1 EGF for unrooted trees with degree constraints 2 EGF for unicycles with degree constraints 3 EGF for graphs of fixed excess (main asymptotics)

D., Ravelomanana Shifing the phase transition 19 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Trees with degree constraints Unicycles with degree constraints Kernel of a graph General symbolic lemma

Local summary

1 EGF for unrooted trees with degree constraints 2 EGF for unicycles with degree constraints 3 EGF for graphs of fixed excess (main asymptotics)

D., Ravelomanana Shifing the phase transition 19 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Outline

1

Problem and Motivation

2

Saddle-point method and analytic lemma

3

Distribution of random parameters

4

Lower bound for 2-SAT

D., Ravelomanana Shifing the phase transition 20 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Expected results

n – number of vertices m – number of edges Framework: m = αn, linear dependence.

1 m = (1 − ε)αn

← − only trees and unicycles

2 m = αn

← − complex components with positive probability

3 m = (1 + ε)αn

← − probability of fixed excess is exponentially small

D., Ravelomanana Shifing the phase transition 21 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Expected results

n – number of vertices m – number of edges Framework: m = αn, linear dependence.

1 m = (1 − ε)αn

← − subcritical phase

2 m = αn

← − cricital phase

3 m = (1 + ε)αn

← − supercritical phase

D., Ravelomanana Shifing the phase transition 21 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Desired probability

Subcritical phase

P (graph g ∈ G(n, m, ∆) consists only of trees and unicycles) = # graphs from G(n, m, ∆) whose components are trees and unicycles # graphs from G(n, m, ∆)

D., Ravelomanana Shifing the phase transition 22 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Number of graphs with degree constraints

1 ∆ = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary ∆ ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,∆| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• D., Ravelomanana

Shifing the phase transition 23 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Number of graphs with degree constraints

1 ∆ = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary ∆ ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,∆| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• D., Ravelomanana

Shifing the phase transition 23 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Number of graphs with degree constraints

1 ∆ = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary ∆ ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,∆| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• D., Ravelomanana

Shifing the phase transition 23 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Number of graphs with degree constraints

1 ∆ = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary ∆ ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,∆| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• D., Ravelomanana

Shifing the phase transition 23 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Number of graphs with degree constraints

1 ∆ = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary ∆ ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,∆| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• D., Ravelomanana

Shifing the phase transition 23 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Number of graphs with degree constraints

1 ∆ = Z≥0. Stirling approximation:

n! (n − m)! (n

2)

m

∼ √ 4πnα · 2mnnmm n2m(n − m)n−m × exp

• − n + m

n + m2 n2

• 3/4
• 2 Arbitrary ∆ ([de Panafieu, Ramos ’16])

n! (n − m)!|Gn,m,∆| ∼ √ 4πnα p · 2mnnmm n2m(n − m)n−m × exp

• − n log ω(

z) + 2m log z + 1 2φ0( z) + 1 4φ2

0(

z)

• D., Ravelomanana

Shifing the phase transition 23 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Contour integrals for subcritical phase

Demonstration on the blackboard

n! |Gn,m,∆| 1 2πi U(z)n−m (n − m)!eV(z) dz zn+1 = 1 − O(µ−3) near the critical point m = αn:        2α = φ0( z)

def

= z ω′( z) ω( z) , 1 = φ1( z)

def

= z ω′′( z) ω′( z) .

D., Ravelomanana Shifing the phase transition 24 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Full range of densities

Theorem ( Regime: m = αn(1 − µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,∆ has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,∆ has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,∆ has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 25 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Full range of densities

Theorem ( Regime: m = αn(1 − µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,∆ has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,∆ has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,∆ has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 25 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Full range of densities

Theorem ( Regime: m = αn(1 − µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,∆ has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,∆ has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,∆ has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 25 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Full range of densities

Theorem ( Regime: m = αn(1 − µn−1/3) )

1 if µ → −∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = 1 − Θ(|µ|−3) ;

2 if |µ| = O(1), i.e. µ is fixed, then

P(Gn,m,∆ has only trees and unicycles) → constant ∈ (0, 1) , P(Gn,m,∆ has a complex part with total excess q) → constant ∈ (0, 1) ,

3 if µ → +∞, |µ| = O(n1/12), then

P(Gn,m,∆ has only trees and unicycles) = Θ(e−µ3/6µ−3/4) , P(Gn,m,∆ has a complex part with excess q) = Θ(e−µ3/6µ3q/2−3/4) .

D., Ravelomanana Shifing the phase transition 25 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Geometric statement

Demonstration on the blackboard

1 h(z; r) def

= r log ω′(z) − r log z + (1 − r) log(2ω − zω′)

2 Φ(θ; r) def

= Re h(z0eiθ; r), max

θ∈[0,2π] Φ(θ; r) = Φ(θ; r)

• θk= 2πk

p D., Ravelomanana Shifing the phase transition 26 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Good old hypergeometric (Airy) function [FlJaKnŁuPi]

A(y, µ) = eµ3/6 3(y+1)/3

• k≥0

1

232/3µ

k k!Γ((y + 1 − 2k)/3)

1 As µ → −∞,

A(y, µ) = 1 √ 2π|µ|y−1/2

• 1 − 3y2 + 3y − 1

6|µ|3 + O(µ−6)

• 2 As µ → +∞,

A(y, µ) = e−µ3/6 2y/2|µ|1−y/2

• 1

Γ(y/2) + 4µ−3/2 3 √ 2Γ(y/2 − 3/2) + O(µ−2)

• D., Ravelomanana

Shifing the phase transition 27 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

In-class excercise

Excercise

Trees and unicycles: y = 1/2. Complex component: y ≥ 1 + 1/2. As µ → −∞, A(y, µ) = 1 √ 2π|µ|y−1/2

• 1 − 3y2 + 3y − 1

6|µ|3 + O(µ−6)

• What is the asymptotics of A(y, µ)?

D., Ravelomanana Shifing the phase transition 28 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

In-class excercise

Excercise

Trees and unicycles: y = 1/2. Complex component: y ≥ 1 + 1/2. As µ → −∞, A(y, µ) = 1 √ 2π|µ|y−1/2

• 1 − 3y2 + 3y − 1

6|µ|3 + O(µ−6)

• Asymptotics of A(y, µ):

A(1/2, µ) ∼ 1 √ 2π

• 1 −

5 24|µ|3

• A(3/2, µ) ∼

1 √ 2π|µ|

• 1 + O(|µ|−3)
• D., Ravelomanana

Shifing the phase transition 28 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

More than just probability: Analytic lemma

Excercise

As m = αn(1 − µn−1/3), n! (n − m)!|Gn,m,∆|[zn] U(z)n−m (1 − T2(z))y ∼ √ 2πCy · A(y, Cµ)ny/3−1/6 Excercise: P of 1 bicycle inside critical phase. Asymptotics? 1 2

D., Ravelomanana Shifing the phase transition 29 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

More than just probability: Analytic lemma

Excercise

As m = αn(1 − µn−1/3), n! (n − m)!|Gn,m,∆|[zn] U(z)n−m (1 − T2(z))y ∼ √ 2πCy · A(y, Cµ)ny/3−1/6 Excercise: P of 1 bicycle inside critical phase. Asymptotics? 1 2 2-core contains 3 paths.

D., Ravelomanana Shifing the phase transition 29 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

More than just probability: Analytic lemma

Excercise

As m = αn(1 − µn−1/3), n! (n − m)!|Gn,m,∆|[zn] U(z)n−m (1 − T2(z))y ∼ √ 2πCy · A(y, Cµ)ny/3−1/6 Excercise: P of 1 bicycle inside critical phase. Asymptotics? 1 2 2-core contains 3 paths. “Mnemonic rule” y = − 1

2 ⇒ y = 3 − 1 2.

D., Ravelomanana Shifing the phase transition 29 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

More than just probability: Analytic lemma

Excercise

As m = αn(1 − µn−1/3), n! (n − m)!|Gn,m,∆|[zn] U(z)n−m (1 − T2(z))y ∼ √ 2πCy · A(y, Cµ)ny/3−1/6 Excercise: P of 1 bicycle inside critical phase. Asymptotics? 1 2 C n(y+3)/3−1/6 ny/3−1/6 · 1 n − m + 1 = O(1)

D., Ravelomanana Shifing the phase transition 29 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Local summary

1 Airy function from [FlJaKnŁuPi] 2 Contour integrals for m = linear(n) αn 3 Core technical statement (due to Petrov): global max of real

part complex function

4 Analytic lemma will be used afer.

We gain additional n1/3 for each additional 1 1 − T2(z).

D., Ravelomanana Shifing the phase transition 30 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Local summary

1 Airy function from [FlJaKnŁuPi] 2 Contour integrals for m = linear(n) αn 3 Core technical statement (due to Petrov): global max of real

part complex function

4 Analytic lemma will be used afer.

We gain additional n1/3 for each additional 1 1 − T2(z).

D., Ravelomanana Shifing the phase transition 30 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Local summary

1 Airy function from [FlJaKnŁuPi] 2 Contour integrals for m = linear(n) αn 3 Core technical statement (due to Petrov): global max of real

part complex function

4 Analytic lemma will be used afer.

We gain additional n1/3 for each additional 1 1 − T2(z).

D., Ravelomanana Shifing the phase transition 30 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Desired probability Number of graphs with degree constraints Contour integrals More than just probability: Analytic lemma

Local summary

1 Airy function from [FlJaKnŁuPi] 2 Contour integrals for m = linear(n) αn 3 Core technical statement (due to Petrov): global max of real

part complex function

4 Analytic lemma will be used afer.

We gain additional n1/3 for each additional 1 1 − T2(z).

D., Ravelomanana Shifing the phase transition 30 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Outline

1

Problem and Motivation

2

Saddle-point method and analytic lemma

3

Distribution of random parameters

4

Lower bound for 2-SAT

D., Ravelomanana Shifing the phase transition 31 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Length of a 2-path

Excercise

q — excess (condition and then sum over q) ← − marked 2-path inside complex component of some graph E[uPn] ∝ [zn] U(z)n−m+q (n − m + q)!eV(z) 1 − T2(z) 1 − uT2(z) Qestion: asymptotics of length of 2-path? Hint: analytic lemma.

D., Ravelomanana Shifing the phase transition 32 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Length of a 2-path

Excercise

q — excess (condition and then sum over q) ← − marked 2-path inside complex component of some graph E[uPn] ∝ [zn] U(z)n−m+q (n − m + q)!eV(z) 1 − T2(z) 1 − uT2(z) EPn ⇐ d du(∗)

• u=1

⇐

Analytic lemma EPn ∼ C · n1/3

D., Ravelomanana Shifing the phase transition 32 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Bivariate EGF for tree height

[Flajolet, Odlyzko ’82]

F(z, u) =

• n≥0

zn n!

n

• h=0

A[h]

n ↑ # trees height h

· uh

1

d duF(z, u)

• u=1

∼ C1 log

• 1 − z

ρ , 2

d2 du2 F(z, u)

• u=1

∼ C2

• 1 − z

ρ

−1/2

3 Modify analytic lemma for log(1 − φ1(z)).

D., Ravelomanana Shifing the phase transition 33 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Diameter, circumference and longest path of complex component

All of order Θ(n1/3)

D., Ravelomanana Shifing the phase transition 34 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Planarity

Demonstration on the blackboard

Let p(µ) be the probability that Gn,m,∆ is planar.

1 p(µ) = 1 − Θ(|µ|−3), as µ → −∞; 2 p(µ) → constant ∈ (0, 1), as |µ| = O(1), and p(µ) is

computable;

3 p(µ) → 0, as µ → +∞.

D., Ravelomanana Shifing the phase transition 35 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Experimental results

(1/3)

D., Ravelomanana Shifing the phase transition 36 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Experimental results

(2/3)

D., Ravelomanana Shifing the phase transition 36 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Experimental results

(3/3)

D., Ravelomanana Shifing the phase transition 36 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Local summary

1 Phase transition of diameter, circumference, longest path. 2 Phase transition for planarity. 3 Diameter of trees and unicycles — open question? 4 Largest component — open question? 5 Stay tuned for 2-SAT!

D., Ravelomanana Shifing the phase transition 37 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Local summary

1 Phase transition of diameter, circumference, longest path. 2 Phase transition for planarity. 3 Diameter of trees and unicycles — open question? 4 Largest component — open question? 5 Stay tuned for 2-SAT!

D., Ravelomanana Shifing the phase transition 37 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Local summary

1 Phase transition of diameter, circumference, longest path. 2 Phase transition for planarity. 3 Diameter of trees and unicycles — open question? 4 Largest component — open question? 5 Stay tuned for 2-SAT!

D., Ravelomanana Shifing the phase transition 37 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Local summary

1 Phase transition of diameter, circumference, longest path. 2 Phase transition for planarity. 3 Diameter of trees and unicycles — open question? 4 Largest component — open question? 5 Stay tuned for 2-SAT!

D., Ravelomanana Shifing the phase transition 37 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT Length of a 2-path Diameter, circumference and longest path Planarity Experimental results

Local summary

1 Phase transition of diameter, circumference, longest path. 2 Phase transition for planarity. 3 Diameter of trees and unicycles — open question? 4 Largest component — open question? 5 Stay tuned for 2-SAT!

D., Ravelomanana Shifing the phase transition 37 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Outline

1

Problem and Motivation

2

Saddle-point method and analytic lemma

3

Distribution of random parameters

4

Lower bound for 2-SAT

D., Ravelomanana Shifing the phase transition 38 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

2-CNF formula and digraph model

Digraph representation and sum-representation of a 2-sat formula (x1 ∨ x2)(x2 ∨ x3)(x2 ∨ x1)(x4 ∨ x3)(x4 ∨ x2)(x4 ∨ x4)

1 2 3 1 4 3 2 4 + 1 2 3 1 4 3 2 4 = 1 2 3 4 1 2 3 4

D., Ravelomanana Shifing the phase transition 39 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Excercise on 2-CNF representation construction

Excercise

2 1 2 3 3 1 + 2 1 2 3 3 1 = ???

D., Ravelomanana Shifing the phase transition 40 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Excercise on 2-CNF representation construction

Excercise

2 1 2 3 3 1 + 2 1 2 3 3 1 = 1 2 3 1 2 3

D., Ravelomanana Shifing the phase transition 40 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Digraph model from graph with degree constraint

Lemma: CNF is UNSAT iff it contains a circuit x x and x x. Random CNF: G(n, m, ∆) ⊕ G(n, m, ∆) How to control the probability of circuit 1 1 1? Idea: probability the same as for any other circuit.

D., Ravelomanana Shifing the phase transition 41 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Complicated excercise on cycles

Excercise

Fix nodes x, y (say, x = 1 and y = 2). Condider random directed graphs with vertex degrees from ∆, subcritical phase, condition on trees and unicycles.

• ℓ≥1

2ℓP(x, y ∈ circuit of length ℓ) =?

D., Ravelomanana Shifing the phase transition 42 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Complicated excercise on cycles

Excercise

Fix nodes x, y (say, x = 1 and y = 2). Condider random directed graphs with vertex degrees from ∆, subcritical phase, condition on trees and unicycles.

• ℓ≥1

2ℓP(x, y ∈ circuit of length ℓ) Step 1. Mark the circuit + expectation of 2L.

• V •(z) =
• z d

dz z d dz

• 2 markings

circuit>2(uz)

• z→

T2(z),u=2

D., Ravelomanana Shifing the phase transition 42 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Complicated excercise on cycles

Excercise

Fix nodes x, y (say, x = 1 and y = 2). Condider random directed graphs with vertex degrees from ∆, subcritical phase, condition on trees and unicycles.

• ℓ≥1

2ℓP(x, y ∈ circuit of length ℓ) Step 2. Count the coefficient by analytic lemma. ∝ 1 n(n − 1)[zn] U(z)n−m exp

• V(z)
• V •(z) ,

D., Ravelomanana Shifing the phase transition 42 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Complicated excercise on cycles

Excercise

Fix nodes x, y (say, x = 1 and y = 2). Condider random directed graphs with vertex degrees from ∆, subcritical phase, condition on trees and unicycles.

• ℓ≥1

2ℓP(x, y ∈ circuit of length ℓ) Step 3. Final expression. 2 markings → n2/3 (subcritical ×|µ|−2): ∼ n2/3|µ|−2 n(n − 1) ∼ O(n−4/3|µ|−2)

D., Ravelomanana Shifing the phase transition 42 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Case without complex component

Demonstration on the blackboard

Subcritical case: n = αm(1 − µn−1/3), µ → −∞. P(Fn,m is SAT) ≥ 1 − 5 24|µ|3 + O(|µ|−6)

1 2 3 1 3 2 5 4 5 4

=

1 2 3 1 5 4 3 2 5 4

+

1 3 2 1 5 4 2 3 5 4

D., Ravelomanana Shifing the phase transition 43 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Full statement of the theorem

1 P(Fn,m,∆ is sat) ≥ 1 − O(|µ|−3) as µ → −∞, 2 P(Fn,m,∆ is sat) ≥ Θ(1) as |µ| = O(1), 3 P(Fn,m,∆ is sat) ≥ exp(−Θ(µ3)) as µ → +∞.

D., Ravelomanana Shifing the phase transition 44 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Local summary

1 No complex component, contradictory circuit.

Done by marking + analytic lemma.

2 Correction for non-uniformity? 3 Upper bound — what happens inside complex components?

D., Ravelomanana Shifing the phase transition 45 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Local summary

1 No complex component, contradictory circuit.

Done by marking + analytic lemma.

2 Correction for non-uniformity? 3 Upper bound — what happens inside complex components?

D., Ravelomanana Shifing the phase transition 45 / 51

Problem and Motivation Saddle-point method and analytic lemma Distribution of random parameters Lower bound for 2-SAT 2-CNF formula and digraph model Case without complex component Full statement of the theorem

Local summary

1 No complex component, contradictory circuit.

Done by marking + analytic lemma.

2 Correction for non-uniformity? 3 Upper bound — what happens inside complex components?

D., Ravelomanana Shifing the phase transition 45 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Proportion of vertices of certain degree Distribution of parameters from Hatami-Molloy

Outline

5

Sequence of degrees vs. set of degrees. Batle!

D., Ravelomanana Shifing the phase transition 46 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Proportion of vertices of certain degree Distribution of parameters from Hatami-Molloy

Proportion of vertices of certain degree

Excercise

Given ∆ = {d1, d2, . . .}, 1 ∈ ∆. How many vertices of degree d?

D., Ravelomanana Shifing the phase transition 47 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Proportion of vertices of certain degree Distribution of parameters from Hatami-Molloy

Proportion of vertices of certain degree

Excercise

Given ∆ = {d1, d2, . . .}, 1 ∈ ∆. How many vertices of degree d? Idea. T(z, u) = z ·

• ω + (u − 1)zd

d!

• T(z, u)

Distribution is easily obtained through marking method.

D., Ravelomanana Shifing the phase transition 47 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Proportion of vertices of certain degree Distribution of parameters from Hatami-Molloy

Distribution of parameters from Hatami-Molloy

Sequence of degrees D = (dv)v∈G. Q := Q(D) :=

• v∈G d2

v

2|E| − 2, R := R(D) :=

• v∈G dv(dv − 2)2

2|E| Mark the vertex degree → G(z, u). Ed2

1 =

[zn]

• u d

du 2 G(z, u)

• u=1

[zn]G(z, 1) EQ(D) = 4nα − O(n2/3) 2m − 2 = O(n−1/3)

D., Ravelomanana Shifing the phase transition 48 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Results and Methods

1 Analytic description of phase transition in model with degree

constraints

2 Fedor Petrov: help in the proof of geometric statement at

mathoverflow

3 Study of distribution of parameters. 4 Height of sprouting tree – new result? 5 Lower bound for 2-SAT + improvement of bounds by Bollobas. 6 Degree constraints ≈ degree sequence in certain small window.

D., Ravelomanana Shifing the phase transition 49 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Results and Methods

1 Analytic description of phase transition in model with degree

constraints

2 Fedor Petrov: help in the proof of geometric statement at

mathoverflow

3 Study of distribution of parameters. 4 Height of sprouting tree – new result? 5 Lower bound for 2-SAT + improvement of bounds by Bollobas. 6 Degree constraints ≈ degree sequence in certain small window.

D., Ravelomanana Shifing the phase transition 49 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Results and Methods

1 Analytic description of phase transition in model with degree

constraints

2 Fedor Petrov: help in the proof of geometric statement at

mathoverflow

3 Study of distribution of parameters. 4 Height of sprouting tree – new result? 5 Lower bound for 2-SAT + improvement of bounds by Bollobas. 6 Degree constraints ≈ degree sequence in certain small window.

D., Ravelomanana Shifing the phase transition 49 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Results and Methods

1 Analytic description of phase transition in model with degree

constraints

2 Fedor Petrov: help in the proof of geometric statement at

mathoverflow

3 Study of distribution of parameters. 4 Height of sprouting tree – new result? 5 Lower bound for 2-SAT + improvement of bounds by Bollobas. 6 Degree constraints ≈ degree sequence in certain small window.

D., Ravelomanana Shifing the phase transition 49 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Results and Methods

1 Analytic description of phase transition in model with degree

constraints

2 Fedor Petrov: help in the proof of geometric statement at

mathoverflow

3 Study of distribution of parameters. 4 Height of sprouting tree – new result? 5 Lower bound for 2-SAT + improvement of bounds by Bollobas. 6 Degree constraints ≈ degree sequence in certain small window.

D., Ravelomanana Shifing the phase transition 49 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Results and Methods

1 Analytic description of phase transition in model with degree

constraints

2 Fedor Petrov: help in the proof of geometric statement at

mathoverflow

3 Study of distribution of parameters. 4 Height of sprouting tree – new result? 5 Lower bound for 2-SAT + improvement of bounds by Bollobas. 6 Degree constraints ≈ degree sequence in certain small window.

D., Ravelomanana Shifing the phase transition 49 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Open problems

1 The case 1 /

∈ ∆.

2 Upper bound for 2-SAT. 3 Size of the largest component. 4 Statistics of complex component done – what about trees and

unicycles?

5 Non-uniform 2-SAT model. How to prove that it is equivalent

to the classical one?

D., Ravelomanana Shifing the phase transition 50 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Open problems

1 The case 1 /

∈ ∆.

2 Upper bound for 2-SAT. 3 Size of the largest component. 4 Statistics of complex component done – what about trees and

unicycles?

5 Non-uniform 2-SAT model. How to prove that it is equivalent

to the classical one?

D., Ravelomanana Shifing the phase transition 50 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Open problems

1 The case 1 /

∈ ∆.

2 Upper bound for 2-SAT. 3 Size of the largest component. 4 Statistics of complex component done – what about trees and

unicycles?

5 Non-uniform 2-SAT model. How to prove that it is equivalent

to the classical one?

D., Ravelomanana Shifing the phase transition 50 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Open problems

1 The case 1 /

∈ ∆.

2 Upper bound for 2-SAT. 3 Size of the largest component. 4 Statistics of complex component done – what about trees and

unicycles?

5 Non-uniform 2-SAT model. How to prove that it is equivalent

to the classical one?

D., Ravelomanana Shifing the phase transition 50 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

Open problems

1 The case 1 /

∈ ∆.

2 Upper bound for 2-SAT. 3 Size of the largest component. 4 Statistics of complex component done – what about trees and

unicycles?

5 Non-uniform 2-SAT model. How to prove that it is equivalent

to the classical one?

D., Ravelomanana Shifing the phase transition 50 / 51

Sequence of degrees vs. set of degrees. Batle! Summary Results and Methods Open problems

That’s all!

D., Ravelomanana Shifing the phase transition 51 / 51