Chapter 3: The Second Moment The Probabilistic Method Summer 2020 - PowerPoint PPT Presentation

Chapter 3: The Second Moment The Probabilistic Method Summer 2020 Freie Universität Berlin

Chapter Overview • Introduce the second moment method • Survey applications in graph theory and number theory

§1 Concentration Inequalities Chapter 3: The Second Moment The Probabilistic Method

What Does the Expectation Mean? Basic fact • 𝑌 ≤ 𝔽 𝑌 and {𝑌 ≥ 𝔽 𝑌 } have positive probability • Often want more quantitative information • What are these positive probabilities? • How much below/above the expectation can the random variable be? Limit laws • Law of large numbers • Average of independent trials will tend to the expectation • Central limit theorem • Average will be normally distributed Not always applicable • We often only have a single instance, or lack independence • Can still make use of more general bounds

Markov’s Inequality Theorem 3.1.1 (Markov’s Inequality) Let 𝑌 be a non-negative random variable, and let 𝑏 > 0 . Then ℙ 𝑌 ≥ 𝑏 ≤ 𝔽 𝑌 . 𝑏 Proof • Let 𝑔 be the density function for the distribution of 𝑌 ∞ 𝑦𝑔 𝑦 ⅆ𝑦 = ׬ 𝑏 𝑦𝑔 𝑦 ⅆ𝑦 + ׬ ∞ 𝑦𝑔 𝑦 ⅆ𝑦 • 𝔽 𝑌 = ׬ 0 0 𝑏 ∞ 𝑦𝑔 𝑦 ⅆ𝑦 ≥ ׬ ∞ 𝑏𝑔 𝑦 ⅆ𝑦 = 𝑏 ׬ ∞ 𝑔 𝑦 ⅆ𝑦 = 𝑏ℙ(𝑌 ≥ 𝑏) ≥ ׬ ∎ 𝑏 𝑏 𝑏 Moral: 𝔽 𝑌 small ⇒ 𝑌 typically small

Chebyshev’s Inequality Converse? Does 𝔽 𝑌 large ⇒ 𝑌 typically large? • Not necessarily; e.g. 𝑌 = 𝑜 2 with probability 𝑜 −1 , 0 otherwise • But such random variables have large variance… Theorem 3.1.2 (Chebyshev’s Inequality) Let 𝑌 be a random variable, and let 𝑏 > 0 . Then ≥ 𝑏 ≤ Var 𝑌 ℙ 𝑌 − 𝔽 𝑌 . 𝑏 2 Proof 2 ≥ 𝑏 2 • 𝑌 − 𝔽 𝑌 ≥ 𝑏 = 𝑌 − 𝔽 𝑌 2 • Let 𝑍 = 𝑌 − 𝔽 𝑌 • Then 𝔽 𝑍 = Var 𝑌 • Apply Markov’s Inequality ∎

Using Chebyshev Moral • 𝔽 𝑌 large and Var 𝑌 small ⇒ 𝑌 typically large • Special case: showing 𝑌 nonzero Corollary 3.1.3 If Var 𝑌 = 𝑝 𝔽 𝑌 2 , then ℙ 𝑌 = 0 = 𝑝 1 . Proof • 𝑌 = 0 ⊆ 𝑌 − 𝔽 𝑌 ≥ 𝔽 𝑌 ≤ Var 𝑌 • Chebyshev ⇒ ℙ 𝑌 − 𝔽 𝑌 ≥ 𝔽 𝑌 𝔽 𝑌 2 = 𝑝(1) ∎ • In fact, in this case 𝑌 = 1 + 𝑝 1 𝔽[𝑌] with high probability

Typical application Set-up • 𝐹 𝑗 events, occurring with probability 𝑞 𝑗 • 𝑌 𝑗 = 1 𝐹 𝑗 their indicator random variables • 𝑌 = σ 𝑗 𝑌 𝑗 their sum, the number of occurring events Goal • Show that with high probability, some event occurs Applying Chebyshev • Need to show Var 𝑌 = 𝑝 𝔽 𝑌 2 Expand the variance • Var 𝑌 = Var σ 𝑗 𝑌 𝑗 = σ 𝑗 Var(𝑌 𝑗 ) + σ 𝑗≠𝑘 Cov(𝑌 𝑗 , 𝑌 𝑘 )

Some Simplification Estimating the summands • Var 𝑌 = Var σ 𝑗 𝑌 𝑗 = σ 𝑗 Var(𝑌 𝑗 ) + σ 𝑗≠𝑘 Cov(𝑌 𝑗 , 𝑌 𝑘 ) • Var 𝑌 𝑗 = 𝑞 𝑗 1 − 𝑞 𝑗 ≤ 𝑞 𝑗 • ∴ σ 𝑗 Var(𝑌 𝑗 ) ≤ σ 𝑗 𝑞 𝑗 = σ 𝑗 𝔽 𝑌 𝑗 = 𝔽[𝑌] • Cov 𝑌, 𝑍 = 𝔽 𝑌𝑍 − 𝔽 𝑌 𝔽 𝑍 • Cov 𝑌, 𝑍 = 0 if 𝑌 and 𝑍 are independent • Otherwise Cov 𝑌 𝑗 , 𝑌 𝑘 ≤ 𝔽 𝑌 𝑗 𝑌 𝑘 = ℙ 𝐹 𝑗 ∧ 𝐹 𝑘 Corollary 3.1.4 Let 𝐹 𝑗 be a sequence of events with probabilities 𝑞 𝑗 , and let 𝑌 count the number of events that occur. Write 𝑗 ∼ 𝑘 if the events 𝐹 𝑗 and 𝐹 𝑘 are not independent, and let Δ = σ 𝑗∼𝑘 ℙ 𝐹 𝑗 ∧ 𝐹 𝑘 . If 𝔽 𝑌 → ∞ and Δ = 𝑝 𝔽 𝑌 2 , then 𝑄 𝑌 = 0 = 𝑝 1 .

Any questions?

§2 Thresholds Chapter 3: The Second Moment The Probabilistic Method

Monotone properties Graph properties • Say a graph 𝒬 is monotone (increasing) if adding edges preserves 𝒬 • e.g.: containing a subgraph 𝐼 ⊆ 𝐻 , having 𝛽 𝐻 < 𝑙 , connectivity, … Lemma 3.2.1 If 𝒬 is a monotone increasing graph property, then ℙ 𝐻 𝑜, 𝑞 ∈ 𝒬 is monotone increasing in 𝑞 . Proof (Coupling) • Sampling 𝐻 𝑜, 𝑞 • Assign to each pair of vertices 𝑣, 𝑤 an independent uniform 𝑍 𝑣,𝑤 ~Unif 0,1 • Add edge 𝑣, 𝑤 to 𝐻 iff 𝑍 𝑣,𝑤 ≤ 𝑞 • Each edge appears independently with probability 𝑞 • If 𝑞 ≤ 𝑞′ , then 𝐻 𝑜, 𝑞 ⊆ 𝐻 𝑜, 𝑞 ′ ⇒ if 𝐻 𝑜, 𝑞 ∈ 𝒬 , then 𝐻 𝑜, 𝑞 ′ ∈ 𝒬 ∎

Thresholds Transitions • A monotone property 𝒬 is nontrivial if it is not satisfied by the edgeless graph, and is satisfied by the complete graph • ⇒ ℙ 𝐻 𝑜, 0 ∈ 𝒬 = 0 and ℙ 𝐻 𝑜, 1 ∈ 𝒬 = 1 • Lemma 3.2.1 ⇒ ℙ 𝐻 𝑜, 𝑞 ∈ 𝒬 increases from 0 to 1 as 𝑞 does • How quickly does this increase happen? Definition 3.2.2 (Thresholds) Given a nontrivial monotone graph property 𝒬 , 𝑞 0 (𝑜) is a threshold for 𝒬 if ℙ 𝐻 𝑜, 𝑞 ∈ 𝒬 → ቊ 0 if p ≪ 𝑞 0 𝑜 , 1 if 𝑞 ≫ 𝑞 0 𝑜 .

A Cyclic Example Proposition 3.2.3 1 The threshold for 𝐻(𝑜, 𝑞) to contain a cycle is 𝑞 0 𝑜 = 𝑜 . Proof (lower bound) • Let 𝑌 = # cycles in 𝐻(𝑜, 𝑞) • For ℓ ≥ 3 , let 𝑌 ℓ = # 𝐷 ℓ ⊆ 𝐻 𝑜, 𝑞 𝑜 • ⇒ 𝑌 = σ ℓ=3 𝑌 ℓ • Linearity of expectation: 𝔽 𝑌 ℓ ≤ 𝑜 ℓ 𝑞 ℓ 𝑜𝑞 3 𝑜𝑞 ℓ < 𝑜𝑞 3 σ ℓ=0 𝑜𝑞 ℓ = 𝑜 ∞ • ⇒ 𝔽 𝑌 ≤ σ ℓ=3 1−𝑜𝑞 1 • ⇒ 𝔽 𝑌 = 𝑝(1) if 𝑞 ≪ 𝑜 • Markov: ℙ 𝐻 𝑜, 𝑞 has a cycle = ℙ 𝑌 ≥ 1 ≤ 𝔽 𝑌 → 0 ∎

Cycles Continued Proposition 3.2.3 1 The threshold for 𝐻(𝑜, 𝑞) to contain a cycle is 𝑞 0 𝑜 = 𝑜 . Proof (upper bound) 4 • Let 𝑞 = 𝑜−1 and set 𝑍 = 𝑓 𝐻 𝑜, 𝑞 𝑜 • Then 𝑍 ∼ Bin 2 , 𝑞 𝑜 • ⇒ 𝔽 𝑍 = 2 𝑞 = 2𝑜 𝑜 • ⇒ Var 𝑍 = 2 𝑞 1 − 𝑞 < 2𝑜 • ∴ Var 𝑍 = 𝑝 𝔽 𝑍 2 • Chebyshev: ℙ 𝑍 < 𝑜 → 0 • ℙ 𝐻 𝑜, 𝑞 has a cycle ≥ ℙ 𝑓 𝐻 𝑜, 𝑞 ≥ 𝑜 → 1 ∎

Existence of Thresholds Theorem 3.2.4 (Bollobás-Thomason, 1987) Every nontrivial monotone graph property has a threshold. Proof (upper bound) 1 • Let 𝑞 𝑜 = 𝑞 0 be such that ℙ 𝐻 𝑜, 𝑞 0 ∈ 𝒬 = 2 • Let 𝐻 ∼ 𝐻 1 ∪ 𝐻 2 ∪ ⋯ ∪ 𝐻 𝑛 , where each 𝐻 𝑗 ∼ 𝐻 𝑜, 𝑞 0 is independent • ⇒ 𝐻 ∼ 𝐻 𝑜, 𝑞 for 𝑞 ≔ 1 − 1 − 𝑞 0 𝑛 ≤ 𝑛𝑞 0 • Property is monotone: • ℙ 𝐻 ∈ 𝒬 ≥ ℙ ∪ 𝑗 𝐻 𝑗 ∈ 𝒬 = 1 − ℙ ∩ 𝑗 𝐻 𝑗 ∉ 𝒬 • Graphs are independent: • ℙ ∩ 𝑗 𝐻 𝑗 ∉ 𝒬 = ς 𝑗 ℙ 𝐻 𝑗 ∉ 𝒬 1 • Since 𝐻 𝑗 ∼ 𝐻(𝑜, 𝑞 0 ) , ℙ 𝐻 𝑗 ∉ 𝒬 = 2 • ∴ ℙ 𝐻 ∈ 𝒬 ≥ 1 − 2 −𝑛 → 1 if 𝑛 → ∞ (or if 𝑞 ≫ 𝑞 0 ) ∎

Below the Threshold Theorem 3.2.4 (Bollobás-Thomason, 1987) Every nontrivial monotone graph property has a threshold. Proof (lower bound) 𝑞 0 • Let 𝐻 ∼ 𝐻 1 ∪ 𝐻 2 ∪ ⋯ ∪ 𝐻 𝑛 as before, but with 𝐻 𝑗 ∼ 𝐻 𝑜, 𝑞 for 𝑞 = 𝑛 • ⇒ 𝐻 ∼ 𝐻(𝑜, 𝑟) for 𝑟 = 1 − 1 − 𝑞 𝑛 ≤ 𝑛𝑞 = 𝑞 0 1 • ⇒ ℙ 𝐻 ∉ 𝒬 ≥ 2 • As before, ℙ 𝐻 ∉ 𝒬 ≤ ℙ 𝐻 𝑜, 𝑞 ∉ 𝒬 𝑛 1/m 1 • ⇒ ℙ 𝐻 𝑜, 𝑞 ∉ 𝒬 ≥ 2 1/𝑛 1 • ⇒ ℙ 𝐻 𝑜, 𝑞 ∈ 𝒬 ≤ 1 − → 0 if 𝑛 → ∞ (or if 𝑞 ≪ 𝑞 0 ) ∎ 2

Closing Remarks Random graph theory • Fundamental problem: given a graph property 𝒬 , what is its threshold? At the threshold • We showed what happens for probabilities much smaller than the threshold, and much larger than the threshold • What if 𝑞 = Θ 𝑞 0 𝑜 ? Some properties have a much quicker transition Definition 3.2.5 (Sharp thresholds) We say 𝑞 0 (𝑜) is a sharp threshold for 𝒬 if there are positive constants 𝑑 1 , 𝑑 2 such that ℙ 𝐻 𝑜, 𝑞 ∈ 𝒬 → ቊ0 if 𝑞 ≤ 𝑑 1 𝑞 0 𝑜 , 1 if 𝑞 ≥ 𝑑 2 𝑞 0 𝑜 .

Any questions?

§3 Subgraphs of 𝐻 𝑜, 𝑞 Chapter 3: The Second Moment The Probabilistic Method

Returning to Ramsey Theorem 1.5.7 Given ℓ, 𝑙, 𝑜 ∈ ℕ and 𝑞 ∈ 0,1 , if 𝑜 2 + 𝑜 ℓ 𝑙 2 < 1, ℓ 𝑞 1 − 𝑞 𝑙 then 𝑆 ℓ, 𝑙 > 𝑜 . Choosing parameters • Want to choose 𝑜 as large as possible • Need to avoid large independent sets • ⇒ would like to make edge probability 𝑞 large • Limitation: need to avoid 𝐿 ℓ Question: What is the threshold for K ℓ ⊆ 𝐻(𝑜, 𝑞) ?

A Lower Bound Goal • Let 𝑌 count the number of 𝐿 ℓ in 𝐻(𝑜, 𝑞) • For which 𝑞 do we have ℙ 𝑌 ≥ 1 = 𝑝(1) ? First moment ℓ ℓ 𝑜 2 = Θ 𝑜 ℓ 𝑞 • 𝔽 𝑌 = ℓ 𝑞 2 • Markov’s Inequality: ℙ 𝑌 ≥ 1 ≤ 𝔽 𝑌 Threshold bound ℓ • 𝔽 𝑌 = Θ 𝑜 ℓ 𝑞 ≪ 1 2 ℓ 2 ≪ 𝑜 −ℓ ⇔ 𝑞 ≪ 𝑜 −2/(ℓ−1) • ⇔ 𝑞 • ⇒ 𝑞 0 (𝑜) ≥ 𝑜 −2/(ℓ−1)