Enumeration. Martingales. Random graphs. Mikhail Isaev School of - PowerPoint PPT Presentation

Enumeration. Martingales. Random graphs. Mikhail Isaev School of Mathematical Sciences, Monash University Discrete Maths Research Group talk August 28, 2017 . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . .. . . . . .

Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 2 / 29

It is clear for the following cases: when Z is small; when Z X 1 X n , where X 1 X n are independent and small. For our purposes we needed: when Z f X 1 X n , where X 1 X n are independent; when Z is a complex martingale. Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Expectation of the exponential function We are interested in estimates for E e Z , where Z is a complex random variable. E e Z ≈ e E Z E e Z ≈ e E Z + 1 2 E ( Z − E Z ) 2 . and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 3 / 29

For our purposes we needed: when Z f X 1 X n , where X 1 X n are independent; when Z is a complex martingale. Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Expectation of the exponential function We are interested in estimates for E e Z , where Z is a complex random variable. E e Z ≈ e E Z E e Z ≈ e E Z + 1 2 E ( Z − E Z ) 2 . and It is clear for the following cases: when Z is small; when Z = X 1 + · · · + X n , where X 1 , . . . , X n are independent and small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 3 / 29

Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Expectation of the exponential function We are interested in estimates for E e Z , where Z is a complex random variable. E e Z ≈ e E Z E e Z ≈ e E Z + 1 2 E ( Z − E Z ) 2 . and It is clear for the following cases: when Z is small; when Z = X 1 + · · · + X n , where X 1 , . . . , X n are independent and small. For our purposes we needed: when Z = f ( X 1 , . . . , X n ) , where X 1 , . . . , X n are independent; when Z is a complex martingale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 3 / 29

Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Random vectors with independent components Theorem (I., McKay, 2017) Let X = ( X 1 , . . . , X n ) be a random vector with independent components taking values in Ω = Ω 1 × · · · × Ω n . Let f : Ω → C . Suppose, for any 1 ≤ j ̸ = k ≤ n, x , x j | f ( x ) − f ( x j ) | ≤ α, sup x , x j , x k , x jk | f ( x ) − f ( x j ) − f ( x k ) + f ( x jk ) | ≤ ∆ , sup where the suprema is over all x , x j , x k , x jk ∈ Ω such that x,x j and x k , x jk di�er only in the j-th coordinate, x,x k and x j , x jk di�er only in the k-th coordinate. (A) If α = o ( n − 1 / 2 ) , then E e f ( X ) = e E f ( X ) ( 1 + O ( n α 2 ) ) . (B) If α = o ( n − 1 / 3 ) and ∆ = o ( n − 4 / 3 ) , then E e f ( X ) = e E f + 1 2 E ( f − E f ) 2 ( 1 + O ( n α 3 + n 2 α 2 ∆) e 2 Var ℑ f ( X ) ) 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 4 / 29

Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Random permutations Theorem (Greenhill, I., McKay, 2017+) Let X = ( X 1 , . . . , X n ) be a uniform random element of S n and f : S n → C . Suppose, for any distinct j , a ∈ { 1 , . . . , n } , sup | f ( ω ) − f ( ω ◦ ( ja )) | ≤ α, ω ∈ S n and, for any distinct j , k , a , b ∈ { 1 , . . . , n } , sup | f ( ω ) − f ( ω ◦ ( ja )) − f ( ω ◦ ( kb )) + f ( ω ◦ ( ja )( kb )) | ≤ ∆ , ω ∈ S n (A) If α = o ( n − 1 / 2 ) , then E e f ( X ) = e E f ( X ) ( 1 + O ( n α 2 ) ) . (B) If α = o ( n − 1 / 3 ) and ∆ = o ( n − 4 / 3 ) , then E e f ( X ) = e E f + 1 1 + O ( n α 3 + n 2 α 2 ∆) e 2 E ( f − E f ) 2 ( 2 Var ℑ f ( X ) ) 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 5 / 29

Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Random subsets Let B n , m denote the set of subsets of { 1 , . . . , n } of size m. Theorem (Greenhill, I., McKay, 2017+) Let X be a uniform random element of B n , m , m ≤ n / 2, and f : B n , m → C . Suppose, for any A ∈ B n , m and a ∈ A, j / ∈ A, | f ( A ) − f ( A ⊕ { j , a } ) | ≤ α, and, for any distinct a , b ∈ A, j , k / ∈ A, | f ( A ) − f ( A ⊕ { j , a } ) − f ( A ⊕ { k , b } ) + f ( A ⊕ { j , k , a , b } ) | ≤ ∆ , (A) If α = o ( m − 1 / 2 ) , then E e f ( X ) = e E f ( X ) ( 1 + O ( m α 2 ) ) . (B) If α = o ( m − 1 / 3 ) and ∆ = o ( m − 4 / 3 ) , then E e f ( X ) = e E f + 1 2 E ( f − E f ) 2 ( 1 + O ( m α 3 + m 2 α 2 ∆) e 2 Var ℑ f ( X ) ) 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 6 / 29

Concentration results. For a real random variable X, by the Markov inequality, 1 tc e tX Pr e tX e tc Pr X c e it X X it X X Asymptotic normality. Let Z X , t X . Then, 2 t 2 2 Z 2 1 e Z e Z Z t and e 2 Subgraph counts in random graphs with given degrees. 3 Asymptotic enumeration by complex-analytic methods. 4 Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Four applications in the random graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 7 / 29

For a real random variable X, by the Markov inequality, tc e tX Pr e tX e tc Pr X c e it X X it X X Asymptotic normality. Let Z X , t X . Then, 2 t 2 2 Z 2 1 e Z e Z Z t and e 2 Subgraph counts in random graphs with given degrees. 3 Asymptotic enumeration by complex-analytic methods. 4 Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Four applications in the random graph theory Concentration results. 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 7 / 29

it X X it X X Asymptotic normality. Let Z X , t X . Then, 2 t 2 2 Z 2 1 e Z e Z Z t and e 2 Subgraph counts in random graphs with given degrees. 3 Asymptotic enumeration by complex-analytic methods. 4 Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Four applications in the random graph theory Concentration results. For a real random variable X, by the Markov inequality, 1 Pr ( X > c ) = Pr ( e tX > e tc ) ≤ e − tc E e tX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 7 / 29

it X X it X X Let Z X , t X . Then, t 2 2 Z 2 1 e Z e Z Z t and e 2 Subgraph counts in random graphs with given degrees. 3 Asymptotic enumeration by complex-analytic methods. 4 Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Four applications in the random graph theory Concentration results. For a real random variable X, by the Markov inequality, 1 Pr ( X > c ) = Pr ( e tX > e tc ) ≤ e − tc E e tX . . . Asymptotic normality. 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 7 / 29

Subgraph counts in random graphs with given degrees. 3 Asymptotic enumeration by complex-analytic methods. 4 Introduction Subgraph counts Asymptotic enumeration Cumulant expansion Four applications in the random graph theory Concentration results. For a real random variable X, by the Markov inequality, 1 Pr ( X > c ) = Pr ( e tX > e tc ) ≤ e − tc E e tX . . . ( ) Asymptotic normality. Let Z = it X − E X it X − E X Var X , φ ( t ) = E exp . Then, 2 √ √ Var X 2 E ( Z − E Z ) 2 = e − t 2 / 2 . E e Z = φ ( t ) e E Z + 1 and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mikhail Isaev (Monash University) Discrete Maths talk August 28, 2017 7 / 29